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\usepackage{amsmath}
\usepackage{enumitem}
\usepackage{amssymb,amsfonts,stmaryrd,mathrsfs}
\usepackage{latexsym,amsthm,mathtools,bbm}
\usepackage{subcaption}
\usepackage{hyperref}
\usepackage{todonotes}
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\usepackage{bm}
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\usepackage[capitalize]{cleveref}







\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{conj}{Conjecture}

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\newcommand{\bfp}{\mathbf{p}}
\newcommand{\bfP}{\mathbf{P}}
\newcommand{\bfq}{\mathbf{q}}
\newcommand{\bfr}{\mathbf{r}}
\newcommand{\Jch}{\theta^{(\alpha)}}
\newcommand{\MchJack}{{\theta}^{(\alpha,\gamma)}}
\newcommand{\Mch}{\boldsymbol\theta^{(q,t)}}

\newcommand{\Jla}{J^{(\alpha)}_{\lambda}}
\newcommand{\Jth}{J^{(\alpha)}_{\theta}}
\DeclareMathOperator{\hook}{hook}
\newcommand{\tH}{\widetilde{H}^{(q,t)}}
\newcommand{\J}{{J}^{(q,t)}}
\newcommand{\Jsh}{{J}^{*}}
\newcommand{\JJack}{{\mathfrak J}^{(\alpha,\gamma)}}
\newcommand{\normJ}{{j}^{(q,t)}}
\newcommand{\normJt}{\widetilde{j}^{(\alpha,u)}}

\newcommand{\Op}{\mathcal{O}(\Lambda^{(\alpha)})}
\newcommand{\nablaJ}{\boldsymbol{\nabla}}
\newcommand{\DeltaJ}{\boldsymbol{\Delta}}
\newcommand{\GammaJ}{\boldsymbol{\Gamma}}
\newcommand{\GY}{\Gamma_Y}
\newcommand{\B}{\mathcal B}
\newcommand{\tB}{\widetilde{\mathcal B}}
\newcommand{\C}{\mathcal C}
\newcommand{\tC}{\widetilde{\mathcal C}}
\newcommand{\pstar}{\mathbf{p}^*}
\newcommand{\E}{\mathcal E}
\newcommand{\F}{\mathcal{F}}
\newcommand{\Pop}{\mathcal{P}}
\newcommand{\Top}{\mathcal{T}}
\newcommand{\Gammaplus}[1]{{\Gamma}^{(+)}_{#1}}
\newcommand{\GammaplusJ}[1]{\boldsymbol{\Gamma}^{(+)}_{#1}}
\DeclareMathOperator{\Exp}{Exp}







\usepackage{lipsum}

%% define your title in the usual way
\title[Macdonald characters]{Macdonald characters from a new formula for Macdonald polynomials}


%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[H. Ben Dali and M. D'Adderio]{Houcine Ben Dali\thanks{\href{mailto:houcine.ben-dali@univ-lorraine.fr}{houcine.ben-dali@univ-lorraine.fr}. Supported by the LUE DrEAM project of Université de Lorraine.}\addressmark{1} \and Michele D'Adderio\thanks{\href{mailto:michele.dadderio@unipi.it}{michele.dadderio@unipi.it}. Partially supported by PRIN 2022A7L229 ALTOP and by INDAM research group GNSAGA.}\addressmark{2}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Université de Lorraine, CNRS, IECL, F-54000 Nancy, France\\
\addressmark{1}Université de Paris, CNRS, IRIF, F-75006, Paris, France
\\\addressmark{2}Dipartimento di Matematica, Università di Pisa, Pisa, PI, 56127, Italia}

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\received{\today}

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%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
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\abstract{We introduce a new operator $\GammaJ$ on symmetric functions, which enables us to obtain a creation formula for Macdonald polynomials. This formula makes a connection between the theory of Macdonald operators initiated by Bergeron, Garsia, Haiman and Tesler, and shifted Macdonald polynomials introduced by Knop, Okounkov and Sahi.

We use this formula to introduce a two-parameter generalization of Jack characters. Finally, we provide a change of variables in order to formulate several positivity conjectures related to these generalized characters. Our conjectures extend some important problems on Jack polynomials, including some famous conjectures of Goulden and Jackson.}

%% put your French abstract here, or comment this out if you don't have one
% \resume{}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Macdonald polynomials, Macdonald characters, Matchings-Jack conjecture}

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\usepackage[backend=bibtex]{biblatex}
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\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above






% \section{Introduction}

% \subsection{Characters}


\section{Introduction}
\subsection{Jack and Macdonald polynomials}
% In the last decade, several operators have been introduced to study Macdonald polynomials; Delta operator, Theta operator and many positivity problems have been considered relating Macdonald polynomials to the combinatorics of weighted paths \cite{CarlssonMellit2018,DAdderioMellit2022}.
Jack polynomials are symmetric functions depending on one parameter $\alpha$ which have been introduced by Jack \cite{Jack1970/1971}. 
The combinatorial analysis of Jack polynomials has been initiated by Stanley \cite{Stanley1989}
and a first combinatorial interpretation has been given by Knop and Sahi in terms of tableaux \cite{KnopSahi1997}. 
A second family of combinatorial objects related to Jack polynomials is given by \textit{maps}, which are roughly graphs embedded in surfaces. This connection has first been observed in the conjectures of Goulden and Jackson \cite{GouldenJackson1996} and important progress has recently been made in this direction \cite{ChapuyDolega2022,
BenDaliDolega2023} with a first ``topological expansion'' of Jack polynomials in terms of maps. 

Macdonald polynomials are symmetric polynomials introduced by Macdonald in 1989, which depend on two parameters $q$ and $t$. Jack polynomials can be obtained from Macdonald polynomials by taking an appropriate limit. Several combinatorial results on Jack polynomials have been generalized to the Macdonald case, in particular, an interpretation in terms of tableaux was established in \cite{HaglundHaimanLoehr2005}. However, no connection between Macdonald polynomials and maps is known, even conjecturally. As a first step towards a Macdonald generalization of maps, we introduce in this paper some new tools that make the parallel between the Jack and Macdonald stories more compelling.

First, we prove a creation formula (\cref{eq Gamma-J,eq Gamma-J 2}) for Macdonald polynomials inspired from the one used in \cite{BenDaliDolega2023} to connect Jack polynomials to maps. Second, we use this formula to introduce a Macdonald analog of Jack characters (\cref{ssec intro characters}). Finally, we formulate a Macdonald version of some Jack conjectures, including Goulden and Jackson's Matchings-Jack and b-conjectures.


% For the definitions and notation introduced in \cref{subsec Partitions,sec SymFun} we refer to \cite{Stanley1989,Macdonald1995}.







\subsection{Preliminaries}\label{subsec Partitions}
For the results of this section we refer to \cite{DAdderioRomero2023,Macdonald1995}.
A \textit{partition} $\lambda=[\lambda_1,...,\lambda_\ell]$ is a weakly decreasing sequence of positive integers $\lambda_1\geq...\geq\lambda_\ell>0$. 
We denote by $\mathbb{Y}$ the set of integer partitions. The integer $\ell$ is called the \textit{length} of $\lambda$ and is denoted $\ell(\lambda)$. 
The size of $\lambda$ is the integer $|\lambda|:=\lambda_1+\lambda_2+...+\lambda_\ell.$ If $n$ is the \textit{size} of $\lambda$, we say that $\lambda$ is a partition of $n$ and we write $\lambda\vdash n$. 
The integers $\lambda_1$,...,$\lambda_\ell$ are called the \textit{parts} of $\lambda$. 
For $i\geq 1$, we denote $m_i(\lambda)$ the number of parts of size $i$ in $\lambda$. We then set 
$z_\lambda:=\prod_{i\geq1}m_i(\lambda)!i^{m_i(\lambda)}.$
% We denote by $\leq$ the \textit{dominance partial} ordering on partitions, defined by 
% $\mu\leq\lambda \iff |\mu|=|\lambda| \text{ and }\hspace{0.3cm} \mu_1+...+\mu_i\leq \lambda_1+...+\lambda_i \text{ for } i\geq1.$
We identify a partition  $\lambda$ with its \textit{Young diagram}, defined by 
$\lambda:=\{(i,j),1\leq i\leq \ell(\lambda),1\leq j\leq \lambda_i\}.$
% \textit{The conjugate partition} of $\lambda$, denoted $\lambda'$, is the partition associated to the Young diagram obtained by reflecting the diagram of $\lambda$ with respect to the line $j=i$:
% $$\lambda':=\{(i,j),1\leq j\leq \ell(\lambda),1\leq i\leq \lambda_i\}.$$
Fix a box $\Box:=(i,j)\in\lambda$. Its \textit{arm}, \textit{leg}, \textit{co-arm} and \textit{co-leg} are respectively given by 
$$a_\lambda(\Box):=|\{(i,c)\in\lambda,c>j\}|, \qquad\ell_\lambda(\Box):=|\{(r,j)\in\lambda,r>i\}|,$$
$$a'_\lambda(\Box):=|\{(i,c)\in\lambda,c<j\}|, \text{ and } \ell'_\lambda(\Box):=|\{(r,j)\in\lambda,r<i\}|.$$

\noindent Finally, $n$ and $n'$ denote respectively the two statistics on Young diagram given by 
$$n(\lambda):=\sum_{\Box\in\lambda}\ell_\lambda(\Box) \quad \text{ and } \quad n'(\lambda):=\sum_{\Box\in\lambda}a_\lambda(\Box).$$

% \subsection{Symmetric functions and plethysm} 
We consider the graded algebra $\Lambda=\oplus_{r\geq 0}\Lambda^{(r)}$ of symmetric functions in the alphabet $(x_1,x_2,\dots)$ with coefficients in $\mathbb{Q}(q,t)$. Let $p_\lambda$ and $h_\lambda$ denote  the power-sum and the complete symmetric functions in $(x_i)_{i\geq 1}$, respectively.
We use here a variable  $u$ to keep track of the degree of the functions, and an extra variable $v$; all the functions considered are in $\Lambda[v]\llbracket u\rrbracket$. Consider the \emph{Hall scalar product} defined by $\langle p_\mu,p_\nu\rangle =\delta_{\mu,\nu} z_\mu$.
Let $f^\perp$ denote the adjoint of multiplication by $f\in \Lambda$ with respect to $\langle,\rangle$. 


We will use the \textit{plethystic notation}: if $E(q,t,u,v,x_1,x_2,\dots)\in\Lambda[v]\llbracket u\rrbracket$ and $f\in\Lambda$ then $f[E]$ is the image of $f$ under the algebra morphism defined by
\begin{align*}
 \Lambda[v]\llbracket u\rrbracket\longrightarrow &\Lambda[v]\llbracket u\rrbracket\\
  p_k\longmapsto &E(t^k,q^k,u^k,v^k,x_1^k,\dots).  
\end{align*}
Set $X:=x_1+x_2+\dots$. Notice that $f[X]=f(x_1,x_2,\dots)$ for any $f$.
Moreover, 
$$p_k\left[X\frac{1-q}{1-t}\right]=\frac{1-q^k}{1-t^k}p_k(x_1,x_2,\dots)\text{ and }p_\lambda[-X]=(-1)^{\ell(\lambda)}p_\lambda(x_1,x_2,\dots).$$



% \subsection{A scalar product:}
\noindent We consider the scalar product $\langle,\rangle_{q,t}$ on $\Lambda$ defined by
$$\langle p_\mu[X],p_\nu[X]\rangle_{q,t}=\delta_{\mu,\nu}z_\mu(q,t):=\delta_{\mu,\nu} z_\mu p_\mu\left[\frac{1-q}{1-t}\right].$$
The \emph{integral form of Macdonald polynomials} $\J_\lambda$ can be defined as the unique family of polynomials satisfying a triangularity property in the monomial basis, and orthogonal with respect to $\langle,\rangle_{q,t}$. More precisely
$\left\langle\J_\lambda,\J_\rho\right\rangle_{q,t}=\delta_{\lambda,\rho}\normJ_\lambda$
with
    $$\normJ_\lambda:=\prod_{\Box\in\lambda}\left(1-q^{a_\lambda(\Box)+1}t^{\ell_\lambda(\Box)}\right)\left(1-q^{ a_\lambda(\Box)}t^{\ell_\lambda(\Box)+1}\right).$$
For every $r\in \mathbb{N}$, the set $\{\J_\lambda\mid \lambda\vdash r\}$ is a basis of $\Lambda^{(r)}$.

Finally, let $\Pop_Z$ be the operator such that $\Pop_Z\cdot f[X]=\Exp[ZX]f[X]$, i.e.\ the multiplication by the \emph{plethystic exponential} $\Exp[ZX]:=\sum_{n\geq 0}h_n[ZX]$, and let $\Top_{Z}:=\sum_{\mu \in \mathbb{Y}}z_\mu^{-1}p_\mu[Z]p_\mu^\perp$ be the translation operator, so that $\Top_Z\cdot f[X]=f[X+Z]$. Note that 
$\Pop_{Z+Z'}=\Pop_Z\cdot\Pop_{Z'}$.



\subsection{A new formula for Macdonald polynomials}
Consider the operators\footnote{We use boldface symbols to distinguish these operators from their relatives from Section~\ref{ssec modified Macdonald}.} $\nablaJ$ and $\DeltaJ_v$ on symmetric functions defined by
$$\nablaJ\cdot J^{(q,t)}_\lambda\! =\! (-1)^{|\lambda|}\left(\prod_{\Box\in\lambda}q^{a'(\Box)}t^{-\ell'(\Box)}\right) J^{(q,t)}_\lambda,
\quad 
\DeltaJ_v\cdot J^{(q,t)}_\lambda\! =\! \prod_{\Box\in\lambda}\left(1-v\cdot q^{a'(\Box)}t^{-\ell'(\Box)}\right) J^{(q,t)}_\lambda.$$
These are integral versions of some known operators on modified Macdonald polynomials (see \cref{ssec modified Macdonald}).
% =(-1)^{|\lambda|}q^{n'(\lambda)}t^{-n(\lambda)}J^{(q,t)}_\lambda



We finally introduce the following operator\footnote{This operator is a close relative of the Theta operator in \cite{DAdderioRomero2023}, first introduced in \cite{DAdderioIraciVWyngaerdTheta}.} on $\Lambda[v]\llbracket u\rrbracket$
$$\GammaJ(u,v):=\DeltaJ_{1/v} \mathcal{P}_{\frac{uv(1-t)}{1-q}}
\DeltaJ_{1/v}^{-1}\ \ .$$ 
The fact that $\GammaJ(u,v)$ is a polynomial in $v$ is a consequence of the Pieri rule.

We can state our new formula for Macdonald polynomials.

\begin{thm} \label{thm creationJqt}
	For any partition $\lambda=[\lambda_1,\lambda_2,\dots,\lambda_k]$, we have
	\begin{equation}\label{eq Gamma-J 2}
		\GammaplusJ{\lambda_1}\GammaplusJ{\lambda_2}\cdots\GammaplusJ{\lambda_k}\cdot1=\J_{\lambda}\quad \text{ where }\quad  \GammaplusJ{m}:=[u^m]\nablaJ^{-1}\GammaJ(u,q^m)\nablaJ.
	\end{equation}        
\end{thm}



It turns out that \cref{thm creationJqt} is an easy consequence of the following \emph{creation formula}.

\begin{thm}\label{thm GammaJ-J}
For any partition $\lambda=[\lambda_1,\lambda_2,\dots,\lambda_k]$, we have
 \begin{align}\label{eq Gamma-J}
\GammaJ(u,q^{\lambda_1})\GammaJ(t^{-1}u,q^{\lambda_2})\cdots \GammaJ(t^{-(k-1)}u,q^{\lambda_k})\cdot 1 & =t^{-n(\lambda)}\nablaJ \Top_{\frac{1}{u(1-t)}} \J_\lambda[uX].
	\end{align}
\end{thm}
\noindent In \cref{sec Proof}, we prove an analogous result for modified Macdonald polynomials \cref{thm formula modified} from which we deduce \cref{thm GammaJ-J}.

In addition to giving a direct construction of Macdonald polynomials, \cref{thm GammaJ-J} provides a dual approach to study the structure of these polynomials. Indeed,  \cref{eq Gamma-J} allows to think of $\J_\lambda$ as a function in the partition $\lambda$ described by the alphabet $(q^{\lambda_1},q^{\lambda_2},\dots)$. This dual approach plays a key role in this paper and is used in \cref{sec characters} to introduce a $q,t$-deformation of Jack characters.





\subsection{Macdonald characters}\label{ssec intro characters}
Jack characters, introduced by Lassalle in \cite{Lassalle2008b}, can be seen as a one parameter deformation of the symmetric group characters, and are directly related to the coefficients of Jack polynomials in the power-sum basis. 
Jack characters have been useful to understand asymptotic behavior of large Young diagram sampled with respect to a Jack deformed Plancherel measure \cite{CuencaDolegaMoll2023,DolegaFeray2016}. Moreover, Goulden and Jackson's Mathings-Jack conjecture has a natural interpretation in terms of structure coefficients of Jack characters. Recently, a combinatorial interpretation of Jack characters in terms of maps on non orientable surfaces has been proved in \cite{BenDaliDolega2023}, answering a positivity conjecture of Lassalle. 

Using the operator $\GammaJ$, we introduce a two parameter deformation $\Mch_\mu$ of Jack characters. These characters have a structure of shifted symmetric functions and are related to shifted Macdonald polynomials, see \cite{Lassalle1998,Okounkov1998}. Macdonald characters $\Mch_\mu$ can be thought of as a natural generalization of the coefficients of Macdonald polynomials in the power-sum basis (see \cref{eq J characters}) which are hard to guess without the new operator $\Gamma$. 

In \cref{sec Macdonald conjectures}, we make several positivity conjectures related to the new characters $\Mch_\mu$.  These conjectures suggest that the characters $\Mch_\mu$ have a combinatorial structure  which generalizes the one given by maps and that we hope to investigate in future works. 
% We conjecture several positivity properties about these characters with in particular a two-parameter version of Goulden and Jackson conjectures.
%\subsection{Outline of the paper} 
%In \cref{sec creation formula} we prove \cref{thm GammaJ-J}. In \cref{sec characters} we introduce a two parameter generalization of Jack characters. We formulate some positivity conjectures related to these characters in \cref{sec Macdonald conjectures}.









\section{A new creation formula for Macdonald polynomials}\label{sec creation formula}
\subsection{Modified Macdonald polynomials}\label{ssec modified Macdonald}
In \cite{GarsiaHaiman1993}, Garsia and Haiman introduced a \emph{modified} version of \emph{Macdonald polynomials} 
$$\tH_\lambda=t^{n(\lambda)}J_\lambda^{(q,1/t)}\left[\frac{X}{1-1/t}\right].$$
The operators $\nabla$ and $\Delta_v$  are defined by
$$\nabla \tH_{\lambda}:=(-1)^{|\lambda|}\prod_{\Box\in\lambda}q^{a_\lambda'(\Box)}t^{\ell_\lambda'(\Box)}\tH_\lambda,
\qquad \Delta_{v} \tH_{\lambda}:=\prod_{\Box\in\lambda}\left(1-vq^{a_\lambda'(\Box)}t^{\ell_\lambda'(\Box)}\right)\tH_\lambda.$$
These operators are related by the \textit{five-term relation} of Garsia and Mellit \cite{GarsiaMellit2019}
\begin{equation}\label{eq:fiveterm}
\nabla \Pop_{\frac{u}{M}}\nabla^{-1}\Pop_{\frac{uv}{M}}=\Delta_{1/v}\Pop_{\frac{uv}{M}}\Delta_{1/v}^{-1}\ ,
\end{equation}
where $M:=(1-q)(1-t)$. Let $B_\lambda:=\sum_{\Box \in\lambda}q^{a'_\lambda(\Box)}t^{\ell'_\lambda(\Box)}=\sum_{1\leq i\leq \ell(\lambda)}t^{i-1}\frac{1-q^{\lambda_i}}{1-q}$, and $D_\lambda:=M B_\lambda-1.$ We state another fundamental identity for Macdonald polynomials, due to Garsia, Haiman and Tesler \cite{GarsiaHaimanTesler2001}: for any partition $\lambda$
\begin{equation} \label{thm Tesler}
\nabla \Pop_{-\frac{u}{M}}\Top_{\frac{1}{u}} \widetilde{H}_\lambda[uX]=\Exp\left[-\frac{uXD_\lambda}{M} \right].
\end{equation}

\subsection{Creation formula for modified Macdonald polynomials}
% Recall the definition $\tilde{\Theta}(u;v):=\Delta_v \Pop_{-\frac{u}{M}} \Delta_v^{-1}$.
We start by proving a modified version of \cref{thm GammaJ-J}.
Set $\Gamma(u,v):=\Delta_{1/v}\Pop_{\frac{uv}{1-q}}\Delta_{1/v}^{-1}.$ 
\begin{thm}\label{thm formula modified}
	For $\lambda=[\lambda_1,\lambda_2,\dots,\lambda_{k}]$ partition we have
	\begin{align}\label{eq formula modified}
	\Gamma(u,q^{\lambda_1})\Gamma(tu,q^{\lambda_2})\cdots \Gamma(t^{\ell-1}u,q^{\lambda_{\ell}})\cdot 1 & =\nabla \Top_{\frac{1}{u}} \widetilde{H}_\lambda[uX]=\nabla \widetilde{H}_\lambda[uX+1].
	\end{align}
\end{thm}

\begin{lem}\label{lem Gamma}
We have
$$\Gamma(u,v)=\nabla \Pop_{\frac{u}{M}}\nabla^{-1}\Pop_{\frac{uv}{1-q}}\nabla \Pop_{\frac{-tu}{M}}\nabla^{-1}.$$
\begin{proof}
    The operator $\Gamma$ can be rewritten as follows 
$$\Gamma(u,v)=\left(\Delta_{1/v}\Pop_{\frac{uv}{M}}\Delta_{1/v}^{-1}\right)\left(\Delta_{1/v}\Pop_{\frac{-tuv}{M}}\Delta_{1/v}^{-1}\right)=\left(\Delta_{1/v}\Pop_{\frac{uv}{M}}\Delta_{1/v}^{-1}\right)\left(\Delta_{1/v}\Pop_{\frac{tuv}{M}}\Delta_{1/v}^{-1}\right)^{-1}.$$
Using the five-term relation \eqref{eq:fiveterm} on each one of the two factors, we obtain 
$$\Gamma(u,v)=\nabla \Pop_{\frac{u}{M}}\nabla^{-1}\Pop_{\frac{uv}{M}}\Pop_{\frac{-utv}{M}}\nabla \Pop_{\frac{-tu}{M}}\nabla^{-1}=\nabla \Pop_{\frac{u}{M}}\nabla^{-1}\Pop_{\frac{uv}{1-q}}\nabla \Pop_{\frac{-tu}{M}}\nabla^{-1}.\qedhere$$
%which finishes the proof of the lemma.
\end{proof}
\end{lem}
%\noindent We need a lemma, see e.g \cite[Eq. (1.47)]{DAdderioRomero2023} for a proof.
%\begin{lem}\label{lem nabla P}
%    $$\nabla \Exp\left[\frac{-uX}{M}\right]\cdot 1=\Exp\left[\frac{uX}{M}\right].
%    $$
%\end{lem}


% We also introduce for any $m\geq 1$ the operator
% $$\Gammaplus{m}:=[z^m]\nabla^{-1}\Gamma(z,q^m)\nabla=[z^m]\nabla^{-1}\Delta_{1/q^m}\Pop_{\frac{zq^m}{1-q}}\Delta_{1/q^m}^{-1}\nabla.$$
% \begin{thm}\label{thm formula2 modified}
% For any partition $\lambda=[\lambda_1,\lambda_2,\dots,\lambda_\ell]$, we have 
% $$t^{-n(\lambda)}\tH_\lambda=\Gammaplus{\lambda_1} \Gammaplus{\lambda_2}\cdots\Gammaplus{\lambda_\ell}\cdot1.$$
% \end{thm}


\noindent We now prove \cref{thm formula modified}.
% \begin{thm}\label{thm Gamma operator}
\begin{proof}[Proof of \cref{thm formula modified}]
It follows, using \cref{lem Gamma}, that 
\begin{align*}
\Gamma(u,v_1)\Gamma(tu,v_2)\cdots \Gamma(t^{k-1}u,v_k) \cdot 1 
& =\nabla \Pop_{\frac{u}{M}}\nabla^{-1}\Pop_{\frac{uv_1}{1-q}}\Pop_{\frac{utv_2}{1-q}}\cdots\Pop_{\frac{ut^{k-1}v_k}{1-q}}\nabla \Pop_{\frac{-ut^{k}}{M}}\nabla^{-1}\cdot 1\ \ .%\\
%&=\nabla \Pop_{\frac{u}{M}}\nabla^{-1}\Pop_{\frac{uv_1}{1-q}}\Pop_{\frac{utv_2}{1-q}}\cdots\Pop_{\frac{ut^{k-1}v_k}{1-q}}\nabla \Pop_{\frac{-ut^{k}}{M}}\cdot 1
\end{align*}

\noindent Using $\nabla^{-1}\cdot 1=1$ and $\nabla \Pop_{-\frac{z}{M}}\cdot 1=\Pop_{\frac{z}{M}}\cdot 1$ (see e.g. \cite[Eq. (1.47)]{DAdderioRomero2023} with $k=n$), we get
\begin{align*}
\Gamma(u,v_1)\Gamma(tu,v_2)\cdots\Gamma(t^{k-1}u,v_k)  \cdot 1 
&=\! \nabla \Pop_{\frac{u}{M}}\nabla^{-1}\Pop_{\frac{uv_1}{1-q}}\Pop_{\frac{utv_2}{1-q}}\cdots\Pop_{\frac{ut^{k-1}v_k}{1-q}}\Pop_{\frac{ut^{k}}{M}}\cdot 1\\
&=\!\nabla \Pop_{\frac{u}{M}}\! \nabla^{-1}\Exp\!\! \left[\frac{ut^{k}X}{M}+\frac{uX}{1-q}\sum_{1\leq i\leq k}t^{i-1}v_i\right]\\
& =\! \nabla \Pop_{\frac{u}{M}}\! \nabla^{-1}\Exp\!\! \left[\frac{uX}{M}-\frac{uX}{M}(1-t)\sum_{1\leq i\leq k}t^{i-1}(1-v_i)\right]\!.
\end{align*}
% \end{thm}

% Using the five term relations
% \[\nabla \Pop_{\frac{z}{M}}\nabla^{-1}=\Delta_{1/v_1}\Pop_{\frac{zv_1}{M}}\Delta_{1/v_1}^{-1}\Pop_{-\frac{zv_1}{M}}\quad \text{and}\quad \Pop_{-\frac{tzv_1}{M}}\Delta_{1/v_1}^{-1}\nabla\Pop_{\frac{tz}{M}}\nabla^{-1}=\Delta_{1/v_1}^{-1}\Pop_{-\frac{tzv_1}{M}}\]

% we have
% \begin{align*}
% 	 & \nabla \Pop_{\frac{z}{M}}\nabla^{-1}\Exp\left[\frac{zX}{M}-\frac{zX}{M}(1-t)\sum_{i\geq 1}t^{i-1}(1-v_i)\right]\\
% 	 &= \nabla \Pop_{\frac{z}{M}}\nabla^{-1}  \Pop_{\frac{z}{M}}\Pop_{-\frac{z(1-v_1)}{1-q}}\Pop_{-\frac{zt(1-v_2)}{1-q}}\Pop_{-\frac{zt^2(1-v_3)}{1-q}}\cdots\\
% 	 &= \Delta_{1/v_1}\Pop_{\frac{zv_1}{M}}\Delta_{1/v_1}^{-1}\Pop_{-\frac{zv_1}{M}}  \textcolor{red}{\Pop_{\frac{z}{M}}\Pop_{-\frac{z(1-v_1)}{1-q}}}\Pop_{-\frac{zt(1-v_2)}{1-q}}\Pop_{-\frac{zt^2(1-v_3)}{1-q}}\cdots\\
% 	  &= \Delta_{1/v_1}\Pop_{\frac{zv_1}{M}}\Delta_{1/v_1}^{-1}\textcolor{blue}{\Pop_{-\frac{zv_1}{M}} \Pop_{-\frac{z(1-t)(1-v_1)}{M}}} \Pop_{\frac{z}{M}}\Pop_{-\frac{zt(1-v_2)}{1-q}}\Pop_{-\frac{zt^2(1-v_3)}{1-q}}\cdots\\
% 	  &= \Delta_{1/v_1}\Pop_{\frac{zv_1}{M}}\Delta_{1/v_1}^{-1}\Pop_{-\frac{tzv_1}{M}}\textcolor{magenta}{ \Pop_{-\frac{z}{1-q}} \Pop_{\frac{z}{M}}}\Pop_{-\frac{zt(1-v_2)}{1-q}}\Pop_{-\frac{zt^2(1-v_3)}{1-q}}\cdots\\
% 	  &= \Delta_{1/v_1}\Pop_{\frac{zv_1}{M}}\Delta_{1/v_1}^{-1}\Pop_{-\frac{tzv_1}{M}} \Pop_{\frac{tz}{M}} \Pop_{-\frac{zt(1-v_2)}{1-q}}\Pop_{-\frac{zt^2(1-v_3)}{1-q}}\cdots\\
% 	  &= \Delta_{1/v_1}\Pop_{\frac{zv_1}{M}}\Pop_{-\frac{tzv_1}{M}}\Delta_{1/v_1}^{-1}\nabla \Pop_{\frac{tz}{M}}\nabla^{-1}  \Pop_{\frac{tz}{M}}\Pop_{-\frac{zt(1-v_2)}{1-q}}\Pop_{-\frac{zt^2(1-v_3)}{1-q}}\cdots\\
% 	  &= \Gamma(z,v_1)\nabla \Pop_{\frac{tz}{M}}\nabla^{-1}  \Pop_{\frac{tz}{M}}\Pop_{-\frac{zt(1-v_2)}{1-q}}\Pop_{-\frac{zt^2(1-v_3)}{1-q}}\cdots
% \end{align*}
% and now iteratively the thesis follows.
\noindent    Fix now a partition $\lambda$.	Applying the previous equation, we get
\begin{align*}
\Gamma(u,q^{\lambda_1})\Gamma(tu,q^{\lambda_2})\cdots \Gamma(t^{\ell-1}u,q^{\lambda_{\ell}})\cdot 1 & =\! \nabla \Pop_{\frac{u}{M}}\! \nabla^{-1}\Exp\!\! \left[\frac{uX}{M}-\frac{uX}{M}(1-t)\sum_{i\geq 1}t^{i-1}(1-q^{\lambda_i})\right]\\
	& =\! \nabla \Pop_{\frac{u}{M}}\!\nabla^{-1}\Exp\!\! \left[-\frac{uXD_\lambda}{M} \right],
\end{align*}

\noindent Now applying \cref{thm Tesler} concludes the proof of the theorem.
\end{proof}



% Using the five term relation we obtain a second expression for $\GammaJ$
% $$\GammaJ(z,u)=\left(\nablaJ\mathcal{P}_{\frac{-zt}{1-q}}\nablaJ^{-1}\right)\cdot\mathcal{P}_{\frac{uz(1-t)}{1-q}}\left(\nablaJ\mathcal{P}_{\frac{z}{1-q}}\nablaJ^{-1}\right).$$



% \begin{thm}\label{thm GammaJ operator}
% We have
% \begin{align}
% \GammaJ(z,v_1)\GammaJ(t^{-1}z,v_2)&\cdots \GammaJ(t^{k-1} z,v_k)  \cdot 1 \nonumber\\
% =&\nablaJ \Pop_{\frac{-tz}{1-q}}\nablaJ^{-1}\Exp\left[\frac{-ztX}{1-q}-\frac{z(1-t)X}{1-q}\sum_{1\leq i\leq k}t^{1-i}(1-v_i)\right]\label{eq GammaJ operator}\\
% =&\nablaJ \Pop_{\frac{-tz}{1-q}}\nablaJ^{-1}\Exp\left[\frac{-zt^{k-1}X}{1-q}-\frac{z(1-t)X}{1-q}\sum_{1\leq i\leq k}t^{1-i}v_i\right]\label{eq GammaJ operator2}.
% \end{align}
% \begin{proof}
%     The required formula is obtained from \cref{thm Gamma operator} by conjugating by $\phi^{-1}$.
% \end{proof}
% \end{thm}





% We have the following plethystic substitution of Macdonald polynomials.
% \begin{thm}\label{thm plethystic subt}
%     $$\J_\lambda\left[\frac{1-v}{1-t}\right]=\prod_{\Box\in\lambda}\left(t^{\ell_\lambda'(\Box)}-vq^{a'_\lambda(\Box)}\right)$$
% \end{thm}


% \begin{thm}[Pieri rule]\label{thm Pieri rule}
%     Let $f$ be a symmetric function of degree $k$ and let $\lambda$ be a partition. Then 
%     $$f\cdot \J_\lambda=\sum_{\lambda\subset_k \xi}d^f_{\lambda,\xi} \J_\xi,$$
%     for some coefficients $d^f_{\lambda,\xi}$, where the sum is taken over the partitions $\xi$ obtained by adding $k$ cells to $\lambda$.
% \end{thm}





\subsection{Proof of \cref{thm GammaJ-J}} \label{sec Proof}
In this subsection we deduce \cref{thm GammaJ-J} from \cref{thm formula modified}.
 
 Consider the transformation $\phi$ on $\Lambda$ defined by
$$f=\sum_{\mu}d^f_\mu(q,t) p_\mu[X] \longmapsto\phi(f):=\sum_{\mu}d^f_\mu(q,1/t) p_\mu\left[\frac{X}{1-1/t}\right],$$ 
where $d^f_\mu$ are the coefficients of $f$ in the power-sum basis. Notice that $\phi$ is invertible and $$\phi^{-1}(f)=\sum_{\mu}d^f_\mu(q,1/t) p_\mu\left[X(1-t)\right] \text{for any $f$.}$$ 
With this definition, one has
$\tH_\lambda=t^{n(\lambda)}\phi(\J_\lambda).$ Moreover, $\nablaJ=\phi^{-1}\cdot \nabla \cdot\phi$ and $\DeltaJ_v=\phi^{-1}\cdot \Delta_v \cdot\phi$.
Finally
$\phi^{-1}\cdot \Pop_{\frac{ut^i}{1-q}}\cdot\phi=\Pop_{\frac{t^{-i}(1-t)u}{1-q}}, \text{ for any }i\geq 0.$
We deduce that $\phi^{-1}\cdot \Gamma(t^iu,v)\cdot\phi=\GammaJ(t^{-i}u,v), \text{ for any }i\geq 0.$ 

On the other hand, one can check that $\phi^{-1}\cdot \Top_{\frac{1}{u}}\cdot\phi=\Top_{\frac{1}{u(1-t)}}$.
Hence, applying $\phi^{-1}$ on \cref{eq formula modified}, we obtain \cref{eq Gamma-J}. 



%  \subsection{Preliminary conjectures}
%  \begin{conj}
%  Let $\mu,\nu\vdash n$.
%      $$t^{n(n-1)/2}\left\langle(-1)^n\cdot\nabla_J\cdot p_\mu\left[X\frac{1-t}{1-q}\right],p_\nu\left[X\frac{1-q}{1-t}\right]\right\rangle$$
%      is positive after the shift $q\rightarrow q+1$, $t\leftarrow t+1$.
%  \end{conj}


% \begin{conj}
%  Let $\mu,\nu\vdash n$.
%      $$q^{n(n-1)/2}\left\langle(-1)^n\cdot\nabla_J^{-1}\cdot \omega(p_\mu)\left[X\frac{1-t}{1-q}\right],\omega(p_\nu)\left[X\frac{1-q}{1-t}\right]\right\rangle$$
%      is positive after the shift $q\rightarrow q+1$, $t\leftarrow t+1$.
%  \end{conj}



 
%  Checked for $n\leq 8$.








\section{A two-parameter generalization of Jack characters}\label{sec characters}


The results of this section will be proved in the long version of the paper.
\subsection{Shifted symmetric functions}
\begin{defi}
    We say that a \emph{polynomial} in $k$ variables $f(u_1,\dots,u_k)$ is \emph{shifted symmetric} if it is symmetric in the variables $u_1,u_2t^{-1},\dots,u_k t^{1-k}.$ 
    A \emph{shifted symmetric function} $f(u_1,u_2,\dots)$ is a sequence of polynomials $(f_k)_{k\geq1}$ of bounded degrees, such that $f_k$ is a shifted symmetric polynomial in $k$ variables for each $k$, and
$f_{k+1}(v_1,\dots,v_k,1)=f_{k}(v_1,\dots,v_k)$.
\end{defi}

If $f$ is a shifted symmetric function, we consider its evaluation on a Young diagram $\lambda=[\lambda_1,\dots,\lambda_k]$ defined by $f(\lambda):=f(q^{\lambda_1},q^{\lambda_2},\dots ,q^{\lambda_k},1,1,\dots ).$
It is well known that the \emph{space of shifted symmetric functions} $\Lambda^*$ can be identified with a subspace of the space of functions on Young diagrams through the map $f\longmapsto (f(\lambda))_{\lambda\in\mathbb Y}.$


\begin{thm}[Shifted Macdonald polynomials]\cite{Okounkov1998}\label{thm shifted Macdonal}
Let $\mu$ be a partition.  There exists a unique function  $\Jsh_\mu(v_1,v_2,\dots)$ such that 
\begin{enumerate}
    \item $\Jsh_\mu$ is shifted symmetric of degree $|\mu|$.
    \item $\Jsh_\mu(\mu)=(-1)^{|\mu|}q^{n'(\mu)}t^{-2n(\mu)}\normJ_\mu$ (normalization property).
    \item for any partition $\lambda\not\supset\mu$ one has
$\Jsh_\mu(\lambda)=0$ (vanishing property).
\end{enumerate}
Moreover, the top homogeneous part of $J^*_\mu$ is $\J_\mu(v_1,t^{-1}v_2,t^{-2}v_3, \dots)$.
\end{thm}

Since Macdonald polynomials form a basis of $\Lambda$, using a triangularity argument it can be deduced that shifted Macdonald polynomials form a basis of $\Lambda^*$. As a consequence we can extend the map $J_\mu\longmapsto J^*_\mu $ to a linear isomorphism 
% \vspace{-0.5cm}
\begin{equation}\label{eq isomorphism *}
  \begin{array}{ccc}
     \Lambda &\longrightarrow & \Lambda^* \\
     f &\longmapsto & f^*.   
\end{array}
\end{equation}

% \begin{rmq}\label{rmq top hom}
% Note that, by linearity, if $f$ is a homogeneous symmetric function, then the top homogeneous part of $f^*$ is equal to $f(v_1,t^{-1}v_2,t^{-2}v_3,\dots)$.
% \end{rmq}




\subsection{An explicit isomorphism between the spaces of symmetric and shifted-symmetric functions}
The main purpose of this subsection is to give two explicit formulas for the isomorphism \eqref{eq isomorphism *}. The first one is \cref{eq construction 1}, which gives the image of a function $f^*$ as a shifted symmetric function. The second formula is \cref{eq construction 2}, which gives the image as a function on Young diagrams. The proof, that we omit, is based on \cref{eq Gamma-J} and the Pieri rule.

% We start by the following lemma.

% \begin{lem}\label{lem GammaJ-scalar J}
%     For any symmetric function $f$, and any partition $\lambda$ one has
% $$\left\langle f,\GammaJ(1,q^{\lambda_1})\GammaJ(t^{-1},q^{\lambda_2})\GammaJ(t^{-2},q^{\lambda_3})\cdots \cdot 1\right\rangle_{q,t}=\left\langle\Pop_{\frac{1}{1-q}}\nablaJ\cdot f,t^{-n(\lambda)}\J_\lambda\right\rangle _{q,t}.$$
% \begin{proof}
%     From \cref{eq Gamma-J}, we know that for any function $f$
%     $$\left\langle f[X], \GammaJ(z,q^{\lambda_1})\GammaJ(t^{-1}z,q^{\lambda_2}) \GammaJ(t^{-2}z,q^{\lambda_3})\cdots \cdot 1\right\rangle_{q,t} 
%     =\left\langle f[X],t^{-n(\lambda)}\nablaJ \Top_{\frac{1}{z(1-t)}} \J_\lambda[zX]\right\rangle_{q,t}.$$
% Since the operator $\nablaJ$ acts diagonally on the basis of Macdonald polynomials $(\J_\lambda)_{\lambda\in\mathbb Y}$ and this basis is orthogonal with respect to the scalar product $\langle,\rangle_{q,t}$ then $\nablaJ$ is self dual with this scalar product. Moreover, the dual of $\tau_{\frac{1}{z(1-t)}}$ is $\Pop_{\frac{1}{z(1-q)}}$. We conclude by specializing $z=1$.
% \end{proof}
% \end{lem}


\begin{thm}
    For any symmetric function $f$, the following holds 
    \begin{equation}\label{eq construction 1}
      f^*(v_1,\dots,v_k)=\left\langle f, \GammaJ(1,v_1)\GammaJ(t^{-1},v_2)\cdots\GammaJ(t^{-(k-1)},v_k)\cdot 1\right\rangle_{q,t}.  
    \end{equation}
    
\noindent Equivalently, for any Young diagram $\lambda$, 
\begin{equation}\label{eq construction 2}
    f^*(\lambda)=\left\langle\Pop_{\frac{1}{1-q}}\nablaJ\cdot f,t^{-n(\lambda)}J_\lambda\right\rangle _{q,t}.
\end{equation}
    
% \begin{proof}
% First, notice that \cref{eq construction 1} implies \cref{eq construction 2} by \cref{lem GammaJ-scalar J}.
% By definition of the isomorphism $f\mapsto f^*$, we should prove that for any partition $\lambda$, 
% \begin{equation}\label{eq RHS thm star-Gamma}
%   \left\langle J_\lambda^{(q,t)}, \GammaJ(1,v_1)\GammaJ(t^{-1},v_2)\cdots 1\right\rangle_{q,t},  
% \end{equation}
% satisfies the three properties of \cref{thm shifted Macdonal}.
% First, from \cref{eq GammaJ operator}, we know that for any $k$
% $$\GammaJ(z,v_1)\cdots\GammaJ(zt^{-(k-1)},v_k)\GammaJ(zt^{-k},1)\cdot 1
% =\GammaJ(z,v_1)\cdots\GammaJ(zt^{-(k-1)},v_k)\cdot 1.$$
% and that for any $n,k\geq0$ the coefficient 
% $$[z^n]\GammaJ(z,v_1)\cdots\GammaJ(zt^{-(k-1)},v_k)\cdot 1$$
% is a homogeneous symmetric function of degree $n$ in the variables $(x_i)_{i\geq1}$, with coefficients which are shifted symmetric in $(v_i)_{1\leq i\leq k}.$
% This implies that for any symmetric function $f$,  the right-hand side of \cref{eq construction 1} is a well defined shifted symmetric function in the variables $(v_i)_{i\geq 1}$. This gives property $(1)$. 

% In order to obtain property $(2)$, we use \cref{lem GammaJ-scalar J} with $f=\J_\lambda$. We get that 
% \begin{align*}
%   \left\langle \J_\lambda,\GammaJ(1,q^{\lambda_1})\GammaJ(t^{-1},q^{\lambda_2})\GammaJ(t^{-2},q^{\lambda_3})\cdots 1\right\rangle_{q,t}
%   &=\left\langle\Pop_{\frac{1}{1-q}}\nablaJ\cdot \J_\lambda,t^{-n(\lambda)}\J_\lambda\right\rangle _{q,t}\\
%   &=(-1)^{|\lambda|}q^{n(\lambda')}t^{-2n(\lambda)}\normJ_\lambda.
% \end{align*}
% We use here \cref{eq GammaJ}, and the fact that $[z^0]\Pop_{\frac{z}{1-q}}=1$. Hence, 
% \begin{equation}
% \left\langle \J_\lambda,\GammaJ(1,q^{\lambda_1})\GammaJ(t^{-1},q^{\lambda_2})\GammaJ(t^{-2},q^{\lambda_3})\cdots 1\right\rangle_{q,t}
% =J^*_\lambda(\lambda).
% \end{equation}
%  This corresponds to property $(2)$.
% Finally, for any partitions $\mu$ and $\lambda$, one has
% \begin{align*}
%   \left\langle \J_\mu,\GammaJ(1,q^{\lambda_1})\GammaJ(t^{-1},q^{\lambda_2})\GammaJ(t^{-2},q^{\lambda_3})\cdots 1\right\rangle_{q,t}
%   &=\left\langle\Pop_{\frac{1}{1-q}}\nablaJ\cdot \J_\mu,t^{-n(\lambda)}\J_\lambda\right\rangle _{q,t}.
% \end{align*}
% But from \cref{thm Pieri rule}, we know that the coefficient of $J_\lambda$ in $\Pop_{\frac{1}{1-q}}\nablaJ\cdot \J_\mu$ is 0 unless $\mu\subset \lambda$.     This proves that (\ref{eq RHS thm star-Gamma}) satisfies property $(3)$, which finishes the proof of the theorem.
% \end{proof}
\end{thm}


\begin{rmq}
    The isomorphism given in \cref{eq construction 2} has been implicitly described by Lassalle, see \cite[Definition 1]{Lassalle1998}.  However, the formula of \cref{eq construction 1} seems to be new.
    % The equivalence between the two formulas is a consequence of \cref{eq Gamma-J}. These two formulas play complementary roles in the proof; \cref{eq construction 1} gives the shifted symmetry property while \cref{eq construction 2} is used to prove the vanishing condition.
\end{rmq}







\subsection{Macdonald characters}\label{ssec Macdonald characters}







\begin{defi}\label{def Macdonald characters}
The \emph{Macdonald character} $\Mch_{\mu}$ is the function defined by 
% $$\Gamma^J(z,v_1)\Gamma^J(t^{-1}z,v_2)\cdots \Gamma^J(t^{-(k-1)}z,v_k)\cdot1=\sum_{\mu}z^{|\mu|}\Mch_{\mu ,k}p_\mu\left[X\right].$$
% In other terms,
\begin{align*}
    \Mch_{\mu}(v_1,v_2,\dots ):=(p_\mu)^*(v_1,v_2,\dots)
    =\left\langle p_\mu,\GammaJ(1,v_1)\GammaJ(t^{-1},v_2)\cdots \cdot 1\right\rangle_{q,t}\ \ .
\end{align*}

Moreover, for any Young diagram $\lambda$


\begin{equation}\label{eq characters J}
  \Mch_\mu(\lambda)=\left\{
\begin{array}{cc}
     \left\langle p_\mu, \nablaJ h^\perp_{|\lambda|-|\mu|}\left[\frac{X}{1-t}\right]\cdot t^{-n(\lambda)}J^{(q,t)}_\lambda\right\rangle_{q,t} & \text{ if } |\mu|\leq |\lambda| \\
    0  & \text{ otherwise.}
\end{array}\right.  
\end{equation}

\noindent In particular, when $|\mu|=|\lambda|$ the characters $\Mch_\mu(\lambda)$ are given by the power-sum expansion of $\J_\lambda$:
% \vspace{-0.9cm}
\begin{equation}\label{eq J characters}
(-1)^{|\lambda|}q^{n(\lambda')}t^{-2n(\lambda)}\J_\lambda=\sum_{\mu\vdash |\lambda|}\frac{\Mch_\mu(\lambda)}{z_\mu(q,t)} p_\mu.  \end{equation}
\end{defi}



% \subsubsection{Definition 2: with Macdonald polynomials}


% For any partitions $\mu$ and $\lambda$, let 
% $$\Mch_\mu(\lambda):=\Mch_\mu(q^{\lambda_1},q^{\lambda_2},\dots ,q^{\lambda_k}).$$



We give here a characterization of $\Mch_\mu$, which has been observed by Féray in the case of Jack polynomials, and proved very useful in practice in this case (see \cite[Theorem 2.5]{BenDaliDolega2023}). 
% The proof is very similar.
% We start by the following lemma.
% \begin{lem}\label{lem uniqueness}
%     Let $n\geq1$. If $G$ is shifted symmetric function of degree less or equal to $n$ with
%       \begin{equation}\label{eq G van}
%     G(\lambda)=0 \text{ for $|\lambda|\leq n$}. 
% \end{equation}
% Then $G=0$. 
%     \begin{proof}
% We expand $G$ in the $J^*_\xi$ basis 
% \begin{equation}\label{eq G J}
%   G=\sum_\xi c_\xi J^*_\xi.  
% \end{equation}

% As $\deg(G)\leq n,$ the sum can be restricted to partitions $\xi$ of size at most $n$. 
% We will prove by contradiction that $G=0$, \textit{i.e} that $c_\xi=0$
% for all partitions $\xi$ with $|\xi|\leq n$. Assume this is not
% the case and consider a partition $\xi_0$ of minimal size such that
% $c_{\xi_0}\neq 0$.
% \cref{eq G van} gives $G(\xi_0)=0$ since $|\xi_0|\leq n$. On the other
% hand, $J_\xi^*(\xi_0)=0$ if $\xi_0$ does not contain $\xi$ (see
% property (3) of \cref{thm shifted Macdonal}). Therefore the RHS of \cref{eq G J} evaluated on
% $\xi_0$ vanishes for
% all partitions $\xi$ except for $\xi=\xi_0$. Moreover,
% $c_{\xi_0}\neq 0$ by the assumptions and
% $J^*_{\xi_0}(\xi_0)\neq0$ from property (2) of \cref{thm shifted Macdonal}. Therefore
% $G(\xi_0) = c_{\xi_0} J^{*}_{\xi_0}(\xi_0) \neq 0$, and we have reached a contradiction. Hence, $G=0$ as required.    \end{proof}
% \end{lem}
\begin{thm}\label{thm Feray}
Let $\mu$ be a partition. Then $\Mch_\mu$ is the unique shifted symmetric function degree $|\mu|$ whose top homogeneous part is $p_\mu(v_1,t^{-1}v_2,t^{-2}v_3,\dots)$ and such that $\Mch_\mu(\lambda)=0$ for any partition  $|\lambda|<|\mu|$.
 % \begin{proof}
%     We start by proving that $\Mch_\mu$ satisfies these three properties. Since $\Mch_\mu=(p_\mu)^*$, then it is a shifted symmetric function, and from \cref{rmq top hom}, we know that its top homogeneous part is $p_\mu(v_1,t^{-1}v_2,\dots)$. Moreover, $\Mch_\mu$ is linear combination of $J_\xi^{*}$ for $\xi$ of size $|\mu|$. This gives the vanishing property.

%     Let us now prove the uniqueness. Let $F$ be a shifted symmetric function of degree $|\mu|$ with the same top degree part as $\Mch_\mu$, and such that $F(\lambda)=0$ for any $|\lambda |<|\mu|$. Set $G:=F-\Mch_\mu$. Then $G$ is a shifted symmetric function of degree at most $|\mu|-1$ with 
% $G(\lambda)=0 \text{ for $|\lambda|<|\mu|$}.$ Using \cref{lem uniqueness} we get that $G=0$ hence $F=\Mch_\mu$ which gives the uniqueness.
%  \end{proof}
\end{thm}




% \begin{prop}[Stability property]\label{prop stability}
%     For any  $\mu$ and $k\geq1$, one has
% $$\Mch_{\mu,k+1}(v_1,\dots,v_k,1)=\Mch_{\mu,k}(v_1,\dots,v_k).$$
% \begin{proof}
%     This is direct from \cref{eq GammaJ operator}.
% \end{proof}
% \end{prop}


% \begin{prop}[Shifted symmetry property]\label{prop Shifted symmetry}
%     For any partition $\mu$ and positive integer $k$, one has $\Mch_{\mu,k}$ is shifted symmetric.
%     \begin{proof}
% This is a consequence of \cref{eq GammaJ operator2}.     
%     \end{proof}
% \end{prop}

% From \cref{prop stability} and \cref{prop Shifted symmetry}, we deduce that $(\Mch_{\mu,k})_{k\geq 1}$ defines a shifted symmetric function $\Mch_\mu.$



% \begin{prop}
%     The Macdonald characters defined in \cref{def Macdonald characters} satisfy the conditions of \cref{thm Feray}.
%     \begin{proof}
%         The fact that $\Mch_\mu$ is shifted symmetric has been proved in Propositions \ref{prop stability} and \ref{prop Shifted symmetry}:
        
%          The vanishing property is direct from \cref{thm characters and J}.
%         Finally, since the top degree part in the operator $\GammaJ(z,u)$ as a polynomial in $u$ is $\cP_{zu\frac{1-t}{1-q}}$, we deduce that the top degree part in $\Mch_\mu(v_1,v_2,\dots,v_k)$ corresponds to 
%         \begin{align*}
%           [z^{|\mu|}p_\mu]\cP_{zv_1\frac{1-t}{1-q}}\cP_{zt^{-1}v_2\frac{1-t}{1-q}}\dots\cP_{zt^{-(k-1)}v_k\frac{1-t}{1-q}}\cdot 1
%           &=[z^{|\mu|}p_\mu]\cP_{z(v_1+v_2\cdot t^{-1}+\dots v_k\cdot t^{-(k-1)})\frac{1-t}{1-q}}\\
%           &=\frac{1}{z_\mu}p_\mu\left[(v_1+v_2\cdot t^{-1}+\dots v_k\cdot t^{-(k-1)})\frac{1-t}{1-q}\right]\\
%          &=\frac{1}{z_\mu}p_\mu\left[\frac{1-t}{1-q}\right]p_\mu(v_1,v_2t^{-1},\dots, v_kt^{-(k-1)}).
%         \end{align*}
%     \end{proof}
% \end{prop}

% \subsection{Positivity conjectures about the characters \texorpdfstring{$\Mch_\mu$}{}}
% \begin{conj}
% Let the shifted parameters $q':=q-1$ and $u:=t-1$.
%     Then, $(-1)^{|\mu|}t^{(k-1)|\mu|}\Mch_\mu$ is a polynomial in the variables $-v_1,\cdots,-v_k$ and the parameters $q'$ and $u$ with non-negative integer coefficients.
% \end{conj}

% \begin{conj}\label{conj Gamma-p}
%     The operator $\GammaJ(z,v)$ is positive on the basis $p_\mu\left[X\frac{1-t}{1-q}\right]$ in the variables $q',u, -v, -z$. More precisely, if $\mu$ and $\nu$ are two partitions such that $|\nu|-|\mu|=n$, then
%     $$(-1)^nt^{|\mu|}\left\langle[z^n]\GammaJ(z,v)\cdot p_\mu\left[X\frac{1-t}{1-q}\right],p_\nu\left[X\frac{1-q}{1-t}\right]\right\rangle$$
%     is a polynomial in the variables $-v,q',u$ with non-negative integer coefficients.
% \end{conj}
% This conjecture has been tested for $|\nu|\leq 9$.





% % % \begin{thm}
% % %     There exists a unique shifted polynomial $\Mch_\mu(v_1,v_2,\dots ,v_k)$ such that for every partition $\lambda$
% % %     $$\Mch_\mu(q^{\lambda_1},q^{\lambda_2},\dots ,q^{\lambda_k})=t^{-n(\lambda)}\left\langle\nabla_J h^\perp_{|\lambda|-|\mu|}\left[X/(1-t)\right]\cdot J^{(q,t)}_\lambda,p_\mu\left[X\frac{1-q}{1-t}\right]\right\rangle,$$
% % % \end{thm}

% % % \begin{prop}
% % %     $$\Mch_\mu(v_1,v_2,\dots ,v_k)=\left\langle\nabla\cdot\Gamma(z,v_1)\Gamma(tz,v_2)\cdots \Gamma(t^{k-1}z,v_k)\cdot 1_{\big|t=1/t},p_\mu\left[X\cdot(1-q)\right]\right\rangle.$$
% % % \end{prop}
% % % Hence \cref{conj pos Gamma} implies that the shifted characters $\theta_\mu^{(q,t+1)}$, multiplied by $(1+t)^{|\mu|}$ are positive polynomials in the variables $-v_1,-v_2,\dots, -v_k$.



% \begin{prop}
%     \cref{conj Gamma-p} implies that $(-1)^{|\mu|}t^{2(k-1)|\mu|}\Mch_\mu(v_1,v_2,\dots,v_k)$ is a polynomial in the variables $-v_1,\cdots,-v_k$ and the parameters $q'$ and $t'$ with non-negative coefficients.

%     \begin{proof}
%     Let us assume that \cref{conj Gamma-p} holds.
%         We prove by induction on $k$ that for any $n$
%         $$(-1)^nt^{2(k-1)n}[z^n]\GammaJ(z,v_1)\GammaJ(t^{-1}z,v_2)\cdots \GammaJ(t^{-(k-1)}z,v_k)\cdot1$$ has 
%         a positive polynomial expansion on the basis $\left(p_\mu\left[X\frac{1-t}{1-q}\right]\right)_{\mu\vdash n}.$
%         For $k=1$ this is direct from \cref{conj Gamma-p}. If the induction assumption holds for $k$, then 
%         $$t^{(2k-1)n}[z^n]\GammaJ(t^{-1}z,v_2)\GammaJ(t^{-2}z,v_3)\cdots \GammaJ(t^{-k}z,v_{k+1})\cdot1,$$
%         also has positive expansion. By multiplying by $t^n$ and applying $\Gamma^J(t^{-1}z,v_1)$ on the left we obtain that 
%         $$\left([z^m]\GammaJ(t^{-1}z,v_1)\right)t^{2(k-1)n}[z^n]\GammaJ(t^{-1}z,v_2)\GammaJ(t^{-2}z,v_3)\cdots \GammaJ(t^{-k}z,v_{k+1})\cdot1,$$
%         has positive expansion on $\left(p_\mu\left[X\frac{1-t}{1-q}\right]\right)_{\mu\vdash (n+m)},$
%         which implies the induction hypothesis for $k+1$.
%     \end{proof}
% \end{prop}








\section{Macdonald generalization of some Jack conjectures}\label{sec Macdonald conjectures}

\subsection{A normalization related to Jack polynomials}
Jack polynomials can be obtained from the integral form of Macdonald polynomials as follows (see \cite[Chapter VI, eq (10.23)]{Macdonald1995})
\begin{equation}\label{eq Jack from Macdonald}
    \lim_{t\rightarrow 1}\frac{J_\lambda^{(q=1+\alpha(t-1),t)}}{(1-t)^{|\lambda|}}=\Jla.
\end{equation}
In the following, we introduce the parameters $(\alpha,\gamma)$  related to $(q,t)$ by
$$\left\{\begin{array}{ll}
   q=1+\gamma\alpha  \\
   t=1+\gamma
\end{array}\right. \longleftrightarrow 
\left\{\begin{array}{ll}
   \alpha=\frac{1-q}{1-t}  \\
   \gamma=t-1
\end{array}\right.\ .$$
We consider the following normalization of Macdonald polynomials $$\JJack_\lambda:=\frac{\J_\lambda}{(1-t)^{|\lambda|}}.$$
Hence, this normalization is directly related to Jack polynomials by $\mathfrak J^{(\alpha,\gamma=0)}_\lambda=J^{(\alpha)}_\lambda.$






 
% \subsection{Jack polynomials}
% \subsubsection{Preliminaries}
% We denote the $\langle,\rangle_\alpha$, the scalar product defined on power-sum functions by
% $$\langle p_\mu,p_\nu\rangle_\alpha=\delta_{\mu,\nu} z_\lambda \alpha^{\ell(\mu)} .$$
% Jack polynomials are orthogonal with respect to this scalar product. We denote $j^{(\alpha)}_\lambda$ their squared norm;
% $$\langle J^{(\alpha)}_\lambda,J_\mu^{(\alpha)}\rangle_\alpha=\delta_{\lambda,\mu}j^{(\alpha)}_\lambda.$$
% \subsubsection{Stanley's conjecture}
% In his seminal work \cite{Stanley1989}, Stanley formulated the following positivity conjecture about the structure coefficients of Jack polynomials.

% \begin{conj}\cite{Stanley1989}
%     For any partitions $\lambda$, $\mu$ and $\nu$, 
%      $$\langle J^{(\alpha)}_\lambda J^{(\alpha)}_\mu,J^{(\alpha)}_\nu\rangle_{\alpha}$$
% is a polynomial in $\alpha$ with integer non-negative coefficients.
% \end{conj}




% \subsubsection{Lassalle's conjecture}


% Lassalle's conjecture, proved in \cite{BenDaliDolega2023}, suggests that the character $\Jch_\mu(\lambda)$ is a positive polynomial in $b:=\alpha-1$, and some coordinates of $\lambda$ called \textit{multirectangular coordinates}. We are here interested in a weak version of this result, which we generalize to the Macdonald case.
% \begin{thm}\cite{BenDaliDolega2023}
% Fix a partition $\mu$. The normalized Jack character $(-1)^{|\mu|}z_\mu\Jch_\mu(\lambda)$  is a polynomial in the variables $b:=\alpha-1,-\alpha \lambda_1,-\alpha \lambda_2\dots$ with integer non-negative coefficients.  
% \end{thm}

% \subsubsection{Goulden and Jackson's conjectures}
% Goulden and Jackson have considered the coefficients $c^\lambda_{\mu,\nu}(\alpha)$ and $h^\lambda_{\mu,\nu}(\alpha)$, defined by 
% \begin{align*}
%     \sum_{\lambda\in\mathbb Y} z^{|\lambda|} \frac{\Jla[X]\Jla[Y]\Jla[Z]}{j^{(\alpha)}_\lambda}=\sum_{m\geq 0}\sum_{\pi,\mu,\nu\vdash m} \frac{z^m c^\pi_{\mu,\nu}(\alpha)}{z_\pi \alpha^{\ell(\pi)}}p_\pi[X]p_\mu[Y]p_\nu[Z],
% \end{align*}

% \begin{align*}
%   \log\left(\sum_{\lambda\in\mathbb Y} z^{|\lambda|} \frac{\Jla[X]\Jla[Y]\Jla[Z]}{j^{(\alpha)}_\lambda}\right) 
%   =\sum_{m\geq 0}\sum_{\pi,\mu,\nu\vdash m} \frac{z^m h^\pi_{\mu,\nu}(\alpha)}{\alpha m}p_\pi[X]p_\mu[Y]p_\nu[Z].   
% \end{align*}
   


% % where 
% % $$\normJ_\lambda:=\lim _{u\mapsto 0}\frac{\normJt_\lambda}{(1-t)^{2|\lambda|}}=\prod_{\Box\in\lambda}\left(\alpha a(\Box)+
% % \ell(\Box)+1\right)\left(\alpha (a(\Box)+1)+\ell(\Box)\right).$$


%  For $\alpha=0$ (resp. $\alpha=1$), the series above are known to count bipartite graphs on orientable surfaces (resp. surfaces orientable or not) called \textit{maps}, see \cite{GouldenJackson1996a}. Goulden and Jackson have formulated the following two conjectures.

% \begin{conj}[Matchings-Jack conjecture \cite{GouldenJackson1996}]\label{conj MJ}
%     The coefficients $c^\pi_{\mu,\nu}(\alpha)$ are polynomials in the parameter $b:=\alpha-1$ with non-negative integer coefficient.
% \end{conj}

% \begin{conj}[\texorpdfstring{$b$}{}-conjecture \cite{GouldenJackson1996}]\label{b-conj}
%     The coefficients $h^\pi_{\mu,\nu}(\alpha)$ are polynomials in the parameter $b:=\alpha-1$ with non-negative integer coefficient.
% \end{conj}

% These two conjectures are still open despite many partial results \cite{DolegaFeray2016,DolegaFeray2017,ChapuyDolega2022,BenDali2022a,BenDali2023}.
% In particular, the integrality part in the Matchings-Jack conjecture has been proved in \cite{BenDali2023}.
% \begin{thm}\cite{BenDali2023}
%     For any partitions $\pi,\mu,\nu\vdash n\geq 1$, the coefficient $c^\pi_{\mu,\nu}(\alpha)$ is polynomial in $b$ with integer coefficients.  
% \end{thm}






\begin{rmq}
Unlike the integral form $\J_\lambda$, the coefficients of  $\JJack_\lambda$ in the monomial basis are positive in $\gamma$ and $u$. This can be seen using the combinatorial interpretation of Macdonald polynomials given in \cite[Proposition 8.1]{HaglundHaimanLoehr2005}.    
\end{rmq}

We observed that the parameterization $(\alpha,\gamma)$ allows to generalize several conjectures about Jack polynomials to Macdonald polynomials which we now formulate.

% \begin{prop}
%     The coefficient of $\JJack_\lambda$ in the monomial basis are polynomials in $\alpha$ and $u$ with integer non-negative coefficients.
%     \begin{proof}
%         This can be easily obtained from the combinatorial interpretation given in \cite[Proposition 8.1]{HaglundHaimanLoehr2005} for $\J_\lambda$.
%     \end{proof}
% \end{prop}



   



\subsection{Weak Lassalle's conjecture}



We consider the following normalization of Macdonald characters defined in \cref{ssec Macdonald characters}
\begin{equation*}
  \MchJack_\mu(s_1,s_2,\dots):=\frac{1}{z_\mu(q,t)\gamma^{|\mu|}}\Mch_\mu(1+\alpha \gamma s_1,1+\alpha \gamma s_2,\dots).  
\end{equation*}
% Note that $\MchJack_\mu$ is symmetric in the variables $t^{-i}u_i+\frac{t^{-i}}{\alpha u}$.
For any partition $\lambda$, we denote 
\begin{equation}\label{eq def theta alpha gamma}
\MchJack_\mu(\lambda):=\MchJack_\mu\left(\frac{q^{\lambda_1}-1}{\alpha \gamma},\frac{q^{\lambda_2}-1}{\alpha \gamma},\dots\right)=\frac{\Mch_\mu(\lambda)}{z_\mu(q,t)\gamma^{|\mu|}}.  
\end{equation}

% \noindent Hence, \cref{eq characters J} can be rewritten as 
% \begin{equation}\label{eq charactersJack J}
%   \MchJack_\mu(\lambda)=\left\{
% \begin{array}{cc}
%       [p_\mu](-1)^{|\mu|} \nablaJ (1-t)^{|\lambda|-|\mu|} h^\perp_{|\lambda|-|\mu|}\left[\frac{X}{1-t}\right]\cdot t^{-n(\lambda)}\JJack_\lambda& \text{ if } |\mu|\leq |\lambda| \\
%     0  & \text{ otherwise.}
% \end{array}\right.  
% \end{equation}
% In particular,
% $$q^{n(\lambda')}t^{-2n(\lambda)}\JJack_\lambda=\sum\limits_{\mu\vdash |\lambda|}\MchJack_\mu(\lambda) p_\mu.$$



% \begin{defi}[Jack characters]Fix a partition $\mu$. The Jack character $\theta^{(\alpha)}_\mu$ is the function on Young diagrams $\lambda$ defined by 
%     $$ \theta^{(\alpha)}_\mu(\lambda):=\left\{
%     \begin{array}{ll}
%     \left[p_\mu\right]\frac{1}{(|\lambda|-|\mu|)!}(\frac{\partial}{\partial p_1})^{|\lambda|-|\mu|}J^{(\alpha)}_\lambda &  \mbox{ if $|\mu|\leq|\lambda|$}\\
%        0 & \mbox{ if } |\lambda|<|\mu|,
%         \end{array}
% \right.$$
% \end{defi}
% Jack characters have been defined by Lassalle \cite{Lassalle2008b}.

% The Jack normalization of Macdonald characters are related to Jack characters introduced in\cite{Lassalle2008b} by the following proposition.
% \begin{prop}\label{prop Jack characters}
% For any partitions $\mu$ and $\lambda$, we have
%     $\lim_{\gamma\mapsto 0}\MchJack_\mu(\lambda)=\theta^{(\alpha)}_\mu(\lambda).$
% %     \begin{proof}
% % Since $t\rightarrow1$ when $u\mapsto0$, and the operator $\nablaJ$ acts on a homogeneous functions of degree $n$ as a multiplication by $(-1)^n$. Moreover,
% % $$\lim_{t\rightarrow 1}h_r^\perp\left[\frac{X}{1-t}\right](1-t)^{r}=\lim_{t\rightarrow 1}(1-t)^{r}\sum_{\nu\vdash r}\prod_{i\in\nu}\frac{p_{\nu_i}^\perp}{(1-t^{\nu_i})z_\nu}=\frac{(p_1^\perp)^r}{r!}.$$
% % \cref{eq charactersJack J} allows to conclude.\qedhere
% % \end{proof}
%     \end{prop}

Jack characters, introduced by Lassalle \cite{Lassalle2008b}, are obtained from the characters $\MchJack_\mu$ by specializing $\gamma=0$. We formulate the following conjecture\footnote{This is a generalization of a weak version of Lassalle's conjecture on Jack characters, in which we keep one alphabet $(s_1,s_2,\dots)$ instead of two alphabets associated to the multirectangular coordinates of $\lambda$.}, tested for $k\leq 3$ and $|\mu|\leq 7$.
\begin{conj}\label{conj Lassalle generalized}
Fix $k\geq 1$ and a partition $\mu$. Then, 
    $(-1)^{|\mu|}t^{(k-1)|\mu|}z_\mu(q,t)\MchJack_\mu(s_1,s_2,\dots, s_k)$ is a polynomial in $\gamma,b:=\alpha-1, -\alpha s_1, -\alpha s_2,\dots, -\alpha s_k$ with non-negative integer coefficients.
\end{conj}

\noindent Computer tests also suggest that the action of the operator $\GammaJ$ on the power-sum basis satisfies a positivity property that would imply the positivity part in \cref{conj Lassalle generalized}.


\subsection{Structure coefficients of \texorpdfstring{$\MchJack_\mu$}{}}\label{ssec structure coefficients}

It follows from the definition of the characters $\Mch_\mu$ that they form a basis of the space of shifted symmetric functions $\Lambda^*$. Moreover, $\MchJack_\mu(\lambda)$ is obtained from $\Mch_\mu$ by a normalization by a scalar and a change of variables (see \cref{eq def theta alpha gamma}) and by consequence their structure coefficients are well defined:
 \begin{equation}\label{eq structure coefficients}
  \MchJack_\mu\MchJack_\nu=\sum_{\pi}g^\pi_{\mu,\nu}(\alpha,\gamma)\MchJack_\pi.   
 \end{equation}

The coefficients $g^\pi_{\mu,\nu}(\alpha,\gamma)$ are a two parameter generalization of structure coefficients of Jack characters $\Jch_\mu$ introduced by Do\l{}e\k{}ga and Féray in \cite{DolegaFeray2016} (see also \cite{Sniady2019}).

% $$\Jch_\mu\Jch_\nu=\sum_{\pi}g^\pi_{\mu,\nu}(\alpha,\gamma=0)\Jch_\pi.$$

% In the following, we will prove that in the case $|\pi|=|\mu|=|\nu|$ the coefficients $g^\pi_{\mu,\nu}(\alpha,u)$ coincide with the coefficients $c^\pi_{\mu,\nu}(\alpha,u)$  defined in \cref{ssec generalized conjectures}. The proof is very similar to the one given in \cite{DolegaFeray2016} for the Jack case. We start by proving some properties of the coefficients $g^\pi_{\mu,\nu}$.

% \begin{lem}\label{lem bounds}
%     The coefficient 
%     $g^\pi_{\mu,\nu}$ is 0 unless 
%     $$\max(|\mu|,|\nu|)\leq|\pi|\leq |\mu|+|\nu|.$$

%     \begin{proof}
%       The upper bound is a consequence of the fact that $\MchJack_\mu \MchJack_\nu$ is a shifted symmetric function of degree $|\mu|+|\nu|$. On the other hand, for any partition $\lambda$ such that $|\lambda|<\max(|\mu|,|\nu|)$, one has by the vanishing condition that 
%       $\MchJack_\mu(\lambda)\MchJack_\nu(\lambda)=0$, and also that
%       $$\sum_{|\pi|\geq \max(\mu,\nu)}g^\pi_{\mu,\nu}(\alpha,u)\MchJack_\pi(\lambda)=0.$$ Combining these two equations with \cref{eq structure coefficients}, we get that 
%       $$\sum_{|\pi|<\max(|\mu|,|\nu|)}g^\pi_{\mu,\nu}(\alpha,u)\MchJack_\pi(\lambda)=0,\quad \text{ for any }|\lambda|<\max(|\mu|,|\nu|).$$
%       But $\sum_{|\pi|< \max(\mu,\nu)}g^\pi_{\mu,\nu}(\alpha,u)\MchJack_\pi$ is a shifted symmetric function of degree smaller than $\max(|\mu|,|\nu|)$. Using \cref{lem uniqueness}, we deduce that it is identically equal to 0, and by consequence $g^\pi_{\mu,\nu}(\alpha,u)=0$ for any $|\pi|< \max(\mu,\nu)$.
%     \end{proof}
% \end{lem}


% We deduce the following corollary.

% \begin{cor}\label{cor g}
%     Fix a positive integer $n$ and three partitions $\lambda,\mu,\nu\vdash n$. Then 
%     $$\MchJack_\mu(\lambda) \MchJack_\nu(\lambda)=\sum_{\pi\vdash n}g^\pi_{\mu,\nu}\MchJack_\pi(\lambda).$$ 

%     \begin{proof}
%         We start by evaluating \cref{eq structure coefficients} in $\lambda$. From the vanishing condition we know that partitions $\pi$ of size larger than $n$ do not contribute to the sum. But  applying \cref{lem bounds} we get that $g^\pi_{\mu,\nu}=0$ if $|\pi|<n$. This concludes the proof. 
%     \end{proof}
% \end{cor}








Let $f$ be the function defined on triplets of non-negative integers by
$$f(n_1,n_2,k):=(N-n)(N+n-k)+n(n-1)-(k-N)(k-N-1),$$
where 
$N:=\max(n_1,n_2) \text{ and } n=\min(n_1,n_2).$
% A priori, the coefficients are rational functions in $\alpha$ and $u$. 
We make the following conjecture which extends \cite[Conjecture 2.2]{Sniady2019}.
\begin{conj}\label{conj structure coeff}
Let $\pi,\mu,\nu$ be three partitions.
    Then, the coefficients 
    $(1+\gamma)^{f(|\mu|,|\nu|,|\pi|)} z_\mu z_\nu g^\pi_{\mu,\nu}$
    are polynomials in $b:=\alpha-1$ and $\gamma$ with non-negative integer coefficients.
\end{conj}






% \begin{rmq}
%     \'{S}niady has formulated a positivity conjecture about to the structure coefficients of Jack characters \cite[Conjecture 0.1]{Sniady2016}. This conjecture is related to the case $\alpha=0$ in \cref{conj structure coeff} but the normalizations are differents.
% \end{rmq}





% From the formula given in \cref{eq characters J}, we know that 
% $$\lim_{u\rightarrow 0} \frac{\Mch_\mu(q^\lambda_1,\dots,q^{\lambda_{\ell(\lambda)}})}{u^{|\mu|}\langle p_\mu,p_\mu\rangle_{q,t}}
% =\left\{
% \begin{array}{cc}
%      \lim_{t\rightarrow 1}[p_\mu](-1)^{|\mu|}\nablaJ (1-t)^{|\lambda|-|\mu|}h^\perp_{|\lambda|-|\mu|}\left[\frac{X}{1-t}\right]\cdot t^{-n(\lambda)}\frac{\J_\lambda}{(1-t)^{|\lambda|}} & \text{ if } |\mu|\leq |\lambda|, \\
%     0  & \text{ otherwise.}
% \end{array}\right.$$
% But as $t\rightarrow1$, the operator $\nablaJ$ acts on a homogeneous functions of degree $n$ as a multiplication by $(-1)^n$, and with the change of variable $q=1+\alpha(t-1)$, 
% $$\lim_{t\rightarrow 1}\frac{\J_\lambda}{(1-t)^{|\lambda|}}=\Jla.$$ Moreover,
% $$\lim_{t\rightarrow 1}h_r^\perp\left[\frac{X}{1-t}\right](1-t)^{r}=\lim_{t\rightarrow 1}(1-t)^{r}\sum_{\nu\vdash r}\prod_{i\in\nu}\frac{p_{\nu_i}^\perp}{(1-t^{\nu_i})z_\nu}=\frac{(p_1^\perp)^r}{r!}.$$
% Hence
% \begin{equation}\label{eq lim theta}
%   \lim_{u\rightarrow 0} \frac{\Mch_\mu(q^\lambda_1,\dots,q^{\lambda_{\ell(\lambda)}})}{u^{|\mu|}\langle p_\mu,p_\mu\rangle_{q,t}}= \left\{
% \begin{array}{ll}
%  \left[p_\mu\right]\frac{1}{(|\lambda|-|\mu|)!}(p_1^\perp)^{|\lambda|-|\mu|}J^{(\alpha)}_\lambda &  \mbox{ if $|\mu|\leq|\lambda|$},\\
%        0 & \mbox{ if } |\lambda|<|\mu|.
%        \end{array}
% \right.  
% \end{equation}


% On the other hand, from \cref{eq def theta alpha u}, we have 
% $$\lim_{u\rightarrow 0} \frac{\Mch_\mu(q^\lambda_1,\dots,q^{\lambda_{\ell(\lambda)}})}{u^{|\mu|}}
% =\lim_{u\rightarrow 0}\theta^{(\alpha,u)}_\mu\left(\frac{q^{\lambda_1}-1}{\alpha u},\frac{q^{\lambda_2}-1}{\alpha u},\dots\right).$$
% Notice that 
% $$\lim_{u\rightarrow 0}\frac{q^{\lambda_i}-1}{\alpha u}=\lambda_i,$$
% hence
% \todo[inline]{explain why $\MchJack_\mu$ has no pole in 0 as function in the $s_i$'s.}
% \begin{equation}\label{eq theta tilde-theta}
%     \lim_{u\rightarrow 0} \frac{\Mch_\mu(q^\lambda_1,\dots,q^{\lambda_{\ell(\lambda)}})}{u^{|\mu|}}
% =\theta_\mu^{(\alpha,u=0)}\left(\lambda_1,\lambda_2,\dots\right).
% \end{equation}
% We conclude by combining the equations \ref{eq lim theta} and \ref{eq theta tilde-theta}.
%  




% \subsection{Link to Jack characters}
% \begin{prop}
%     For any partition $\lambda$, one has 
%     $$\lim_{t\rightarrow 1}\frac{\theta^{(q,t)}_\mu (q^{\lambda_1},q^{\lambda_2},\dots ,q^{\lambda_k})_{|q=t^\alpha}}{(1-t)^{|\mu|}}=\Jch_\mu(\lambda)=\left\langle \exp\left(\frac{\partial}{\partial p_1}\right) J^{(\alpha)}_\lambda,p_\mu\right\rangle_\alpha.$$
%     \begin{proof}
%         We use the following facts:
%         \begin{itemize}
%             \item $$\lim_{t\rightarrow 1}\frac{J_\lambda^{(t^\alpha,t)}}{(1-t)^{|\lambda|}}=\Jla.$$
%             \item 
%             \item $$\lim_{t\rightarrow 1}p_\mu\left[X\frac{1-t^\alpha}{1-t}\right]=\alpha^{\ell(\mu)}p_\mu.$$
%             \item $$\lim_{t\rightarrow 1}\nabla_J=1.$$
%         \end{itemize}
%     \end{proof}
% \end{prop}

% \begin{cor}
%     The Jack characters are obtained from Macdonald characters as follows 
%     $$\Jch_\mu(r_1,r_2,\dots,r_k)=\lim_{t\rightarrow 1}\frac{\Mch_\mu\left( 1+(q-1)r_1,1+(q-1)r_2,\dots ,1+(q-1)r_k\right)_{|q=t^\alpha}}{(1-t)^{|\mu|}}.$$
%     \begin{proof}
%         If $v_i=q^{\lambda_i}$ and $r_i=\frac{1-v_i}{1-q}$ then 
%         $$\lim_{t\rightarrow1}{r_i}_{\big|q=t^\alpha}=\lambda_i.$$
%     \end{proof}
% \end{cor}








\subsection{Generalized Goulden and Jackson's conjectures}\label{ssec generalized conjectures}

We define the coefficients $c^\pi_{\mu,\nu}$ and $h^\pi_{\mu,\nu}$ for partitions $\pi,\mu$ and $\nu$ of the same size by 
 \begin{multline*}%\label{eq def c}
    \sum_{\lambda\in\mathbb Y}\! u^{|\lambda|} t^{-2n(\lambda)}q^{n'(\lambda)}\frac{\JJack_\lambda[X]\JJack_\lambda[Y]\JJack_\lambda[Z]}{\normJ_\lambda \gamma^{-2|\lambda|}}
    \!=\!\!\sum_{n\geq 0}\sum_{\pi,\mu,\nu\vdash n} \!\!\! \frac{u^n c^\pi_{\mu,\nu}(\alpha,\gamma)}{z_\pi(q,t)}p_\pi[X]p_\mu[Y]p_\nu[Z],
\end{multline*}
% and 
\begin{multline*}%\label{eq def h}
    \log\left(\sum_{\lambda\in\mathbb Y} u^{|\lambda|} t^{-2n(\lambda)}q^{n'(\lambda)}\frac{\JJack_\lambda[X]\JJack_\lambda[Y]\JJack_\lambda[Z]}{\normJ_\lambda \gamma^{-2|\lambda|}}\right)\\
    =\sum_{n\geq 0}\sum_{\pi,\mu,\nu\vdash n} \frac{u^n h^\pi_{\mu,\nu}(\alpha,u)}{\alpha [n]_q }p_\pi[X]p_\mu[Y]p_\nu[Z],
\end{multline*}
where 
% \begin{equation}\label{eq norm j t}
% \normJt_\lambda:=u^{-2|\lambda|}\normJ_\lambda=\left\langle\JJack_\lambda,\JJack_\lambda\right\rangle_{q,t},  
% \end{equation}
$[n]_q:=1+q+\dots+q^{n-1}.$
By taking $\gamma=0$, the coefficients $c^\pi_{\mu,\nu}(\alpha,\gamma)$ and $h^\pi_{\mu,\nu}(\alpha,\gamma)$ give the coefficients of the celebrated  Matchings-Jack and $b$-conjectures formulated by Goulden and Jackson in \cite{GouldenJackson1996}. 
The coefficients $c^\pi_{\mu,\nu}$ are actually a special case of $g^\pi_{\mu,\nu}$.
\begin{prop}\label{prop g c}
    Let  $\pi,\mu$ and $\nu$ be three partitions of the same size. Then 
    $$c^\pi_{\mu,\nu}(\alpha,\gamma)= g^\pi_{\mu,\nu}(\alpha,\gamma).$$
%     \begin{proof}
%     % \todo[inline]{Adapt the proof of \cite{DolegaFeray2016} for the Jack case.}

% We introduce for each $\pi\vdash n$ the two generating series
% $$C_{\mu,\nu}:=\sum_{\pi\vdash m}\frac{c^\pi_{\mu,\nu}}{z_\pi(q,t)}p_\pi[X],$$
% and 
% $$G_{\mu,\nu}:=\sum_{\pi\vdash m}\frac{g^\pi_{\mu,\nu}}{z_\pi(q,t)}p_\pi[X].$$
% We want to prove that these two series are equal.
% From the definition of the coefficients $c^\pi_{\mu,\nu}$ and \cref{eq JJack characters} we have
% \begin{align*}
%     C_{\mu,\nu}
%     =\sum_{\lambda\vdash m}\frac{\MchJack_\mu(\lambda) \MchJack_\nu(\lambda)}{t^{-2n(\lambda)} q^{n(\lambda')}\normJt_\lambda}\JJack_\lambda[X].
% \end{align*}
%     Using \cref{cor g}, we get that 
%     \begin{equation}\label{eq C}
%       C_{\mu,\nu}=\sum_{\pi,\lambda\vdash m} \frac{g^\pi_{\mu,\nu}\MchJack_\pi(\lambda)}{t^{-2n(\lambda)} q^{n(\lambda')}\normJt_\lambda}\JJack_\lambda[X].
%     \end{equation}
    
%     On the other hand, using the fact that both Macdonald polynomials and the power-sum function are orthogonal families, \cref{eq JJack characters} can be inverted as follows
%     $$\frac{p_\pi}{z_\pi(q,t)}=\sum_{\lambda\vdash m}\frac{\MchJack_\pi(\lambda)}{t^{-2n(\lambda)}q^{n(\lambda')}\normJt_\lambda}\JJack_\lambda.$$
%     Hence, \cref{eq C} becomes 
%     $$C_{\mu,\nu}=\sum_{\pi\vdash m}\frac{g^\pi_{\mu,\nu}}{z_\pi(q,t)}p_\pi[X].$$
    
%     This is precisely the series $G_{\mu,\nu}$, which concludes the proof of the proposition.
% \end{proof}
\end{prop}


We consider here a Macdonald analog of Goulden and Jackson conjectures.
\begin{conj}[Macdonald generalization of the Matchings-Jack conjecture]\label{conj MJ generalized}
For any positive integer $n$ and partitions $\pi,\mu,\nu$ of $n$, the quantity $(1+\gamma)^{n(n-1)}z_\mu z_\nu c^\pi_{\mu,\nu}(\alpha,\gamma)$ is a polynomial in $b$ and $\gamma$ with non-negative integer coefficients.
\end{conj}
\begin{conj}[Macdonald generalization of the \texorpdfstring{$b$}{}-conjecture]\label{b-conj  generalized}
For any positive integer $n$ and partitions $\pi,\mu,\nu$ of $n$, the quantity
    $(1+\gamma)^{n(n-1)}z_\pi z_\mu z_\nu h^\pi_{\mu,\nu}(\alpha,\gamma)$
    is a polynomial in $b$ and $\gamma$ with non-negative integer coefficients.
\end{conj}
\cref{conj MJ generalized} has been tested for $n\leq 8$ and \cref{b-conj  generalized} for $n\leq 9$.
Notice that by \cref{prop g c}, \cref{conj MJ generalized} is a special case of \cref{conj structure coeff}.

\begin{rmq}
Given the integrality result of \cite{BenDali2023}, it is easy to see that \cref{conj MJ generalized} implies the Matchings-Jack conjecture \cite[Conjecture 4.2]{GouldenJackson1996}. Similarly, \cref{b-conj  generalized} implies the positivity in the $b$-conjecture \cite[Conjecture 6.2]{GouldenJackson1996}.
\end{rmq}

% \begin{rmq}
% In Equations \ref{eq def c} and \cref{eq def h} it seems possible to change the factor $t^{-2n(\lambda)}q^{n(\lambda')}$ by  $t^{-n(\lambda)}(t^{-n(\lambda)}q^{n(\lambda')})^r$ for some $r\geq 0$ and the conjectures above still hold. However, we will prove in \cref{ssec structure coefficients} that the coefficients $c^\pi_{\mu,\nu}(\alpha,u)$ are a special case of the structure coefficients of the characters $\MchJack_\mu$. This implies that in some sense the coefficients $c^\pi_{\mu,\nu}(\alpha,u)$ are a natural two parameter generalization of the coefficients considered by Goulden and Jackson and justifies the factor $t^{-2n(\lambda)}q^{n(\lambda')}$ which appears in the previous definitions.
% \end{rmq}


% \begin{prop}
%     \cref{conj MJ generalized} implies \cref{conj MJ}.
% \begin{proof}
%      \cref{conj MJ generalized} implies that $c^\pi_{\mu,\nu}(\alpha)$ are polynomials in $\alpha$ with positive coefficients. But  these polynomials have integer coefficients by \cite{BenDali2023}. This concludes the proof. 
% \end{proof}
% \end{prop}

% Note also that the positivity in \cref{b-conj  generalized} implies the positivity in \cref{b-conj}.

% Since $f(n,n,n)=n(n-1)$, and given $\cref{prop g c}$, it is easy to check that \cref{conj MJ generalized} is a special case of \cref{conj structure coeff}.

\subsection{A generalization of Stanley's conjecture}
We conclude with a generalization of Stanley's conjecture about the structures coefficients of Jack polynomials. While not directly related to Macdonald characters, this conjecture is also obtained from the new parameterization in $\alpha$ and $\gamma$.
\begin{conj}[Macdonald version of Stanley's conjecture]\label{conj Stanley-Macdonald}
Given $\lambda, \mu,\nu$ partitions, the quantity
$\langle\JJack_\lambda\JJack_\mu,\JJack_\nu\rangle_{q,t}$
    is a polynomial in the parameters $\alpha$ and $\gamma$ with integer non-negative coefficients.
\end{conj}

This conjecture has been tested for $|\nu|\leq 9$. Stanley's conjecture \cite[Conjecture 8.5]{Stanley1989} corresponds to the case $\gamma=0$ of \cref{conj Stanley-Macdonald}.




% \subsection{Reformulation with the super Nabla operator}
% The super Nabla  has been recently introduced in \cite{BergeronHaglundIraciRomero2023} defined by its action on modified Macdonald polynomials 
% $$\nabla_Y \widetilde H_\lambda[X]=\widetilde H_\lambda[X] \widetilde H_\lambda[Y],$$
% where $Y$ is a second alphabet of variables. We consider here the integral version of this operator $\nablaJ_Y$, defined as in \cref{eq OJ}.  It follows from the definitions that 
% $$\nablaJ_Y \J_\lambda[X]=t^{-n(\lambda)}\J_\lambda[X] \J_\lambda\left[\frac{Y}{1-t}\right].$$
% \begin{prop}
% Let $\pi \vdash n$.
%     $$\frac{1}{u^n}\nablaJ\nablaJ_{Y(1-t)}\cdot p_\pi[X]=\sum_{\mu,\nu\vdash n}c^\pi_{\mu,\nu}p_\mu[X] p_\nu[Y].$$
%     \begin{proof}
%         Let $Z$ be a third alphabet of variables, and let $\widetilde \nablaJ$ and $\widetilde \nablaJ_Y$ denote respectively the Nabla and the super Nabla operators acting on the space of symmetric functions in the alphabet $Z$.   
%         Using the fact that Macdonald polynomials are an orthogonal basis, one has
%         \begin{align*}
%             \frac{1}{u^n}\nablaJ\nablaJ_{Y(1-t)}\cdot p_\pi[X]
%             &=\left\langle\frac{1}{u^n}\widetilde\nablaJ\widetilde\nablaJ_{Y(1-t)}\cdot p_\pi[Z],\sum_{\lambda\vdash n}\frac{\J_\lambda[X]\J_\lambda[Z]}{\normJ_\lambda}\right\rangle_{q,t},
%         \end{align*}
%     where the scalar product is taken with respect to the alphabet Z, and the operators on the right hand-side act on symmetric functions in $Y$. Using the fact that the nabla operators are self-dual, we get 
% \begin{align*}
%    \frac{1}{u^n}\nablaJ\nablaJ_{Y(1-t)}\cdot p_\pi[X]
%    &= \left\langle\frac{1}{u^n} p_\pi[Z],\widetilde\nablaJ\widetilde\nablaJ_{Y(1-t)}\cdot \sum_{\lambda\vdash n}\frac{\J_\lambda[X]\J_\lambda[Z]}{\normJ_\lambda}\right\rangle_{q,t}\\
%    &=\left\langle\frac{1}{u^n} p_\pi[Z],(-1)^n \sum_{\lambda\vdash n}q^{n(\lambda')}t^{-2n(\lambda)}\frac{\J_\lambda[X]\J_\lambda[Z]\J_\lambda[Y]}{\normJ_\lambda}\right\rangle_{q,t}\\
% &=\left\langle p_\pi[Z],\sum_{\lambda\vdash n}q^{n(\lambda')}t^{-2n(\lambda)}\frac{\JJack_\lambda[X]\JJack_\lambda[Z]\JJack_\lambda[Y]}{\normJt_\lambda}\right\rangle_{q,t}\\
% &=\sum_{\mu,\nu\vdash n}c^\pi_{\mu,\nu}p_\mu[X] p_\nu[Y]
% \end{align*}
    
%     \end{proof}
% \end{prop}


% \subsection{Special cases in \cref{conj MJ generalized}}

% \subsubsection{Marginal sums}
% We recall the usual $q$-notation.
% We set 
% $$[n]_q:=1+q+\dots+q^{n-1} \text{ for any }  n\geq1,$$
% $$[0]_q!=1 \text{ and }[n]_q!:=[n]_q[n-1]_q\dots[1]_q,\text{ for any }  n\geq1,$$

% $$\text{and } \left[\begin{array}{c}
%      n  \\
%     k_1,k_2,\dots,k_l 
% \end{array}\right]_q:=\frac{[n]_q!}{[k_1]_q!\dots [k_l]_q!} \text{ for any } n,k\geq 0.$$
% It is well known that all these quantities are polynomials in $q$ with non-negative integer coefficients. Finally, if $a$ is a parameter then
% $$(a;q)_n:=(1-a)(1-qa)\cdots(1-q^{n-1}a) \text{ for any }n\geq 1.$$
% \begin{lem}\label{lem J specialized in 1}
% Let $\lambda$ be a partition of size $n$.
%     $$\sum_{\mu\vdash n}\left[p_\mu\right]\J_\lambda=J_\lambda[1]=\left\{\begin{array}{cc}
%          (t;q)_n&\text{ if }\lambda=[n],  \\
%          0& otherwise. 
%     \end{array}\right.$$
%     \begin{proof}
%         The first part of the equation is direct from the definitions. To obtain the second one we take $v=t$ in \cref{thm plethystic subt}. 
%     \end{proof}
% \end{lem}
% For a proof of the following lemma, see \cite[Chapter VI, Section 8, Example 1]{Macdonald1995}.
% \begin{lem}\label{lem J_m}
%     For every $m\geq1$,
%     $$\J_m[X]=(q;q)_m\cdot h_m\left[X\frac{1-t}{1-q}\right]=(q;q)_m\sum_{\pi\vdash n}\frac{1}{z_\pi}p_\pi\left[X\frac{1-t}{1-q}\right].$$
% \end{lem}

% \begin{prop}\label{prop marginal sums}
% Let $\pi,\mu\vdash m.$
%     $$\sum_{\nu\vdash m}c^\pi_{\mu,\nu}=(1-t)^{-m}\frac{1}{z_\mu}q^{\binom{m}{2}}(q;q)_m p_\mu\left[\frac{1-t}{1-q}\right].$$
%     \begin{proof}
%     We adapt here the proof given in \cite{GouldenJackson1996} for the Jack polynomials setting. Using \cref{lem J specialized in 1}, we write
%     \begin{align*}
%   \sum_{\nu\vdash m} c^\pi_{\mu,\nu}
%   &=(1-t)^{-m}[p_\mu[Y]]\left\langle p_\pi[X],\sum_{\lambda\vdash m}t^{-2n(\lambda)}q^{n(\lambda')}\frac{\J_\lambda[X]\J_\lambda[Y]\J_\lambda[1]}{\normJ_\lambda}\right\rangle_{q,t}.
%   \end{align*}
%     But we know from \cref{lem J specialized in 1} that only the term corresponding to $\lambda=[m]$ contributes to the sum. Hence,
%   \begin{align*}
%   \sum_{\nu\vdash m} c^\pi_{\mu,\nu}&=(1-t)^{-m}[p_\mu[Y]]\left\langle p_\pi[X],t^{-2n([m])}q^{n([m])}\frac{\J_{[m]}[X]\J_{[m]}[Y]\J_{[m]}[1]}{\normJ_\lambda}\right\rangle_{q,t}\\
%   &=(1-t)^{-m}[p_\mu[Y]]\left\langle p_\pi[X],q^{\binom{m}{2}}\frac{\J_{[m]}[X]\J_{[m]}[Y]\cdot(t;q)_m}{(q;q)_m(t;q)_m}\right\rangle_{q,t}
%     \end{align*}
% We conclude using \cref{lem J_m}.
%     \end{proof}
% \end{prop}



% \begin{cor}
% For any $m\geq 1$ and $\pi,\mu,\nu\vdash m$. Then,
% $$z_\mu\sum_{\nu\vdash m} c^\pi_{\mu,\nu}$$
%      is a polynomial with non negative integer coefficients in $u$ and $\alpha$.
%      \begin{proof}
%      From \cref{prop marginal sums}, we get
%      \begin{align*}
%        z_\mu\sum_{\nu\vdash m} c^\pi_{\mu,\nu}
%        &=\frac{q^{\binom{m}{2}}(q;q)_m }{(1-t)^m}\prod_{1\leq i\leq \ell(\mu)}\frac{1-t^{\mu_i}}{1-q^{\mu_i}}  \\
%        &=\frac{q^{\binom{m}{2}}(q;q)_m }{(1-q)^{\ell(\mu)}(1-t)^{m-\ell(\mu)}} \prod_{1\leq i\leq \ell(\mu)}\frac{[\mu_i]_t}{[\mu_i]_q}
%      \end{align*}
     
% %          It is clear that $q^{\binom{m}{2}}$ is a polynomial in $u$ and $\alpha$ with integer non-negative coefficients.
% %          Moreover, 
% %          \begin{align*}
% %              (-1)^m(q;q)_m p_\mu\left[\frac{1-t}{1-q}\right]
% %              =(-1)^m(q;q)_m .
% % \end{align*}   

%           But $(q;q)_m=[m]_q!(1-q)^m.$ Hence,
%      \begin{align*}
%        z_\mu\sum_{\nu\vdash m} c^\pi_{\mu,\nu}
%        &=q^{\binom{m}{2}}[m]_q! \frac{(1-q)^{m-\ell(\mu)}}{(1-t)^{m-\ell(\mu)}} \prod_{1\leq i\leq \ell(\mu)}\frac{[\mu_i]_t}{[\mu_i]_q}\\
%        &=q^{\binom{m}{2}}[m]_q! \alpha^{m-\ell(\mu)}\prod_{1\leq i\leq \ell(\mu)}\frac{[\mu_i]_t}{[\mu_i]_q}.
%      \end{align*}


%          Note that 
%          $[m]!_q\prod_{1\leq i \leq\ell(m)}\frac{1}{[\mu_i]_q}$ is divisible by the binomial 
%          $\left[\begin{array}{c}
%               m  \\
%               \mu_1,\mu_2,\dots,\mu_{\ell(\mu)}
%          \end{array}\right]_q.$
%          Hence, $[m]!_q\prod_{1\leq i \leq\ell(m)}\frac{1}{[\mu_i]_q}$ is a positive 
%  polynomial in $q$, and by consequence in $u$ and $\alpha$. This finishes the proof.
%      \end{proof}
% \end{cor}



% \subsubsection{The case $q=t$}


% When $q=t$ (equivalently $\alpha=1)$, Macdonald polynomials are Schur functions up to a scalar factor, (see \cite[Chapter VI, Remark 8.4]{Macdonald1995})

% $$\mathfrak J^{(\alpha=1,u)}_\lambda=H_\lambda(t) s_\lambda,$$

% where 
% $$H_\lambda(t)=\prod_{\Box\in \lambda} [a_\lambda(\Box)+\ell_\lambda(\Box)+1]_t $$
% is a $t$-deformation of the hook product and where $t=u+1$ as usual..
% Moreover, it is direct from \cref{eq j norm} and \cref{eq norm j t} that
% $$\widetilde j^{(\alpha=1,u)}_\lambda=H_\lambda(t)^2.$$

% We recall that the expansion of Schur functions in the power-sum basis are given by the irreducible characters of the symmetric group $\chi^\lambda$, see e.g \cite[Chapter I]{Macdonald1995}
% $$s_\lambda=\sum_{\mu\vdash |\lambda|}\frac{\chi^\lambda(\mu)}{z_\mu}p_\mu.$$



% Hence we obtain the following formula for the coefficient of the generalized Matchings-Jack conjecture.
% \begin{prop}\label{prop c alpha=1}
%     For any partitions $\pi,\mu$ and $\nu$, one has
%     $$c^\pi_{\mu,\nu}(\alpha=1,u)=\frac{1}{z_\mu z_\nu} \sum_{\lambda\vdash |\pi|} t^{n(\lambda')-2n(\lambda)}H_\lambda(t)\chi^\lambda(\pi)\chi^\lambda(\mu)\chi^\lambda(\nu).$$
% \end{prop}

% We deduce the integrality of the coefficients in the parameter $u$.
% \begin{cor}
%  For any partitions $\pi,\mu$ and $\nu$ of size $m$, the normalized coefficient \\
%     $(1+u)^{m(m-1)}z_\mu z_\nu c^\pi_{\mu,\nu}(\alpha=1,u)$ is a polynomial with integer coefficients in $t$.
%     \begin{proof}
%         We use \cref{prop c alpha=1} and the fact that $n(\lambda')-2n(\lambda)$ is minimal when $\lambda=[1^m]$ and the corresponding minimum is $m(m-1).$
%     \end{proof}
% \end{cor}


% \acknowledgements{}

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