Georgian Mathematical Journal Volume 10 (2003), Number 3, 481–494

LAGUERRE-TYPE EXPONENTIALS, AND THE RELEVANT L-CIRCULAR AND L-HYPERBOLIC FUNCTIONS

GIUSEPPE DATTOLI AND PAOLO E. RICCI

Dedicated to the memory of Prof. V. D. Kupradze

Abstract. Some classes of entire functions which are eigenfunctions of generalizations of the Laguerre derivative operator are considered.

Since this property is an analog of the one characterizing the exponential function, we refer to such functions as Laguerre-type exponentials, or shortly L-exponentials. The definition of L-circular and L-hyperbolic functions easily follows.

Applications in the framework of generalized evolution problems are also mentioned.

2000 Mathematics Subject Classification: 33C45, 30D05, 33B10. Key words and phrases: Laguerre derivative, Tricomi functions, Laguerretype exponential functions, L-circular and L-hyperbolic functions.

1. Introduction

The exponential function e^{ax} is an eigenfunction of the derivative operator since

De^{ax} = ae^{ax},  (1.1)

where D := d/dx, and a denotes a real or complex arbitrary constant.

Another interesting differential operator exists in literature, namely the Laguerre derivative denoted in the following by D_{L} and defined by

D_{L} := DxD = \frac{d}{dx}x\frac{d}{dx}.  (1.2)

In the preceding articles, we have shown the role of the Laguerre derivative in the framework of the so-called monomiality principle and its application to the multidimensional Hermite (Hermite–Kampé de Fériet or Gould–Hopper polynomials, see [1], [2],[3]) or Laguerre polynomials [4], [5], [6].

It is easily seen, by induction, that the Laguerre derivative verifies

(DxD)^{n} = D^{n}x^{n}D^{n}.  (1.3)

Furthermore, introducing the Tricomi function of order zero or the relevant Bessel function

C_{0}(x) := ∑_{k=0}^{∞}(−1)^{k} \frac{x^{k}}{(k!)^{2}} = J_{0}(2\sqrt{x}),  (1.4)

we obtain

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

Theorem 1.1. The function

e_{1}(ax) := C_{0}(−ax)  (1.5)

is an eigenfunction of the Laguerre derivative operator, i.e.

D_{L} e_{1}(ax) = ae_{1}(ax).  (1.6)

Proof. Note that

D_{L} = D + xD^{2},  (1.7)

and consequently

D_{L} e_{1}(ax) = (D + xD^{2}) ∑_{k=0}^{∞} a^{k} \frac{x^{k}}{(k!)^{2}}  (1.8)

= ∑_{k=1}^{∞} (k + k(k − 1)) a^{k} \frac{x^{k−1}}{(k!)^{2}} = ∑_{k=1}^{∞} k^{2} a^{k} \frac{x^{k−1}}{(k!)^{2}}

= a ∑_{k=0}^{∞} a^{k} \frac{x^{k}}{(k!)^{2}} = ae_{1}(ax).

□

Note that the preceding conclusion depends on the coefficients of the combination expressing the Laguerre derivative D_{L} in terms of D, and xD, so that it turns out the identity (k + k(k − 1)) = k^{2}.

In the following we will show that the above technique can be iterated, producing Laguerre classes of exponential-type functions called L-exponentials and the relevant L-circular, L-hyperbolic, L-Gaussian functions.

Further extensions are given in the concluding section, and applications to the solution of generalized evolution problems is touched on.

2. Generalizations of the Laguerre Derivative and L-Exponential Functions

In this section, we generalize the Laguerre derivative and define the relevant L-exponential functions.

We start by considering the operator

D_{2L} := DxDxD = D (xD + x^{2}D^{2}) = D + 3xD^{2} + x^{2}D^{3},  (2.1)

and the function

e_{2}(x) := ∑_{k=0}^{∞} \frac{x^{k}}{(k!)^{3}} .  (2.2)

The following theorem holds true:

Theorem 2.1. The function e_{2}(ax) is an eigenfunction of the operator D_{2L}, i.e.

D_{2L} e_{2}(ax) = ae_{2}(ax)  (2.3)

Proof. Note that

D_{2L} e_{2}(ax) = (D + 3xD^{2} + x^{2}D^{3}) ∑_{k=0}^{∞} a^{k} \frac{x^{k}}{(k!)^{3}}

= ∑_{k=1}^{∞} (k + 3k(k − 1) + k(k − 1)(k − 2)) a^{k} \frac{x^{k−1}}{(k!)^{3}} = ∑_{k=1}^{∞} k^{3} a^{k} \frac{x^{k−1}}{(k!)^{3}}

= a ∑_{k=0}^{∞} a^{k} \frac{x^{k}}{(k!)^{3}} = ae_{2}(ax).  (2.4)

This completes the proof. □

Even in this case, the conclusion depends on the identity k + 3k(k − 1) + k(k − 1)(k − 2) = k^{3} so that it can be recognized that the coefficients of the combination expressing the 2L-derivative D_{2L} in terms of D, xD^{2}, and x^{2}D^{3} are the Stirling numbers of second kind, S(3, 1), S(3, 2), S(3, 3), (see [7], and [8], p. 835 for an extended table).

We can consequently extend the above results as follows.

Considering the operator

D_{(n−1)L} := Dx ⋯ DxDxD = D (xD + x^{2}D^{2} + ⋯ + x^{n−1}D^{n−1})

= S(n, 1)D + S(n, 2)xD^{2} + ⋯ + S(n, n)x^{n−1}D^{n}  (2.5)

and the function

e_{n}(x) := ∑_{k=0}^{∞} \frac{x^{k}}{(k!)^{n+1}} ,  (2.6)

we can state the following theorem:

Theorem 2.2. The function e_{n}(ax) is an eigenfunction of the operator D_{nL}, i.e.

D_{nL} e_{n}(ax) = ae_{n}(ax).  (2.7)

Proof. Proceeding by induction, i.e. assuming equation (2.5) to be true, and recalling the above remarks, it is sufficient to prove that the coefficients of the combination expressing the nL-derivative D_{nL} in terms of D, xD^{2}, …, and x^{n}D^{n+1}, verify the same induction property as the Stirling numbers of second kind, namely (see [7]):

S(n + 1, h) = S(n, h − 1) + hS(n, h).  (2.8)

This is clearly true since, considering in the equation

D_{nL} := D (S(n, 1)xD + S(n, 2)x^{2}D^{2} + ⋯ + S(n, n)x^{n}D^{n}) ,  (2.9)

the general terms, i.e.

D (S(n, h − 1)x^{h−1}D^{h−1} + S(n, h)x^{h}D^{h}) ,  (2.10)

we find

(h − 1)S(n, h − 1)x^{h−2}D^{h−1}+ S(n, h − 1)x^{h−1}D^{h}
+ hS(n, h)x^{h−1}D^{h} + S(n, h)x^{h}D^{h+1}

so that the coefficient of x^{h−1}D^{h} is given by S(n, h − 1) + hS(n, h) and must coincide with S(n + 1, h) and then recursion (2.8) holds true. □

Remark 2.1. The above results show that, for every positive integer n, we can define a Laguerre-exponential function, satisfying an eigenfunction property, which is an analog of the elementary property (1.1) of the exponential. This function, denoted by e_{n}(x) := ∑_{k=0}^{∞} \frac{x^{k}}{(k!)^{n+1}} , reduces to the exponential function when n = 0 so that we can set by definition:

e_{0}(x) := e^{x}, D_{0L} := D.

Obviously, D_{1L} := D_{L}.

For this reason we will refer to such functions as L-exponential functions or, shortly, L-exponentials.

Examples of the above functions are given in Fig. 1.

Fig. 1

The above definitions allow us to introduce an extension of the Gauss function e^{−x^{2}} by means of

Definition 2.1. The L-Gaussian functions are given by the entire even functions

e_{1}(−x^{2}) = ∑_{k=0}^{∞}(−1)^{k} \frac{x^{2k}}{(k!)^{2}} ,  (2.11)

LAGUERRE-TYPE EXPONENTIALS 485

and, in general,

e_{n}(−x^{2}) = ∑_{k=0}^{∞}(−1)^{k} \frac{x^{2k}}{(k!)^{n+1}} .  (2.12)

Examples of the above functions are given in Fig. 2.

Fig. 2

3. L-Circular and L-Hyperbolic Functions

Write the 1L-exponential as follows:

e_{1}(ix) = ∑_{k=0}^{∞}(i)^{k} \frac{x^{k}}{(k!)^{2}}  (3.1)

= ∑_{h=0}^{∞}(−1)^{h} \frac{x^{2h}}{((2h)!)^{2}} + i ∑_{h=0}^{∞}(−1)^{h} \frac{x^{2h+1}}{((2h + 1)!)^{2}} .

Then we can formulate the definition

Definition 3.1. The 1L-circular functions are given by

cos_{1}(x) := ℜ (e_{1}(ix)) = E (e_{1}(ix)) = ∑_{h=0}^{∞}(−1)^{h} \frac{x^{2h}}{((2h)!)^{2}} ,  (3.2)
sin_{1}(x) := ℑ (e_{1}(ix)) = \frac{1}{i}O (e_{1}(ix)) = ∑_{h=0}^{∞}(−1)^{h} \frac{x^{2h+1}}{((2h + 1)!)^{2}} ,  (3.3)

where E(f) or O(f) are the even part or the odd part of f.

Obviously,

e_{1}(ix) = cos_{1}(x) + i sin_{1}(x),  e_{1}(−ix) = cos_{1}(x) − i sin_{1}(x),  (3.4)

so that the Euler-type formulas

cos_{1}(x) = \frac{e_{1}(ix) + e_{1}(−ix)}{2},  sin_{1}(x) = \frac{e_{1}(ix) − e_{1}(−ix)}{2i}  (3.5)

hold true.

Recalling equation (1.3), we find the following result:

Theorem 3.1. The 1L-circular functions (3.2), (3.3) are solutions of the differential equation

D_{L}^{2} v + v = (D^{2}x^{2}D^{2}) v + v = 0.  (3.6)

Proof. Equation (3.6) is an easy consequence of Theorem 1.1, since

D_{L}^{2} e_{1}(ix) = D_{L} (ie_{1}(ix)) = −e_{1}(ix),  (3.7)

so that

D_{L}^{2} (cos_{1}(x) + i sin_{1}(x)) = − cos_{1}(x) − i sin_{1}(x).  (3.8)

Then separating the real from the imaginary part in the above equation, the proclaimed result follows. □

Write now the nL-exponential in the form

e_{n}(ix) = ∑_{k=0}^{∞}(i)^{k} \frac{x^{k}}{(k!)^{n+1}}  (3.9)

= ∑_{h=0}^{∞}(−1)^{h} \frac{x^{2h}}{((2h)!)^{n+1}} + i ∑_{h=0}^{∞}(−1)^{h} \frac{x^{2h+1}}{((2h + 1)!)^{n+1}} .

Then we can formulate the definition

Definition 3.2. The nL-circular functions are given by

cos_{n}(x) := ℜ (e_{n}(ix)) = E (e_{n}(ix)) = ∑_{h=0}^{∞}(−1)^{h} \frac{x^{2h}}{((2h)!)^{n+1}} ,  (3.10)

sin_{n}(x) := ℑ (e_{n}(ix)) = \frac{1}{i}O (e_{n}(ix)) = ∑_{h=0}^{∞}(−1)^{h} \frac{x^{2h+1}}{((2h + 1)!)^{n+1}} . (3.11)

Obviously,

e_{n}(ix) = cos_{n}(x) + i sin_{n}(x),  e_{n}(−ix) = cos_{n}(x) − i sin_{n}(x) (3.12)

so that we find again the Euler-type formulas:

cos_{n}(x) = \frac{e_{n}(ix) + e_{n}(−ix)}{2},  sin_{n}(x) = \frac{e_{n}(ix) − e_{n}(−ix)}{2i}.  (3.13)

The same method used in Theorem 3.1 yields the more general result

LAGUERRE-TYPE EXPONENTIALS 487

Theorem 3.2. The nL-circular functions (3.10), (3.11) are solutions of the differential equation

D_{nL}^{2} v + v = 0.

and satisfy the conditions

cos_{n}(0) = 1, sin_{n}(0) = 0.

Then, using the same proof as in Theorem 2.2, we obtain Theorem 3.3. The nL-circular functions satisfy

D_{nL} cos_{n}(x) = − sin_{n}(x), D_{nL} sin_{n}(x) = cos_{n}(x). (3.14)

Examples of the above functions are given in Fig. 3.

Fig. 3

We consider now the L-hyperbolic functions. Definition 3.3. The nL-circular functions are given by

cosh_{n}(x) := ℜ (e_{n}(x)) = E (e_{n}(x)) = ∑_{h=0}^{∞} \frac{x^{2h}}{((2h)!)^{n+1}} ,  (3.15)
sinh_{n}(x) := ℑ (e_{n}(x)) = O (e_{n}(x)) = ∑_{h=0}^{∞} \frac{x^{2h+1}}{((2h + 1)!)^{n+1}} .  (3.16)

Obviously,

e_{n}(x) = cos_{n}(x) + sin_{n}(x),  e_{n}(−x) = cos_{n}(x) − sin_{n}(x)  (3.17)

so that we find again the Euler-type formulas:

cos_{n}(x) = \frac{e_{n}(x) + e_{n}(−x)}{2},  sin_{n}(x) = \frac{e_{n}(x) − e_{n}(ix)}{2}.  (3.18)

Furthermore, we have

Theorem 3.4. The nL-hyperbolic functions (3.15), (3.16) are solutions of the differential equation

D_{nL}^{2} w − w = 0,  (3.19)

and satisfy the conditions

cosh_{n}(0) = 1, sinh_{n}(0) = 0.

Theorem 3.5. The nL-hyperbolic functions satisfy

D_{nL} cosh_{n}(x) = sinh_{n}(x),  D_{nL} sinh_{n}(x) = cosh_{n}(x).  (3.20)

We omit graphics of the L-hyperbolic functions, whose shapes are similar to those of ordinary hyperbolic functions.

3.1. Pseudo-L-circular and pseudo-L-hyperbolic functions. According to the results stated in [9], [10] it is easy to find the functions decomposing the L-exponentials with respect to the cyclic group of a given (integral) order r.

The relevant functions are denoted by [10]

Π_{[r,k]}e_{n}(x) := ∑_{h=0}^{∞} \frac{x^{rh+k}}{((rh + k)!)^{n+1}} ,  (k = 0, 1, …, r − 1),  (3.21)

and called pseudo-nL-hyperbolic functions of order r, while the pseudo-nL-circular functions of order r are defined by the position

σ_{0}^{−k}Π_{[r,k]}e_{n}(σ_{0} x) = ∑_{h=0}^{∞} (−1)^{h} \frac{x^{rh+k}}{((rh + k)!)^{n+1}} , (k = 0, 1, …, r − 1), (3.22)

where σ_{0} denotes any complex r-th root of −1.

The following statement is valid.

Theorem 3.6. The pseudo-nL-circular functions of order r, (3.22) are solutions of the differential equation

D_{nL}^{r} w + w = 0.  (3.23)

The pseudo-nL-hyperbolic functions of order r, (3.21) are solutions of the differential equation

D_{nL}^{r} w − w = 0.  (3.24)

Remark 3.1. Note that equation (1.3) can be easily generalized as follows:

D_{nL}^{r} = \underbrace{(DxDx ⋯ DxD)^{r}}_{(n+1) Derivatives} = D^{r}x^{r}D^{r}x^{r} ⋯ D^{r}x^{r}D^{r}.

4. Consideration of the Above Results, and Further Properties

We prove the following

Theorem 4.1. The only polynomial solutions of the differential equation D_{nL} p = 0 are given by constants.

Proof. Note that D_{L} p(x) = D (xD p(x)) = 0 implies xD p(x) = constant and consequently p(x) = c_{1} + c_{2} log x, c_{h} = constants (h = 1, 2) so that the unique polynomial solution is a constant. Furthermore, D_{2L} p(x) = D_{L}xD p(x) = 0 implies xD p(x) = c_{1} + c_{2} log x so that D p(x) = c_{1} + c_{2} log x + c_{3} ∫ \frac{log x}{x}dx, and the unique polynomial solution is again a constant. Proceeding by induction, we find that D_{nL} p(x) = D_{(n−1)L} xD p(x) = 0 implies that xD p(x) is a linear combination of a set of functions obtained by adding to the preceding set the primitive of each function divided by x. In any case, the only polynomial solution derived in such a way is always a constant. □

It was previously noted (see, e.g., [5]) that, considering in the space of polynomial functions the correspondence

D → D_{L},  x· → D_{x}^{−1},  (4.1)

where

D_{x}^{−n}(1) := \frac{x^{n}}{n!} ,  (4.2)

a differential isomorphism is determined.

In such an isomorphism, the exponential function is transformed into the function e_{1}(x), the Hermite polynomials H_{n}^{(1)}(x, y) := (x − y)^{n} become the Laguerre polynomials L_{n}(x, y) := n! ∑_{r=0}^{n} \frac{(−1)^{r}y^{n−r}x^{r}}{(n−r)!(r!)^{2}} and the monomiality principle ensures that all the relations proved in the initial polynomial space still hold after performing the substitutions stated in equations (4.1).

Note that an iterative application of equations (4.1) to the exponential function gives subsequently functions e_{1}(x), e_{2}(x), e_{3}(x), …, and so on.

Accordingly, the derivative operator is transformed into

D_{L} = DxD,  D_{2L} = D_{L}D_{x}^{−1}D_{L},  D_{3L} = D_{L}D_{x}^{−1}D_{L}D_{x}^{−1}D_{L}, (4.3)

and so on.

Thus we can conclude that the L-exponentials (and the relevant L-circular and L-hyperbolic functions) are determined by an iterative application of the above mentioned-differential isomorphism.

This gives an explanation of the above results, since the subsequent derivatives are transformed into the powers of the Laguerre derivative, determining e.g., the validity of Theorems 3.1, 3.2, 3.3, 3.4, 3.5, 3.6.

5. Applications

5.1. Diffusion equations.

Theorem 5.1. For any fixed integral n, consider the problem

\{ \begin{matrix}D_{nL} S(x, t) = \frac{∂}{∂t} S(x, t), in the half − plane t > 0, \\ S(0, t) = s(t), \end{matrix}  (5.1)

with analytic boundary condition s(t).

The operational solution of equation (5.1) is given by

S(x, t) = e_{n} (x\frac{∂}{∂t}) s(t) = ∑_{k=0}^{∞} \frac{x^{k}}{(k!)^{n+1}} \frac{d^{k}}{dt^{k}} s(t).  (5.2)

Representing s(t) = ∑_{k=0}^{∞} a_{k}t^{k}, from equation (5.2), we find, in particular:

S(x, 0) = ∑_{k=0}^{∞} a_{k} \frac{x^{k}}{(k!)^{n}} .  (5.2_{0})

Note that the operational solution becomes an effective solution whenever the series in equation (5.2) is convergent. The validity of this condition depends on the growth of the coefficients a_{k} of the boundary data s(t), but it is usually satisfied in physical problems.

More generally, the following results holds.

Theorem 5.2. Let Ω̂_{x} be a differential operator with respect to the variable x, and denote by ψ(x) an eigenfunction of Ω̂_{x} such that

Ω̂_{x} ψ(ax) = a ψ(ax),  ψ(0) = 1,  (5.3)

then the evolution problem

\{ \begin{matrix}Ω̂_{x} S(x, t) = \frac{∂}{∂t} S(x, t), in the half − plane t > 0, \\ S(0, t) = s(t), \end{matrix}  (5.4)

with analytic boundary condition s(t) admits an operational solution

S(x, t) = ψ (x\frac{∂}{∂t}) s(t).  (5.5)

Proof. The eigenfunction property of ψ implies:

Ω̂_{x} S(x, t) = Ω̂_{x} ψ (x\frac{∂}{∂t}) s(t) = \frac{∂}{∂t} ψ (x\frac{∂}{∂t}) s(t) = \frac{∂}{∂t} S(x, t),

since \frac{∂}{∂t} commutes with ψ (x\frac{∂}{∂t}).

Furthermore, as a consequence of the hypothesis ψ(0) = 1, the boundary condition, for x = 0, is trivially satisfied. □

5.2. L-hyperbolic-type problems.

Theorem 5.3. Let Ω̂_{x} be a 2nd order differential operator with respect to the variable x, D_{nL} := (D_{nL})_{t} the nL-derivative with respect to the t variable, and denote by ψ(t) and χ(t) two functions such that

D_{nL} ψ(t) = χ(t),  D_{nL} χ(t) = ψ(t),  (5.6)
ψ(0) = 1,  χ(0) = 0;  (5.7)

then the abstract L-hyperbolic-type problem

\{ \begin{matrix}Ω̂_{x}^{2} S(x, t) = D_{nL}^{2} S(x, t), in the half − plane t > 0, \\ S(x, 0) = q(x), \\ D_{nL} S(x, t)|_{t=0} = v(x) \end{matrix}  (5.8)

with analytic initial condition q(x), v(x), admits an operational solution

S(x, t) = ψ (tΩ̂_{x}) q(x) + χ (tΩ̂_{x}) w(x),  (5.9)

where w(x) := Ω̂_{x}^{−1}v(x).

Proof. Since Ω̂_{x} commutes with ψ (tΩ̂_{x}) and χ (tΩ̂_{x}), by using conditions (5.6), and the chain rule with respect to the Laguerre derivative (see Remark 5.1 below), we can write

D_{nL} S(x, t) = Ω̂_{x} χ (tΩ̂_{x}) q(x) + Ω̂_{x} ψ (tΩ̂_{x}) w(x),
D_{nL}^{2} S(x, t) = Ω̂_{x}^{2} ψ (tΩ̂_{x}) q(x) + Ω̂_{x}^{2} χ (tΩ̂_{x}) w(x) = Ω̂_{x}^{2} S(x, t).

Furthermore, for the initial conditions, by using equations (5.7) and the definition of w we find

S(x, 0) = ψ(0)q(x) + χ(0)w(x) = q(x),
D_{nL} S(x, t)|_{t=0} = Ω̂_{x} χ(0)q(x) + Ω̂_{x} ψ(0)w(x) = Ω̂_{x} w(x) = v(x).

Note that conditions (5.6)-(5.7) are satisfied, fixing an arbitrary integral n and assuming

ψ(x) := cosh_{nL}(x), χ(x) := sinh_{nL}(x).
□

5.3. L-elliptic-type problems.

Theorem 5.4. Let Ω̂_{x} be a 2nd order differential operator with respect to the variable x, D_{nL} := (D_{nL})_{y} the nL-derivative with respect to the variable y, and denote by φ(y) and τ(y) two functions such that

D_{nL} φ(y) = −τ(y),  D_{nL} τ(y) = φ(y),  (5.10)
φ(0) = 1,  τ(0) = 0;  (5.11)

then the abstract L-elliptic-type problem

\{ \begin{matrix}Ω̂_{x}^{2} S(x, y) + D_{nL}^{2} S(x, y) = 0, in the half − plane y > 0, \\ S(x, 0) = q(x), \end{matrix}  (5.12)

with analytic boundary condition q(x), admits the operational solution

S(x, y) = φ (yΩ̂_{x}) q(x).  (5.13)

Proof. Since Ω̂_{x} commutes with φ (yΩ̂_{x}), by using conditions (5.10) we can write

D_{nL} S(x, y) = −Ω̂_{x} τ (yΩ̂_{x}) q(x),
D_{nL}^{2} S(x, y) = −Ω̂_{x}^{2} φ (yΩ̂_{x}) q(x) = −Ω̂_{x}^{2} S(x, y).

Furthermore, for the boundary conditions, by using equations (5.11) we find:

S(x, 0) = φ(0)q(x) = q(x).

□

Note that conditions (5.10)–(5.11) are satisfied, fixing an arbitrary integral n and assuming

φ(x) := cos_{nL}(x), τ(x) := sin_{nL}(x).

Remark 5.1. Note that for the Laguerre derivative, the chain rule

\frac{d}{dt} = \frac{d}{dx} \frac{dx}{dt}

becomes

\frac{d}{dt}t\frac{d}{dt} = \frac{d}{dx} \frac{d}{dt}t\frac{dx}{dt},  ⇔  (D_{L})_{t} = \frac{d}{dx} (D_{L})_{t} x

and, in general,

(D_{nL})_{t} = \frac{d}{dx} (D_{nL})_{t} x.

6. Concluding Remarks

The concepts we have so far developed can be further generalized. Limiting ourselves to the case of second order operators, we note that the function

e_{1,m}(x) := ∑_{k=0}^{∞} \frac{x^{k}}{k!(k + m)!}  (6.1)

is such that e_{1,m}(ax) is an eigenfunction of the operator

Ω̂_{m} := D_{L} + mD = \frac{d}{dx}x\frac{d}{dx} + m\frac{d}{dx}  (6.2)

and, indeed,

Ω̂_{m} e_{1,m}(ax) = a e_{1,m}(ax).  (6.3)

It is therefore clear that we can introduce more general forms of L-circular functions as follows:

cos_{1,m}(x) := \frac{e_{1,m}(ix) + e_{1,m}(−ix)}{2},  (6.4)

sin_{1,m}(x) := \frac{e_{1,m}(ix) − e_{1,m}(−ix)}{2i},  (6.5)

and analogous forms for the hyperbolic case.

It is also evident that the above functions satisfy the differential equation

Ω̂_{m}^{2} v(x) = −v(x).  (6.6)

It is furthermore interesting to note that e_{1,m}(x) can be viewed as the n-th order Tricomi function defined by the generating function

∑_{m=−∞}^{+∞} t^{m} e_{1,m}(x) = exp (t + \frac{x}{t})  (6.7)

and that the formalism we have just envisaged can be exploited in a more general framework, as, e.g., that associated with partial differential equations. Namely, the solution of the evolution problem

\{ \begin{matrix}Ω̂_{m}F (x, t) = \frac{∂}{∂t}F (x, t) \\ F (0, t) = g(t) \end{matrix}  (6.8)

can be solved in the form

F (x, t) = m! e_{1,m} (x\frac{∂}{∂t}) g(t)  (6.9)

which, once expanded in a series, yields

F (x, t) = m! ∑_{k=0}^{∞} \frac{x^{k}}{k!(k + m)!}g^{(k)}(t).  (6.10)

Different solutions, expressed in terms of integral transforms, will be discussed elsewhere.

Before concluding, let us note that more in general we can introduce the functions

e_{r,m_{1},...,m_{r}}(x) := ∑_{k=0}^{∞} \frac{x^{k}}{k!(k + m_{1})! … (k + m_{r})!}  (6.11)

which are essentially the multi-index Bessel functions discussed in ref. [11]. Their role in the definition of more general forms of generalized circular and hyperbolic functions will be discussed elsewhere.

References

1. P. Appell and J. Kampé de Fériet, Fonctions Hypergéométriques et Hypersphériques. Polynômes d’Hermite, Gauthier-Villars, Paris, 1926.

2. H. W. Gould and A. T. Hopper, Operational formulas connected with two generalizations of Hermite polynomials. Duke Math. J. 29(1962), 51–62.

3. H. M. Srivastava and H. L. Manocha,A treatise on generating functions. Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1984.

4. G. Dattoli, Hermite–Bessel and Laguerre–Bessel functions: a by-product of the monomiality principle. Advanced special functions and applications (Melfi, 1999), 147–164, Proc. Melfi Sch. Adv. Top. Math. Phys., 1, Aracne, Rome, 2000.

5. P. E. Ricci, Tecniche operatoriali e Funzioni speciali. Preprint, 2001.

6. G. Dattoli, P. E. Ricci, and C. Cesarano, On a class of polynomials generalizing the Laguerre family. J. Comput. Anal. Appl. (to appear).

7. J. Riordan, An introduction to combinatorial analysis. Wiley Publications in Mathematical Statistics John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958.

8. M. Abramowitz and I. A. Stegun, Handbook of mathematical functions, with formulas, graphs, and mathematical tables. National Bureau of Standards Applied Mathematics Series, 55. U.S. Government Printing Office, Washington, D.C., 1965.

9. P. E. Ricci, Le Funzioni Pseudo-Iperboliche e Pseudo-trigonometriche. Pubbl. Ist. Matematica Applicata 12(1978), 37–49.

10. Y. Ben Cheikh, Decomposition of some complex functions with respect to the cyclic group of order n. Appl. Math. Inform. 4(1999), 30–53.

11. G. Dattoli, P. E. Ricci, and P. Pacciani, Comments on the theory of Bessel functions with more than one index. (Submitted)

(Received 25.12.2002)

Authors’ addresses:

Giuseppe Dattoli Unità Tecnico Scientifica Tecnologie Fisiche Avanzate ENEA – Centro Ricerche Frascati – C.P. 65 Via E. Fermi, 45 00044 - Frascati - Roma Italia E-mail: dattoli@frascati.enea.it

Paolo E. Ricci Dipartimento di Matematica “Guido Castelnuovo” Università degli Studi di Roma “La Sapienza” P.le A. Moro, 2 00185 – Roma Italia E-mail: riccip@uniroma1.it