Ramanujan Cubic Polynomials of the Second Kind

Roman Wituła Institute of Mathematics Silesian University of Technology Kaszubska 23 Gliwice 44-100 Poland

                 roman.witula@polsl.pl

Abstract

We present generalizations of some identities discussed earlier by Shevelev. Moreover, we introduce Ramanujan cubic polynomials of the second kind (RCP2). This new type of cubic polynomial is closely related to the Ramanujan cubic polynomials (RCP) defined by Shevelev. We also give many fundamental properties of RCP2’s.

1 Shevelev type identities

V. Shevelev [2] gave a trigonometric equality of the form

\sqrt{\begin{matrix}cos & \frac{2π}{5} \\ cos & \frac{π}{5} \end{matrix}} + \sqrt{\begin{matrix}cos & \frac{π}{5} \\ cos & \frac{2π}{5} \end{matrix}} = \sqrt{5}.  (1)

The theorem, given below, shows that (1) is a special case of a large class of identities for Fibonacci numbers F_{n}: Theorem 1. We have

\sqrt[r]{\frac{F_{n−1}φ^{r−1}}{φ^{n−1} − F_{n}}} + \sqrt[r]{\frac{φ^{n−1} − F_{n}}{F_{n−1}φ^{r−1}}} = \sqrt{5},  n, r ∈ ℕ,  (2)

where φ denotes the golden ratio (φ = \frac{1+\sqrt{5}}{2}).

Proof. We note that (2) is a consequence of the following identities

φ + φ^{−1} = \sqrt{5}

and

φ^{n} = F_{n}φ + F_{n−1}

(which is proved by a simple induction) or, equivalently,

φ^{r} = \frac{F_{n−1}φ^{r−1}}{φ^{n−1} − F_{n}} .

In the next theorem we present identities (2) for the general Fibonacci sequences

\{ \begin{matrix}F_{0}^{∗} = 0, F_{1}^{∗} = 1, \\ F_{n+1}^{∗} = λ_{1}F_{n}^{∗} + λ_{2}F_{n−1}^{∗}, n ∈ ℕ, \end{matrix}  (3)

where λ_{1}, λ_{2} ∈ ℂ, λ_{1}^{2} + 4λ_{2} ≠ 0, λ_{2} ≠ 0.

Let x_{1}, x_{2} be two roots of the characteristic equation

x^{2} − λ_{1}x − λ_{2} = 0.

We note that x_{1} ≠ x_{2}. Then we have Theorem 2. The following identities hold:

x_{l}^{n} = F_{n}^{∗}x_{l} + λ_{2}F_{n−1}^{∗},  (4)
\frac{λ_{2}F_{n−1}^{∗}}{x_{l}^{n−1} − F_{n}^{∗}} − \frac{x_{l}^{n−1} − F_{n}^{∗}}{F_{n−1}^{∗}} = λ_{1},  (5)
\frac{x_{l}^{n} − λ_{2}F_{n−1}^{∗}}{\sqrt{λ_{2}}F_{n}^{∗}} − \frac{\sqrt{λ_{2}}F_{n}^{∗}}{x_{l}^{n} − λ_{2}F_{n−1}^{∗}} = \frac{λ_{1}}{\sqrt{λ_{2}}} ,  (6)
(\frac{λ_{2}F_{n−1}^{∗}}{x_{1}} + F_{n}^{∗})^{\frac{k}{n−1}} + (\frac{λ_{2}F_{n−1}^{∗}}{x_{2}} + F_{n}^{∗})^{\frac{k}{n−1}} = λ_{1}F_{k}^{∗} + 2λ_{2}F_{k−1}^{∗},  (7)

for any l = 1, 2 and k, n ∈ ℕ. Proof. (4). Equality (4) can be proven by induction with respect to n ∈ ℕ. (5). From (4) we get

x_{l} = \frac{λ_{2}F_{n−1}^{∗}}{x_{l}^{n−1} − F_{n}^{∗}} .

Next, we note that

x_{l}^{2} − λ_{1}x_{l} − λ_{2} = 0 ⇔ x_{l} − λ_{2}x_{l}^{−1} = λ_{1}.  (8)

(6). From (4) we have

x_{l} = \frac{x_{l}^{n} − λ_{2}F_{n−1}^{∗}}{F_{n}^{∗}} .

Hence, by (8) we obtain (6).

(7). From (4) we receive (l = 1, 2):

x_{l}^{n−1} = \frac{λ_{2}F_{n−1}^{∗}}{x_{l}} + F_{n}^{∗} ⇒ x_{l} = \sqrt[n−1]{\frac{λ_{2}F_{n−1}^{∗}}{x_{l}} + F_{n}^{∗}}

(the last one holds for the respective value of (n − 1)-th root of the number \frac{λ_{2}F_{n−1}^{∗}}{x_{l}} + F_{n}^{∗}), which implies the identity

(\frac{λ_{2}F_{n−1}^{∗}}{x_{1}} + F_{n}^{∗})^{\frac{k}{n−1}} + (\frac{λ_{2}F_{n−1}^{∗}}{x_{2}} + F_{n}^{∗})^{\frac{k}{n−1}} = x_{1}^{k} + x_{2}^{k} {(4) \atop =} λ_{1}F_{k}^{∗} + 2λ_{2}F_{k−1}^{∗}.

Corollary 3. For any k, n ∈ ℕ the following identity holds:

(F_{n+1} + \frac{F_{n}}{φ} )^{k/n} + (F_{n+1} − \frac{F_{n}}{φ − 1})^{k/n} =

= (F_{n−1} + φ F_{n})^{k/n} + (F_{n+2} − φ^{2} F_{n})^{k/n} = F_{k} + 2 F_{k−1} = L_{k}, (9)

where L_{k} denotes the k-th Lucas number.

Remark 4. Identities, similar to those discussed in the previous theorem, can be generated for the elements of linear recurrence equations of any order. See in particular the relations defining the so-called quasi-Fibonacci numbers [3, 4, 5, 8]. Remark 5. Shevelev’s intention in the paper [2] was, it seems, to investigate the sum

\sqrt{|\frac{x_{1}}{x_{2}} |} + \sqrt{|\frac{x_{2}}{x_{1}} |},

where x_{1}, x_{2} ∈ ℝ are roots of the polynomial

x^{2} − λ_{1}x − λ_{2}

and x_{1}x_{2} < 0. Then we have

\sqrt{|\frac{x_{1}}{x_{2}} |} + \sqrt{|\frac{x_{2}}{x_{1}} |} = \frac{|x_{1}| + |x_{2}|}{\sqrt{|x_{1}x_{2}|}} = \sqrt{\frac{(|x_{1}| + |x_{2}|)^{2}}{|x_{1}x_{2}|}} =

= \sqrt{\frac{(x_{1} + x_{2})^{2} + 2|x_{1}x_{2}| − 2x_{1}x_{2}}{|x_{1}x_{2}|}} = \sqrt{\frac{λ_{1}^{2} + 4λ_{2}}{λ_{2}}} . (10)

In the particular case of

x^{2} + x − 1 = (x + 2 cos \frac{π}{5} )(x − 2 cos \frac{2π}{5} ),

the identity (1) follows from (10).

The extension of sums (10) to sums for roots of a given cubic polynomial is described in the next section.

Remark 6. We note that (see formula (7)):

λ_{1} F_{k}^{∗} + 2 λ_{2} F_{k−1}^{∗} = F_{k+1}^{∗} + λ_{2} F_{k−1}^{∗} = L_{k}^{∗},  (11)

where L_{k}^{∗} denotes the generalized Lucas sequence

\{ \begin{matrix}L_{0}^{∗} = 2, L_{1}^{∗} = λ_{1}, \\ L_{n+1}^{∗} = λ_{1} L_{n}^{∗} + λ_{2} L_{n−1}^{∗},  n ∈ ℕ. \end{matrix}  (12)

2 Cubic Shevelev sums

Let us assume that ξ_{1}, ξ_{2}, ξ_{3} are complex roots of the following polynomial with complex coefficients

f(z) := z^{3} + p z^{2} + q z + r.

The symbols \sqrt[3]{ξ_{1}}, \sqrt[3]{ξ_{2}}, \sqrt[3]{ξ_{3}} will denote any of the third complex roots of the numbers ξ_{1}, ξ_{2} and ξ_{3}, respectively (only in the case that ξ_{1}, ξ_{2} and ξ_{3} are real numbers we will assume that \sqrt[3]{ξ_{1}}, \sqrt[3]{ξ_{2}} and \sqrt[3]{ξ_{3}} also denote the respective real roots).

Let us set

A := (\sqrt[3]{ξ_{1}} + \sqrt[3]{ξ_{2}} + \sqrt[3]{ξ_{3}})^{3}

and

B := (\sqrt[3]{ξ_{1}} \sqrt[3]{ξ_{2}} + \sqrt[3]{ξ_{1}} \sqrt[3]{ξ_{3}} + \sqrt[3]{ξ_{2}} \sqrt[3]{ξ_{3}})^{3} .

Thus, the numbers

\sqrt[3]{ξ_{1}} + \sqrt[3]{ξ_{2}} + \sqrt[3]{ξ_{3}}  and  \sqrt[3]{ξ_{1}} \sqrt[3]{ξ_{2}} + \sqrt[3]{ξ_{1}} \sqrt[3]{ξ_{3}} + \sqrt[3]{ξ_{2}} \sqrt[3]{ξ_{3}}

belong to the sets of the third complex roots of A and B, respectively, which, for the conciseness of notation, will be denoted by the symbols \sqrt[3]{A} and \sqrt[3]{B}, respectively. In other words, we have

\sqrt[3]{ξ_{1}} + \sqrt[3]{ξ_{2}} + \sqrt[3]{ξ_{3}} ∈ \sqrt[3]{A}

and

\sqrt[3]{ξ_{1}} \sqrt[3]{ξ_{2}} + \sqrt[3]{ξ_{1}} \sqrt[3]{ξ_{3}} + \sqrt[3]{ξ_{2}} \sqrt[3]{ξ_{3}} ∈ \sqrt[3]{B}.

Then we can deduce the relation

27AB = (A + p − 3\sqrt[3]{r})^{3}.  (13)

We note that

ξ_{1}^{1/3}ξ_{2}^{1/3}ξ_{3}^{1/3}(\frac{ξ_{1}^{1/3}}{ξ_{2}^{1/3}} + \frac{ξ_{2}^{1/3}}{ξ_{1}^{1/3}} + \frac{ξ_{2}^{1/3}}{ξ_{3}^{1/3}} + \frac{ξ_{3}^{1/3}}{ξ_{2}^{1/3}} + \frac{ξ_{1}^{1/3}}{ξ_{3}^{1/3}} + \frac{ξ_{3}^{1/3}}{ξ_{1}^{1/3}} ) =

= ξ_{1}^{1/3}ξ_{2}^{1/3}(ξ_{1}^{1/3} + ξ_{2}^{1/3}) + ξ_{2}^{1/3}ξ_{3}^{1/3}(ξ_{2}^{1/3} + ξ_{3}^{1/3}) + ξ_{1}^{1/3}ξ_{3}^{1/3}(ξ_{1}^{1/3} + ξ_{3}^{1/3}) =

= (ξ_{1}^{1/3} + ξ_{2}^{1/3} + ξ_{3}^{1/3})(ξ_{1}^{1/3}ξ_{2}^{1/3} + ξ_{2}^{1/3}ξ_{3}^{1/3} + ξ_{1}^{1/3}ξ_{3}^{1/3}) − 3ξ_{1}^{1/3}ξ_{2}^{1/3}ξ_{3}^{1/3}.

Hence and from (13), for the respective values of \sqrt[3]{A}, \sqrt[3]{B} and \sqrt[3]{r} we get

\frac{ξ_{1}^{1/3}}{ξ_{2}^{1/3}} + \frac{ξ_{2}^{1/3}}{ξ_{1}^{1/3}} + \frac{ξ_{2}^{1/3}}{ξ_{3}^{1/3}} + \frac{ξ_{3}^{1/3}}{ξ_{2}^{1/3}} + \frac{ξ_{1}^{1/3}}{ξ_{3}^{1/3}} + \frac{ξ_{3}^{1/3}}{ξ_{1}^{1/3}} =

= \frac{\sqrt[3]{A}\sqrt[3]{B}}{−\sqrt[3]{r}} − 3 = \frac{1}{−3\sqrt[3]{r}}(A + p) − 2

(from the formula (3.5) in [6])

= \frac{1}{\sqrt[3]{2}\sqrt[3]{r}}(\sqrt[3]{S + \sqrt{τ}} + \sqrt[3]{S − \sqrt{τ}})

= \frac{1}{\sqrt[3]{2}}(\sqrt[3]{S_{1} + \sqrt{τ_{1}}} + \sqrt[3]{S_{1} − \sqrt{τ_{1}}}),

where

S = rS_{1}, τ = r^{2}τ_{1},
S_{1} = \frac{pq}{r} + \frac{6}{r^{2/3}} (q + p\sqrt[3]{r} + 3\sqrt[3]{r^{2}}) − 9,
τ_{1} = (\frac{pq}{r} )^{2} − 4\frac{q^{3}}{r^{2}} − 4\frac{p^{3}}{r} + 18\frac{pq}{r} − 27 =

= (\frac{pq}{r} + 9)^{2} − \frac{4}{r^{2}} (q^{3} + p^{3}r + 27r^{2}).

In consequence, if f(z) is the RCP polynomial (see [2, 7]), then

pr^{1/3} + 3r^{2/3} + q = 0,  (14)

which implies

S_{1} = \frac{pq}{r} − 9 and τ_{1} = (\frac{pq}{r} + 9)^{2} − 36\frac{pq}{r} = (\frac{pq}{r} − 9)^{2},

since

a^{3} + b^{3} + c^{3} = (a + b + c)(a^{2} + b^{2} + c^{2} − ab − ac − bc) + 3abc,  (15)

for a, b, c ∈ ℂ. Hence, we get the Shevelev formula

\frac{ξ_{1}^{1/3}}{ξ_{2}^{1/3}} + \frac{ξ_{2}^{1/3}}{ξ_{1}^{1/3}} + \frac{ξ_{2}^{1/3}}{ξ_{3}^{1/3}} + \frac{ξ_{3}^{1/3}}{ξ_{2}^{1/3}} + \frac{ξ_{1}^{1/3}}{ξ_{3}^{1/3}} + \frac{ξ_{3}^{1/3}}{ξ_{1}^{1/3}} = (\frac{pq}{r} − 9)^{1/3}.

However, if we assume that

q^{3} + p^{3}r + 27r^{2} = 0,  (16)

then we obtain

\sqrt{τ_{1}} = |\frac{pq}{r} + 9|, S_{1} − \frac{pq}{r} − 9 = \frac{6}{r^{2/3}} (q + p\sqrt[3]{r}),

S_{1} + \frac{pq}{r} + 9 = \frac{2pq}{r} + \frac{6}{r^{2/3}} (q + p\sqrt[3]{r} + 3\sqrt[3]{r^{2}}).

Hence, we get the formula

\frac{ξ_{1}^{1/3}}{ξ_{2}^{1/3}} + \frac{ξ_{2}^{1/3}}{ξ_{1}^{1/3}} + \frac{ξ_{2}^{1/3}}{ξ_{3}^{1/3}} + \frac{ξ_{3}^{1/3}}{ξ_{2}^{1/3}} + \frac{ξ_{1}^{1/3}}{ξ_{3}^{1/3}} + \frac{ξ_{3}^{1/3}}{ξ_{1}^{1/3}} =

= \sqrt[3]{\frac{3}{r^{2/3}} (q + p\sqrt[3]{r})} + \sqrt[3]{\frac{pq}{r} + \frac{3}{r^{2/3}} (q + p\sqrt[3]{r} + 3\sqrt[3]{r^{2}})}.

For example, let us set

f(z) = z^{3} + 3z^{2} − 3\sqrt[3]{2}z + 1.  (17)

Then the condition (16) is satisfied and the roots ξ_{1}, ξ_{2} and ξ_{3} of f(z) are real: ξ_{1} = 0.56048, ξ_{2} = 0.445392 and ξ_{3} = −4.00587. Furthermore, the following equality holds

\frac{ξ_{1}^{1/3}}{ξ_{2}^{1/3}} + \frac{ξ_{2}^{1/3}}{ξ_{1}^{1/3}} + \frac{ξ_{2}^{1/3}}{ξ_{3}^{1/3}} + \frac{ξ_{3}^{1/3}}{ξ_{2}^{1/3}} + \frac{ξ_{1}^{1/3}}{ξ_{3}^{1/3}} + \frac{ξ_{3}^{1/3}}{ξ_{1}^{1/3}} = \sqrt[3]{9(1 − \sqrt[3]{2})} + \sqrt[3]{18(1 − \sqrt[3]{2})} =

= \sqrt[3]{9(1 − \sqrt[3]{2})}(1 + \sqrt[3]{2}) = −3. (18)

We note that condition (16), by (15), is a condition of a type different from the condition (14).

3 RCP of the second kind

Shevelev in paper [1] (see also [7]) distinguished polynomials f ∈ ℝ[z] of the form

f(z) = z^{3} + p z^{2} + q z + r,  (19)

having real roots and satisfying the condition (14), and called them Ramanujan cubic polynomials (shortly RCP).

Now we introduce a new family of cubic polynomials of the form (19), having real roots and satisfying the condition (16). We will call them Ramanujan cubic polynomials of the second kind (shortly RCP2). The polynomial (17) is an example of RCP2 which is not RCP. On the other hand, the polynomials (see [7]):

z^{3} − \frac{3}{2} z^{2} − \frac{3}{2} z + 1 = (z − \frac{1}{2}) (z + 1) (z − 2),
z^{3} + z^{2} − 2 z − 1 = (z − 2 cos \frac{2 π}{7} ) (z − 2 cos \frac{4 π}{7} ) (z − 2 cos \frac{8 π}{7} ),

belong to the set RCP\RCP2. The polynomial

z^{3} − 3 z + 1 = (z − 2 cos \frac{2 π}{9} ) (z − 2 cos \frac{4 π}{9} ) (z − 2 cos \frac{8 π}{9} )

belongs to the common part of families of RCP’s and RCP2’s (see [7] and Theorem 7 a) written below). The polynomial

z^{3} − 3 z + \sqrt{3} = (z − 2 sin \frac{2 π}{9} ) (z + 2 sin \frac{4 π}{9} ) (z − 2 sin \frac{8 π}{9} )

is neither RCP nor RCP2.

In the next theorem we present the basic properties of RCP2’s.

Theorem 7. Let f(z) ∈ ℝ[z] and be of the form (19). Then the following facts hold.

a) If f(z) is either RCP or RCP2 and p q r = 0, then f(z) must be RCP and RCP2 simultaneously. Conversely, if f(z) belongs to the intersection of the sets RCP and RCP2 then p q r ≠ 0.

b) If f(z) satisfies (16), then f(z) is RCP2. In other words, the condition (16) implies that all the roots of f(z) are real. Only in the case of p q = −9 r polynomial f(z) possesses double root. In this case we have

g(z) := \frac{1}{p^{3}} f(p z) = z^{3} + z^{2} + \frac{\sqrt{5} − 1}{6} z + \frac{1 − \sqrt{5}}{54}.  (20)

Moreover, if ξ_{1} ξ_{2} and ξ_{3} = ξ_{2} are roots of g(z), then we obtain (see formula (25) below):

\sqrt[3]{ξ_{1}} + 2 \sqrt[3]{ξ_{2}} = \sqrt[3]{−1 + 2 \sqrt[3]{\frac{\sqrt{5} − 1}{2}} − 6 \sqrt[3]{\frac{1}{3} \sqrt[3]{(\frac{\sqrt{5} − 1}{2})^{2}} − \frac{1}{3} \sqrt[3]{(\frac{\sqrt{5} − 1}{2})^{4}}}} (21)

and

\sqrt[9]{\frac{\sqrt{5} − 1}{2}} (1 + \sqrt[3]{\frac{ξ_{1}}{ξ_{2}}} + \sqrt[3]{\frac{ξ_{2}}{ξ_{1}}} ) = \sqrt[3]{\sqrt[3]{(\frac{\sqrt{5} − 1}{2})^{2}} − 1}.  (22)

Next, whenever f(z) satisfies the condition (14), then f(z) is RCP if and only if r > 0.

c) If f(z) is RCP2, then we have

r ≠ 0 ⟹ \frac{p q}{r} ⩽ \frac{9}{\sqrt[3]{4}}.  (23)

If f(z) is RCP, then we have (see [1]):

r ≠ 0 ⟹ \frac{p q}{r} ⩽ \frac{9}{4}.  (24)

d) If f(z) is RCP, then

p^{2} ⩾ 12 q,

whereas, if f(z) is RCP2, then

p^{2} ⩾ 3 \sqrt[3]{4} q.

e) Let f(z) belong to family of RCP2’s and let ξ_{1}, ξ_{2}, ξ_{3} be roots of f(z). Then we have

\sqrt[3]{ξ_{1}} + \sqrt[3]{ξ_{2}} + \sqrt[3]{ξ_{3}} =

= \sqrt[3]{−p − 6 \sqrt[3]{r} − 3 \sqrt[3]{3 \sqrt[3]{r} (q + p \sqrt[3]{r})} − 3 \sqrt[3]{(p + 3 \sqrt[3]{r}) (q + 3 \sqrt[3]{r^{2}})}}. (25)

For example, for the polynomial (17) we obtain

\sqrt[3]{ξ_{1}} + \sqrt[3]{ξ_{2}} + \sqrt[3]{ξ_{3}} = 0.

f) Let f(z) belong to the family of RCP2’s and a, b ∈ ℝ. Suppose that ξ_{1}, ξ_{2}, ξ_{3} are roots of f(z). If a ξ_{1} + b, a ξ_{2} + b, a ξ_{3} + b are also roots of some RCP2, then

b (9 b^{2} − 9 a b p + a^{2} (p^{2} + 6 q))(9 b^{3} − 9 a b^{2} p + a^{2} b (p^{2} + 6 q) − a^{3} (p q + 9 r) = 0. (26)

g) If a, a ϱ, a ϱ^{2} ∈ ℝ are roots of some RCP2, then

2 \sqrt[3]{2} ϱ = −\sqrt[3]{2} − 3 ± \sqrt{3 (3 + 2 \sqrt[3]{2} − \sqrt[3]{4})}.

Moreover, for a = 2 \sqrt[3]{2} we have

a ϱ^{2} = 6 − \sqrt[3]{2} + \frac{9}{\sqrt[3]{2}} ∓ (1 + \frac{3}{\sqrt[3]{2}}) \sqrt{3 (3 + 2 \sqrt[3]{2} − \sqrt[3]{4})}.

h) If f(z) is RCP2, then

f(z) = z^{3} + \sqrt[3]{(α − \frac{27}{2}) r} z^{2} − \sqrt[3]{(α + \frac{27}{2}) r^{2}} z + r,  (27)

for any α, r ∈ ℝ. We note that if g(z) is RCP then from (18) in [7] we have

g(z) = z^{3} + (β − \frac{3}{2}) ϱ^{1/3} z^{2} − (β + \frac{3}{2}) ϱ^{2/3} z + ϱ

for some β, ϱ ∈ ℝ. Proof. a) Both conclusions follow from (14), (15) and (16).

b) Suppose that f(z) satisfies (16). Then

f′(z) = (z + \frac{p − \sqrt{p^{2} − 3 q}}{3} ) (z + \frac{p + \sqrt{p^{2} − 3 q}}{3} )

and

f(\frac{−p + \sqrt{p^{2} − 3 q}}{3} ) f(\frac{−p − \sqrt{p^{2} − 3 q}}{3} ) =^{(16)} −\frac{1}{27} (p q + 9 r)^{2},

which means that all the roots of f(z) are real. Now let p q = −9 r. Then from (16) we get

q^{2} + \frac{3}{9} p^{2} q − \frac{1}{9} p^{4} = 0,

which implies

q = \frac{\sqrt{5} − 1}{6} p^{2}  and  r = \frac{1 − \sqrt{5}}{54} p^{3},

and the relation (20) follows.

The equality (21) can be deduced from formula (25).

c) From (16) we get

\frac{p^{3} q^{3}}{r^{3}} = −\frac{p^{6}}{r^{2}} − 27 \frac{p^{3}}{r} ,

which implies

\frac{9^{3}}{4} − (\frac{p q}{r} )^{3} = (\frac{27}{2} + \frac{p^{3}}{r} )^{2} ⩾ 0,  (28)

i.e.,

\frac{p q}{r} ⩽ \frac{9}{\sqrt[3]{4}}.

d) From (16) we obtain

27 r^{2} + p^{3} r + q^{3} = 0,
Δ_{r} = p^{6} − 4 · 27 · q^{3} ⩾ 0,

i.e,

p^{2} ⩾ 3 \sqrt[3]{4} q.

Similarly, if we have

3 \sqrt[3]{r^{2}} + p \sqrt[3]{r} + q = 0

and p, q, r ∈ ℝ, then

Δ\sqrt[3]{r} = p^{2} − 12 q ⩾ 0 ⇔ p^{2} ⩾ 12 q.

e) We have the formula (see formula (3.5) in [6]):

\sqrt[3]{ξ_{1}} + \sqrt[3]{ξ_{2}} + \sqrt[3]{ξ_{3}} = \sqrt[3]{−p − 6 \sqrt[3]{r} − \frac{3}{\sqrt[3]{2}} (\sqrt[3]{S + \sqrt{T}} + \sqrt[3]{S − \sqrt{T}} )},  (29)

where

S := p q + 6 q \sqrt[3]{r} + 6 p \sqrt[3]{r^{2}} + 9 r,
T := p^{2} q^{2} − 4 q^{3} − 4 p^{3} r + 18 p q r − 27 r^{2}.

Hence, by (16) we get

T = p^{2} q^{2} + 18 p q r + 81 r^{2} = r^{2}(\frac{p q}{r} + 9)^{2},  (30)

which implies

\{S ± \sqrt{T} \} = \{S ± (p q + 9 r)\},
S − p q − 9 r = 6 \sqrt[3]{r} (q + p \sqrt[3]{r}),
S + p q + 9 r = 2 p q + 6 q \sqrt[3]{r} + 6 p \sqrt[3]{r^{2}} + 18 r =
= 2 q (p + 3 \sqrt[3]{r}) + 6 \sqrt[3]{r^{2}} (p + 3 \sqrt[3]{r}) = 2 (p + 3 \sqrt[3]{r}) (q + 3 \sqrt[3]{r^{2}})

and, at last, the formula (25) follows.

In consequence, if f(z) = z^{3} + 3 z^{2} − 3 \sqrt[3]{2} z + 1, then p = 3, q = −3 \sqrt[3]{2}, r = 1 and from (25) we get

(\sqrt[3]{ξ_{1}} + \sqrt[3]{ξ_{2}} + \sqrt[3]{ξ_{3}})^{3} = −9 − 3 \sqrt[3]{9 (1 − \sqrt[3]{2})} − 3 \sqrt[3]{18 (1 − \sqrt[3]{2})} =

= −9 − 3 \sqrt[3]{9 (1 − \sqrt[3]{2})} (\sqrt[3]{2} + 1) =^{(18)} −9 + 9 = 0.

f) We have

(x − a ξ_{1} − b) (x − a ξ_{2} − b) (x − a ξ_{3} − b) = x^{3} + p_{1} x^{2} + q_{1} x + r_{1},

where

p_{1} = a p − 3 b,
q_{1} = a^{2} q + 3 b^{2} − 2 a b p,
r_{1} = a^{3} r − a^{2} b q − b^{3} + a b^{2} p.

If this polynomial is also RCP2, then q_{1}^{3} + p_{1}^{3} r_{1} + 27 r_{1}^{2} = 0, which (with assistance of Mathematica) implies the equation (26).

g) Suppose that a ≠ 0 and

z^{3} + p z^{2} + q z + r = (z − a) (z − a ϱ) (z − a ϱ^{2}).

Then we have the relations

r = −(a ϱ)^{3},
p = −a (1 + ϱ + ϱ^{2}),
q = a^{2} (ϱ + ϱ^{2} + ϱ^{3}),

and the condition (16) has now the form

(ϱ + ϱ^{2} + ϱ^{3})^{3} + 27 ϱ^{6} + ϱ^{3} (1 + ϱ + ϱ^{2})^{3} = 0

or

2 (1 + ϱ + ϱ^{2})^{3} + 27 ϱ^{3} = 0.

Hence \sqrt[3]{2} (1 + ϱ + ϱ^{2}) = −3 ϱ,

ϱ^{2} + (1 + \frac{3}{\sqrt[3]{2}}) ϱ + 1 = 0,

which implies

2 \sqrt[3]{2} ϱ = −\sqrt[3]{2} − 3 ± \sqrt{3 (3 + 2 \sqrt[3]{2} − \sqrt[3]{4})}.

h) Let us set

α := \frac{27}{2} + \frac{p^{3}}{r} .  (31)

Then from (28) we generate the relation

\frac{9^{3}}{4} − (α − \frac{27}{2}) \frac{q^{3}}{r^{2}} = α^{2},

i.e.,

\frac{9^{3}}{4} − α^{2} = (α − \frac{27}{2}) \frac{q^{3}}{r^{2}} ,

q^{3} = −(α + \frac{27}{2}) r^{2}.

From (31) we obtain

p^{3} = (α − \frac{27}{2}) r.

The following theorem, proved by Shevelev for RCP’s [1], holds also for RCP2’s. Theorem 8. If for two RCP2’s of the form

y^{3} + p_{1} y^{2} + q_{1} y + r_{1}, z^{3} + p_{2} z^{2} + q_{2} z + r_{2}

the following condition holds (r_{1} r_{2} ≠ 0):

\frac{p_{1} q_{1}}{r_{1}} = \frac{p_{2} q_{2}}{r_{2}} ,

then for their roots y_{1}, y_{2} y_{3} and z_{1}, z_{2}, z_{3}, respectively, the sequence of numbers

\frac{y_{1}}{y_{2}} , \frac{y_{2}}{y_{1}} , \frac{y_{1}}{y_{3}} , \frac{y_{3}}{y_{1}} , \frac{y_{2}}{y_{3}} , \frac{y_{3}}{y_{2}} ,

is a permutation of the sequence

\frac{z_{1}}{z_{2}} , \frac{z_{2}}{z_{1}} , \frac{z_{1}}{z_{3}} , \frac{z_{3}}{z_{1}} , \frac{z_{2}}{z_{3}} , \frac{z_{3}}{z_{2}} .

Proof. The proof runs like Shevelev’s proof of Theorem 5 in [1]. Only one change is needed, for the case of RCP2 in formula (38) we have

\frac{p^{3} r + q^{3}}{r^{2}} =^{(16)} −27.

4 Acknowledgments

I wish to express my gratitude to the referee for several helpful comments and suggestions concerning the first version of the paper.

References

[1] V. Shevelev, On Ramanujan cubic polynomials, available electronically at

    http://arxiv.org/abs/0711.3420, 2007.

[2] V. Shevelev, Gold ratio and a trigonometric identity, available electronically at

    http://arxiv.org/abs/0909.5072v1, 2009.

[3] R. Wituła, D. Słota and A. Warzyński, Quasi-Fibonacci numbers of the seventh order, J. Integer Seq. 9 (2006), Article 06.4.3.

[4] R. Wituła and D. Słota, New Ramanujan-type formulas and quasi-Fibonacci numbers of order 7, J. Integer Seq. 10 (2007), Article 07.5.6.

[5] R. Wituła and D. Słota, Quasi-Fibonacci numbers of order 11, J. Integer Seq. 10 (2007), Article 07.8.5.

[6] R. Wituła, Ramanujan type trigonometric formulas: the general form for the argument \frac{2 π}{7}, J. Integer Seq. 12 (2009), Article 09.8.5.

[7] R. Wituła, Full description of Ramanujan cubic polynomials, J. Integer Seq. 13 (2010), Article 10.5.7.

[8] R. Wituła and D. Słota, Quasi-Fibonacci numbers of order 13 on the occasion the Thirteenth International Conference on Fibonacci Numbers and their Applications, Congr. Numer. 201 (2010), 89–107.

2000 Mathematics Subject Classification: Primary 11C08; Secondary 11B83, 33B10. Keywords: Ramanujan cubic polynomial.

Received June 17 2010; revised version received July 1 2010. Published in Journal of Integer Sequences, July 9 2010.

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