Quasi-Fibonacci Numbers of the Seventh Order

Roman Wituła, Damian Słota and Adam Warzyński Institute of Mathematics Silesian University of Technology Kaszubska 23 Gliwice 44-100 Poland

                       r.witula@polsl.pl
                       d.slota@polsl.pl

Abstract In this paper we introduce and investigate the so-called quasi-Fibonacci numbers of the seventh order. We discover many surprising relations and identities, and study some applications to polynomials.

1 Introduction

Grzymkowski and Wituła [3] discovered and studied the following two identities:

(1 + ξ + ξ^{4})^{n} = F_{n+1} + F_{n}(ξ + ξ^{4}),  (1.1)
(1 + ξ^{2} + ξ^{3})^{n} = F_{n+1} + F_{n}(ξ^{2} + ξ^{3}),  (1.2)

where F_{n} denote the Fibonacci numbers and ξ ∈ ℂ is a primitive fifth root of unity (i.e., ξ^{5} = 1 and ξ ≠ 1). These identities make it possible to prove many classical relations for Fibonacci numbers as well as to generalize some of them. We may state that these identities make up an independent method of proving such relations, which is an alternative to the methods depending on the application of either Binet formulas or the generating function of Fibonacci and Lucas numbers.

1.1 Example of the application of identities (1.1) and (1.2)

First, we note that

(1 + ξ + ξ^{4})^{u+v} = F_{u+v+1} + F_{u+v} (ξ + ξ^{4})  (1.3)

and

(1 + ξ + ξ^{4})^{u+v} = (1 + ξ + ξ^{4})^{u} (1 + ξ + ξ^{4})^{v} =

= (F_{u+1} + F_{u} (ξ + ξ^{4})) (F_{v+1} + F_{v} (ξ + ξ^{4})) =

(by the identity 1 + ξ + ξ^{2} + ξ^{3} + ξ^{4} = 0)
= F_{u+1} F_{v+1} + F_{u} F_{v} + (F_{u} F_{v+1} + F_{u+1} F_{v} − F_{u} F_{v}) (ξ + ξ^{4}) =

= F_{u+1} F_{v+1} + F_{u} F_{v} + (F_{u} F_{v+1} + F_{u−1} F_{v}) (ξ + ξ^{4}). (1.4)

Replacing u by u − r and v by v + r in (1.4) we obtain

(1 + ξ + ξ^{4})^{u+v} = F_{u−r+1} F_{v+r+1} + F_{u−r} F_{v+r}+ + (F_{u−r} F_{v+r+1} + F_{u−r+1} F_{v+r} − F_{u−r} F_{v+r})(ξ + ξ^{4}). (1.5)

We note that the numbers 1 and ξ + ξ^{4} are linearly independent over ℚ. Hence comparing the parts without (ξ +ξ^{4}) of (1.3) with (1.4) and (1.4) with (1.5) we get two known identities (see [2, 5])

F_{u+v+1} = F_{u+1} F_{v+1} + F_{u} F_{v}

and

F_{u+1} F_{v+1} − F_{u−r+1} F_{v+r+1} = F_{u−r} F_{v+r} − F_{u} F_{v}

(after the next (u − r) iterations) = (−1)^{u−r+1}(F_{r} F_{v−u+r} − F_{0} F_{v−u+2r}) = (−1)^{u−r+1}F_{r} F_{v−u+r}.

1.2 The aim of the paper

In this paper the relations (1.1) and (1.2) will be generalized in the following way:

(1 + δ(ξ + ξ^{6}))^{n} = A_{n}(δ) + B_{n}(δ)(ξ + ξ^{6}) + C_{n}(δ)(ξ^{2} + ξ^{5}),
(1 + δ(ξ^{2} + ξ^{5}))^{n} = A_{n}(δ) + B_{n}(δ)(ξ^{2} + ξ^{5}) + C_{n}(δ)(ξ^{3} + ξ^{4}),

and

(1 + δ(ξ^{3} + ξ^{4}))^{n} = A_{n}(δ) + B_{n}(δ)(ξ^{3} + ξ^{4}) + C_{n}(δ)(ξ + ξ^{6}),

where ξ ∈ ℂ are primitive seventh roots of unity (i.e., ξ^{7} = 1 and ξ ≠ 1), δ ∈ ℂ, δ ≠ 0. New families of numbers created by these identities

{A_{n}(δ)}_{n=1}^{∞},  {B_{n}(δ)}_{n=1}^{∞}, and {C_{n}(δ)}_{n=1}^{∞}  (1.6)

called here “the quasi-Fibonacci numbers of order (7; δ)”, δ ∈ ℂ, δ ≠ 0, are investigated in this paper. The elements of each of the three sequences (1.6) satisfy the same recurrence relation of order three

𝕏_{n+3} + (δ − 3)𝕏_{n+2} + (3 − 2δ − 2δ^{2})𝕏_{n+1} + (−1 + δ + 2δ^{2} − δ^{3})𝕏_{n} = 0,

which enables a direct trigonometrical representation of these numbers (see formulas (3.17)– (3.19)). In consequence, many surprising algebraic and trigonometric identities and summation formulas for these numbers may be generated. Also the polynomials connected with the numbers {C_{n}(δ)}_{n=1}^{∞}

∑_{k=1}^{n} C_{k+1}(δ) x^{n−k}, n ∈ ℕ,

are investigated in this paper.

2 Minimal polynomials, linear independence over ℚ

Let Ψ_{n}(x) be the minimal polynomial of cos(2π/n) for every n ∈ ℕ. W. Watkins and J. Zeitlin described [10] (see also [9]) the following identities:

T_{s+1}(x) − T_{s}(x) = 2^{s} ∏_{d|n} Ψ_{d}(x)

if n = 2s + 1 and

T_{s+1}(x) − T_{s−1}(x) = 2^{s} ∏_{d|n} Ψ_{d}(x)

if n = 2s, where T_{s}(x) denotes the s-th Chebyshev polynomial of the first kind. In the sequel, if n = 2s + 1 is a prime number, we obtain

T_{s+1}(x) − T_{s}(x) = 2^{s}Ψ_{1}(x)Ψ_{n}(x).

For example, we have

Ψ_{7}(x) = \frac{1}{8(x − 1)}(T_{4}(x) − T_{3}(x)) = \frac{1}{8(x − 1)}(8x^{4} − 4x^{3} − 8x^{2} + 3x + 1)

= \frac{1}{8(x − 1)}8(x − 1)(x^{3} + \frac{1}{2}x^{2} − \frac{1}{2}x − \frac{1}{8}) = x^{3} + \frac{1}{2}x^{2} − \frac{1}{2}x − \frac{1}{8},

Lemma 2.1. If n ⩾ 3 then the roots of Ψ_{n}(x) = 0 are cos(2πk/n), for 0 < k ⩽ s and (k, n) = 1 (where n = 2s or n = 2s + 1 respectively).

Below a more elementary proof of this result but only for the polynomial Ψ_{7}(x), will be presented. Our proof is based only on the following simple fact:

Lemma 2.2. Let q ∈ ℚ and cos(qπ) ∈ ℚ. Then cos(qπ) ∈ \{0, ±\frac{1}{2}, ±1\}.

Proof. For the elementary proof of this Lemma see, for example, the Appendix to the Russian translation of Niven’s book [6] (written by I. M. Yaglom). Lemma 2.3. Let ξ = exp(i2π/7). Then the polynomial

p_{7}(x) := (x − ξ − ξ^{6})(x − ξ^{2} − ξ^{5})(x − ξ^{3} − ξ^{4})
= x^{3} + x^{2}(−ξ − ξ^{2} − ξ^{3} − ξ^{4} − ξ^{5} − ξ^{6})
+ x((ξ + ξ^{6})(ξ^{2} + ξ^{5}) + (ξ + ξ^{6})(ξ^{3} + ξ^{4}) + (ξ^{2} + ξ^{5})(ξ^{3} + ξ^{4}))
− (ξ + ξ^{6})(ξ^{2} + ξ^{5})(ξ^{3} + ξ^{4}) = x^{3} + x^{2} − 2x − 1

is a minimal polynomial of the numbers

ξ + ξ^{6} = 2 cos(2π/7), ξ^{2} + ξ^{5} = 2 cos(4π/7), ξ^{3} + ξ^{4} = 2 cos(6π/7).  (2.1)

Moreover, we have the identity

Ψ_{7}(x) = \frac{1}{8}p_{7}(2x).  (2.2)

Proof. Besides Lemma 2.2, each of the numbers in (2.1) is an irrational number, so the polynomial p_{7}(x) is irreducible over ℚ. Corollary 2.4. We have

p_{7}(x − 1) = x^{3} − 2x^{2} − x + 1.

See the identity (3.66) below, where the connection of p_{7}(x − 1) with quasi-Fibonacci numbers of 7-th order is presented. Corollary 2.5. The numbers cos(2kπ/7), k = 0, 1, 2 are linearly independent over ℚ. Proof. If we suppose that

a + b cos(2π/7) + c cos(4π/7) = 0

for some a, b, c ∈ ℚ, then we also have

a + b cos(2π/7) + c(2 cos^{2}(2π/7) − 1) = 0,

i.e.,

a − c + b cos(2π/7) + 2c cos^{2}(2π/7) = 0

so the degree of cos(2π/7) is ⩽ 2, which by Lemma 2.3 means that a = b = c = 0. Corollary 2.6. Every three numbers which belong to the set {1, ξ + ξ^{6}, ξ^{2} + ξ^{5}, ξ^{3} + ξ^{4}} are linearly independent over ℚ. Proof. It follows from the identity 1 + ξ + ξ^{2} + … + ξ^{6} = 0 and from Corollary 2.5.

Corollary 2.7. The following decomposition holds:

f_{n}(𝕏) := (𝕏 − (2 cos \frac{2π}{7} )^{2^{n}}) · (𝕏 − (2 cos \frac{4π}{7} )^{2^{n}}) · (𝕏 − (2 cos \frac{6π}{7} )^{2^{n}})
= 𝕏^{3} − α_{n}𝕏^{2} + β_{n}𝕏 − 1,

where α_{0} = −1, β_{0} = −2 and

\{ \begin{matrix}α_{n+1} = α_{n}^{2} − 2β_{n}, \\ β_{n+1} = β_{n}^{2} − 2α_{n}, n ∈ ℕ. \end{matrix}  (2.3)

For example, we have α_{0} = −1, β_{0} = −2, α_{1} = 5, β_{1} = 6, α_{2} = 13, β_{2} = 26, α_{3} = 117 and β_{3} = 650. (A more general decomposition will be presented later, see Lemma 3.14 a.) We note that 13|α_{n} and 13|β_{n} for every n ∈ ℕ, n ≥ 2.

Proof. We have

(𝕐^{2} − (2 cos \frac{2π}{7})^{2^{n+1}}) · (𝕐^{2} − (2 cos \frac{4π}{7})^{2^{n+1}}) · (𝕐^{2} − (2 cos \frac{6π}{7})^{2^{n+1}}) =

= −f_{n}(𝕐) · f_{n}(−𝕐) = (𝕐^{3} − α_{n}𝕐^{2} + β_{n}𝕐 − 1)(𝕐^{3} + α_{n}𝕐^{2} + β_{n}𝕐 + 1) = = (𝕐^{3} + β_{n}𝕐)^{2} − (α_{n}𝕐^{2} + 1)^{2} = 𝕐^{6} − (α_{n}^{2} − 2β_{n})𝕐^{4} + (β_{n}^{2} − 2α_{n})𝕐^{2} − 1.

It is obvious that

α_{n} = (2 cos \frac{2π}{7} )^{2^{n}} + (2 cos \frac{4π}{7} )^{2^{n}} + (2 cos \frac{6π}{7} )^{2^{n}}

and

β_{n} = (4 cos \frac{2π}{7} cos \frac{4π}{7} )^{2^{n}} + (4 cos \frac{2π}{7} cos \frac{6π}{7} )^{2^{n}} + (4 cos \frac{4π}{7} cos \frac{6π}{7} )^{2^{n}}

for every n ∈ ℕ. Moreover, from (2.3), we deduce that

α_{n+1} + β_{n+1} = (α_{n} − 1)^{2} + (β_{n} − 1)^{2} − 2,
α_{n+1} − β_{n+1} = (α_{n} − 1)^{2} − (β_{n} − 1)^{2} = (α_{n} − β_{n})(α_{n} + β_{n} + 2) =
= (α_{n} − β_{n})((α_{n−1} − 1)^{2} + (β_{n−1} − 1)^{2}),

which yields

β_{n+1} − α_{n+1} = ∏_{k=0}^{n−1} ((α_{k} − 1)^{2} + (β_{k} − 1)^{2}) = 13 ∏_{k=1}^{n−1} ((α_{k} − 1)^{2} + (β_{k} − 1)^{2}) =

= ∏_{k=1}^{n} (α_{k} + β_{k} + 2).

The following extension lemma is the basic technical tool to generating almost all formulas presented in the next sections.

Lemma 2.8. Let a_{1}, a_{2}, …, a_{n} ∈ ℝ be linearly independent over ℚ and let f_{k}, g_{k} ∈ ℚ[δ], k = 1, 2, …, n. If the identity

∑_{k=1}^{n} f_{k}(δ) a_{k} = ∑_{k=1}^{n} g_{k}(δ) a_{k}

holds for every δ ∈ ℚ, then f_{k}(δ) = g_{k}(δ) for every k = 1, 2, …, n and δ ∈ ℂ.

3 Quasi-Fibonacci numbers of order 7

Lemma 3.1. Let δ, ξ ∈ ℂ, ξ^{7} = 1 and ξ ≠ 1. Then the following identities hold:

(1 + δ(ξ + ξ^{6}))^{n} = a_{n}(δ) + b_{n}(δ)(ξ^{1·1} + ξ^{1·6}) + c_{n}(δ)(ξ^{2·1} + ξ^{2·6})
+ d_{n}(δ)(ξ^{3·1} + ξ^{3·6})
= a_{n}(δ) + b_{n}(δ)(ξ + ξ^{6}) + c_{n}(δ)(ξ^{2} + ξ^{5}) + d_{n}(δ)(ξ^{3} + ξ^{4})
= A_{n}(δ) + B_{n}(δ)(ξ + ξ^{6}) + C_{n}(δ)(ξ^{2} + ξ^{5}),  (3.1)

(1 + δ(ξ^{2} + ξ^{5}))^{n} = a_{n}(δ) + b_{n}(δ)(ξ^{1·2} + ξ^{1·5}) + c_{n}(δ)(ξ^{2·2} + ξ^{2·5})
+ d_{n}(δ)(ξ^{3·2} + ξ^{3·5})
= a_{n}(δ) + b_{n}(δ)(ξ^{2} + ξ^{5}) + c_{n}(δ)(ξ^{3} + ξ^{4}) + d_{n}(δ)(ξ + ξ^{6})
= A_{n}(δ) + B_{n}(δ)(ξ^{2} + ξ^{5}) + C_{n}(δ)(ξ^{3} + ξ^{4}),  (3.2)
(1 + δ(ξ^{3} + ξ^{4}))^{n} = a_{n}(δ) + b_{n}(δ)(ξ^{1·3} + ξ^{1·4}) + c_{n}(δ)(ξ^{2·3} + ξ^{2·4})
+ d_{n}(δ)(ξ^{3·3} + ξ^{3·4})
= a_{n}(δ) + b_{n}(δ)(ξ^{3} + ξ^{4}) + c_{n}(δ)(ξ + ξ^{6}) + d_{n}(δ)(ξ^{2} + ξ^{5})
= A_{n}(δ) + B_{n}(δ)(ξ^{3} + ξ^{4}) + C_{n}(δ)(ξ + ξ^{6}),  (3.3)

where

\{ \begin{matrix}a_{0}(δ) = 1, b_{0}(δ) = c_{0}(δ) = d_{0}(δ) = 0, \\ a_{n+1}(δ) = a_{n}(δ) + 2δb_{n}(δ), \\ b_{n+1}(δ) = δa_{n}(δ) + b_{n}(δ) + δc_{n}(δ), \\ c_{n+1}(δ) = δb_{n}(δ) + c_{n}(δ) + δd_{n}(δ), \\ d_{n+1}(δ) = δc_{n}(δ) + (1 + δ)d_{n}(δ) \end{matrix}  (3.4)

for every n ∈ ℕ and

A_{n+1}(δ) := a_{n+1}(δ) − d_{n+1}(δ)

= a_{n}(δ) + 2δb_{n}(δ) − δc_{n}(δ) − (1 + δ)d_{n}(δ) = (a_{n}(δ) − d_{n}(δ)) + 2δ(b_{n}(δ) − d_{n}(δ)) − δ(c_{n}(δ) − d_{n}(δ)) = A_{n}(δ) + 2δB_{n}(δ) − δC_{n}(δ),

B_{n+1}(δ) := b_{n+1}(δ) − d_{n+1}(δ)

= δa_{n}(δ) + b_{n}(δ) + δc_{n}(δ) − δc_{n}(δ) − (1 + δ)d_{n}(δ) = δ(a_{n}(δ) − d_{n}(δ)) + b_{n}(δ) − d_{n}(δ) = δA_{n}(δ) + B_{n}(δ),

C_{n+1}(δ) := c_{n+1}(δ) − d_{n+1}(δ)

= δb_{n}(δ) + c_{n}(δ) + δd_{n}(δ) − δc_{n}(δ) − (1 + δ)d_{n}(δ) = δ(b_{n}(δ) − d_{n}(δ)) + (1 − δ)(c_{n}(δ) − d_{n}(δ)) = δB_{n}(δ) + (1 − δ)C_{n}(δ),

i.e.,

\{ \begin{matrix}A_{0}(δ) = 1, B_{0}(δ) = C_{0}(δ) = 0, \\ A_{n+1}(δ) = A_{n}(δ) + 2δB_{n}(δ) − δC_{n}(δ), \\ B_{n+1}(δ) = δA_{n}(δ) + B_{n}(δ), \\ C_{n+1}(δ) = δB_{n}(δ) + (1 − δ)C_{n}(δ) \end{matrix}  (3.5)

for every n ∈ ℕ.

Proof. For example, we have

(1 + δ(ξ + ξ^{6}))^{n+1} = (1 + δ(ξ + ξ^{6}))^{n}(1 + δ(ξ + ξ^{6}))

= (a_{n}(δ) + b_{n}(δ)(ξ + ξ^{6}) + c_{n}(δ)(ξ^{2} + ξ^{5}) + d_{n}(δ)(ξ^{3} + ξ^{4}))(1 + δ(ξ + ξ^{6})) = a_{n}(δ) + δa_{n}(δ)(ξ + ξ^{6}) + b_{n}(δ)(ξ + ξ^{6}) + δb_{n}(δ)(ξ^{2} + 2 + ξ^{5}) + c_{n}(δ)(ξ^{2} + ξ^{5}) + δc_{n}(δ)(ξ^{3} + ξ + ξ^{6} + ξ^{4}) + d_{n}(δ)(ξ^{3} + ξ^{4}) + δd_{n}(δ)(ξ^{4} + ξ^{5} + ξ^{2} + ξ^{3}) = a_{n}(δ) + 2δb_{n}(δ) + (ξ + ξ^{6})(δa_{n}(δ) + b_{n}(δ) + δc_{n}(δ)) + (ξ^{2} + ξ^{5})(δb_{n}(δ) + c_{n}(δ) + δd_{n}(δ)) + (ξ^{3} + ξ^{4})(δc_{n}(δ) + (1 + δ)d_{n}(δ)) =

(by equality ξ^{3} + ξ^{4} = −1 − ξ − ξ^{6} − ξ^{2} − ξ^{5}) = a_{n}(δ) + 2δb_{n}(δ) − δc_{n}(δ) − (1 + δ)d_{n}(δ) + (ξ + ξ^{6})(δa_{n}(δ) + b_{n}(δ) − (1 + δ)d_{n}(δ)) + (ξ^{2} + ξ^{5})(δb_{n}(δ) + (1 − δ)c_{n}(δ) − d_{n}(δ)).

Definition 3.2. The numbers (3.5) will be called the quasi-Fibonacci numbers of order (7; δ) (see Table 1 at the end of the paper). To simplify notation we write a_{n}, b_{n}, c_{n}, d_{n}, A_{n}, B_{n}, C_{n} instead of a_{n}(1), b_{n}(1), c_{n}(1), d_{n}(1), A_{n}(1), B_{n}(1), C_{n}(1) respectively, and these numbers will be called the quasi-Fibonacci numbers of the seventh order.

Remark 3.3. Note that section III of [8] contains an explicit reference to the sequences A_{n}, B_{n} and C_{n}. This paper is partly based on the properties of −p_{7}(−x) and p_{7}(x − 1). A_{n}, B_{n} and C_{n} also appear in [4]. Corollary 3.4. Adding equalities (3.1–3.3) we obtain the identity

(1 + δ(ξ + ξ^{6}))^{n} + (1 + δ(ξ^{2} + ξ^{5}))^{n} + (1 + δ(ξ^{3} + ξ^{4}))^{n} =

= 3A_{n}(δ) − B_{n}(δ) − C_{n}(δ). (3.6)

Corollary 3.5. There follows from (3.4) and (3.5) two special systems (for δ = 1)

\{ \begin{matrix}a_{0} = 1, b_{0} = c_{0} = d_{0} = 0, \\ a_{n+1} = a_{n} + 2b_{n}, \\ b_{n+1} = a_{n} + b_{n} + c_{n}, \\ c_{n+1} = b_{n} + c_{n} + d_{n}, \\ d_{n+1} = c_{n} + 2d_{n} \end{matrix}  (3.7)

and

\{ \begin{matrix}A_{0} = 1, B_{0} = C_{0} = 0, \\ A_{n+1} = A_{n} + 2B_{n} − C_{n}, \\ B_{n+1} = A_{n} + B_{n}, \\ C_{n+1} = B_{n} \end{matrix}  (3.8)

for every n ∈ ℕ (see Table 2, \underline{A006356} and \underline{A006054} in [7]). Corollary 3.6. Another system may also be derived from (3.5) for δ = −1

\{ \begin{matrix}A_{0}(−1) = 1, B_{0}(−1) = C_{0}(−1) = 0, \\ A_{n+1}(−1) = A_{n}(−1) − 2B_{n}(−1) + C_{n}(−1), \\ B_{n+1}(−1) = −A_{n}(−1) + B_{n}(−1), \\ C_{n+1}(−1) = −B_{n}(−1) + 2C_{n}(−1) \end{matrix}

for every n ∈ ℕ (see Table 3 and \underline{A085810} in [7]). Corollary 3.7. If δ ≠ 0 then from (3.5) the following identities can be generated:

δB_{n}(δ) = C_{n+1}(δ) − (1 − δ)C_{n}(δ),  (3.9)
δA_{n}(δ) = B_{n+1}(δ) − B_{n}(δ),  (3.10)
δ^{2}A_{n}(δ) = δB_{n+1}(δ) − δB_{n}(δ)  (3.11)
= C_{n+2}(δ) − (1 − δ)C_{n+1}(δ) − (C_{n+1}(δ) − (1 − δ)C_{n}(δ))
= C_{n+2}(δ) − (2 − δ)C_{n+1}(δ) + (1 − δ)C_{n}(δ),
A_{n+1}(δ) − A_{n}(δ) = 2δB_{n}(δ) − δC_{n}(δ)  (3.12)
= 2C_{n+1}(δ) − 2(1 − δ)C_{n}(δ) − δC_{n}(δ)
= 2C_{n+1}(δ) − (2 − δ)C_{n}(δ),

hence, we obtain

δ^{2}A_{n+1}(δ) − δ^{2}A_{n}(δ) = 2δ^{2}C_{n+1}(δ) − (2δ^{2} − δ^{3})C_{n}(δ).  (3.13)

On the other hand, by (3.11) we obtain

δ^{2}A_{n+1}(δ) − δ^{2}A_{n}(δ) = C_{n+3}(δ) − (2 − δ)C_{n+2}(δ) + (1 − δ)C_{n+1}(δ)− − C_{n+2}(δ) + (2 − δ)C_{n+1}(δ) − (1 − δ)C_{n}(δ) = = C_{n+3}(δ) − (3 − δ)C_{n+2}(δ) + (3 − 2δ)C_{n+1}(δ) + (δ − 1)C_{n}(δ). (3.14)

From (3.13) and (3.14) we obtain the final recurrence identities for numbers C_{n}(δ)

C_{n+3}(δ) + (δ − 3)C_{n+2}(δ) + (3 − 2δ − 2δ^{2})C_{n+1}(δ) + (−1 + δ + 2δ^{2} − δ^{3})C_{n}(δ) = 0. (3.15)

Remark 3.8. We note that the elements of each of the sequences {A_{n}(δ)} and {B_{n}(δ)} also satisfy this recurrence relation (which follows from (3.9) and (3.10)).

Lemma 3.9. The characteristic polynomial p_{7}(𝕏; δ) of (3.15) has the following decomposition:

p_{7}(𝕏; δ) := 𝕏^{3} + (δ − 3)𝕏^{2} + (3 − 2δ − 2δ^{2})𝕏 + (−1 + δ + 2δ^{2} − δ^{3}) = = (𝕏 − 1 − δ(ξ + ξ^{6}))(𝕏 − 1 − δ(ξ^{2} + ξ^{5}))(𝕏 − 1 − δ(ξ^{3} + ξ^{4})) (3.16)

where ξ = exp(i2π/7). So the relation

C_{n}(δ) = α(1 + δ(ξ + ξ^{6}))^{n} + β(1 + δ(ξ^{2} + ξ^{5}))^{n} + γ(1 + δ(ξ^{3} + ξ^{4}))^{n}

holds for every n ∈ ℕ and for some α, β, γ ∈ ℝ.

Remark 3.10. To find α, β, γ, from Lemma 3.9, it is sufficient to solve the following linear system:

\begin{cases}C_{1}(δ) = 0 = α(1 + δ(ξ + ξ^{6})) + β(1 + δ(ξ^{2} + ξ^{5})) + γ(1 + δ(ξ^{3} + ξ^{4})), \\ C_{2}(δ) = δ^{2} = α(1 + δ(ξ + ξ^{6}))^{2}+ β(1 + δ(ξ^{2} + ξ^{5}))^{2}+ γ(1 + δ(ξ^{3} + ξ^{4}))^{2}, \\ C_{3}(δ) = 3δ^{2}− δ^{3} = α(1 + δ(ξ + ξ^{6}))^{3}+ β(1 + δ(ξ^{2} + ξ^{5}))^{3}+ γ(1 + δ(ξ^{3} + ξ^{4}))^{3}. \end{cases}

After some calculations we obtain the identities

7C_{n}(δ) = (ξ^{2·1} + ξ^{2·6} − ξ^{3·1} − ξ^{3·6})(1 + δ(ξ + ξ^{6}))^{n}  (3.17)

+ (ξ^{2·2} + ξ^{2·5} − ξ^{3·2} − ξ^{3·5})(1 + δ(ξ^{2} + ξ^{5}))^{n} + (ξ^{2·3} + ξ^{2·4} − ξ^{3·3} − ξ^{3·4})(1 + δ(ξ^{3} + ξ^{4}))^{n} = (ξ^{2} + ξ^{5} − ξ^{3} − ξ^{4})(1 + δ(ξ + ξ^{6}))^{n} + (ξ^{3} + ξ^{4} − ξ − ξ^{6}) × (1 + δ(ξ^{2} + ξ^{5}))^{n} + (ξ + ξ^{6} − ξ^{2} − ξ^{5})(1 + δ(ξ^{3} + ξ^{4}))^{n}

= 2( cos \frac{4}{7}π + cos \frac{π}{7} )(1 + 2δ cos \frac{2π}{7} )^{n} − 2( cos \frac{π}{7} + cos \frac{2}{7}π)

× (1 + 2δ cos \frac{4π}{7} )^{n} + 2( cos \frac{2π}{7} − cos \frac{4π}{7} )(1 + 2δ cos \frac{6π}{7} )^{n}.

Hence, by (3.9) we get

7B_{n}(δ) = \frac{1}{δ}(7C_{n+1}(δ) − 7(1 − δ)C_{n}(δ))  (3.18)

= (ξ^{1·1} + ξ^{1·6} − ξ^{3·1} − ξ^{3·6})(1 + δ(ξ + ξ^{6}))^{n} + (ξ^{1·2} + ξ^{1·5} − ξ^{3·2} − ξ^{3·5})(1 + δ(ξ^{2} + ξ^{5}))^{n} + (ξ^{1·3} + ξ^{1·4} − ξ^{3·3} − ξ^{3·4})(1 + δ(ξ^{3} + ξ^{4}))^{n} = (ξ + ξ^{6} − ξ^{3} − ξ^{4})(1 + δ(ξ + ξ^{6}))^{n} + (ξ^{2} + ξ^{5} − ξ − ξ^{6}) × (1 + δ(ξ^{2} + ξ^{5}))^{n} + (ξ^{3} + ξ^{4} − ξ^{2} − ξ^{5})(1 + δ(ξ^{3} + ξ^{4}))^{n}

= 2 ( cos \frac{2π}{7} + cos \frac{π}{7} )(1 + 2δ cos \frac{2π}{7} )^{n} + 2 ( cos \frac{4π}{7} − cos \frac{2π}{7} )

× (1 + 2δ cos \frac{4π}{7} )^{n} − 2 ( cos \frac{π}{7} + cos \frac{4π}{7} )(1 + 2δ cos \frac{6π}{7} )^{n}

and by (3.10)

7A_{n}(δ) = \frac{1}{δ}(7B_{n+1}(δ) − 7B_{n}(δ)) = (ξ^{0·1} + ξ^{0·6} − ξ^{3·1} − ξ^{3·6})(1 + δ(ξ + ξ^{6}))^{n}+

+ (ξ^{0·2} + ξ^{0·5} − ξ^{3·2} − ξ^{3·5})(1 + δ(ξ^{2} + ξ^{5}))^{n}+
+ (ξ^{0·3} + ξ^{0·4} − ξ^{3·3} − ξ^{3·4})(1 + δ(ξ^{3} + ξ^{4}))^{n} =
= (2 − ξ^{3} − ξ^{4})(1 + δ(ξ + ξ^{6}))^{n} + (2 − ξ − ξ^{6})(1 + δ(ξ^{2} + ξ^{5}))^{n}+
+ (2 − ξ^{2} − ξ^{5})(1 + δ(ξ^{3} + ξ^{4}))^{n} =
= 2(1 + cos \frac{π}{7} )(1 + 2δ cos \frac{2π}{7} )^{n} + 2(1 − cos \frac{2π}{7} )(1 + 2δ cos \frac{4π}{7} )^{n}+

+ 2(1 − cos \frac{4π}{7} )(1 + 2δ cos \frac{6π}{7} )^{n}. (3.19)

Remark 3.11. In the sequel, for δ = 1, we obtain from (3.15)

C_{n+3} − 2C_{n+2} − C_{n+1} + C_{n} = 0  (3.20)

for every n ∈ ℕ. Of course, the elements of the sequences {B_{n}} and {A_{n}} satisfy the identity (3.20). The characteristic polynomial of (3.20) has the following decomposition: (see Corollary 2.4)

𝕏^{3} − 2𝕏^{2} − 𝕏 + 1 = (𝕏 − 1 − ξ − ξ^{6})(𝕏 − 1 − ξ^{2} − ξ^{5})(𝕏 − 1 − ξ^{3} − ξ^{4}),

where ξ = exp(i2π/7). Moreover, the decompositions of C_{n} (there are cyclic transformations of the sums ξ + ξ^{6}, ξ^{2} + ξ^{5} and ξ^{3} + ξ^{4} in elements of the first equality) follows immediately from (3.17)

7C_{n} = (ξ^{2} + ξ^{5} − ξ^{3} − ξ^{4})(1 + ξ + ξ^{6})^{n}+ + (ξ^{3} + ξ^{4} − ξ − ξ^{6})(1 + ξ^{2} + ξ^{5})^{n} + (ξ + ξ^{6} − ξ^{2} − ξ^{5})(1 + ξ^{3} + ξ^{4})^{n} =

= (ξ + ξ^{6} − ξ^{3} − ξ^{4})(1 + ξ + ξ^{6})^{n−1} + (ξ^{2} + ξ^{5} − ξ − ξ^{6})(1 + ξ^{2} + ξ^{5})^{n−1}+
+ (ξ^{3} + ξ^{4} − ξ^{2} − ξ^{5})(1 + ξ^{3} + ξ^{4})^{n−1} (3.21)

for every n = 1, 2, 3, … Lemma 3.12. The following summation formulas for the elements of the sequences {C_{n}(δ)}, {B_{n}(δ)} and {A_{n}(δ)} hold:

δ ∑_{n=1}^{N} A_{n}(δ) = B_{N+1}(δ) − δ;  (3.22)

δ ∑_{n=1}^{N} B_{n}(δ) = C_{N+1}(δ) + δ ∑_{n=1}^{N} C_{n}(δ);  (3.23)

hence, from (3.12) we obtain

A_{N+1}(δ) − A_{1}(δ) = 2δ ∑_{n=1}^{N} B_{n}(δ) − δ ∑_{n=1}^{N} C_{n}(δ) = 2C_{N+1}(δ) + δ ∑_{n=1}^{N} C_{n}(δ),  (3.24)

i.e.,

δ ∑_{n=1}^{N} C_{n}(δ) = −1 + A_{N+1}(δ) − 2C_{N+1}(δ);  (3.25)

7δ ∑_{n=1}^{N} C_{n}(δ) = −7 − 7C_{N+1}(δ) − 7B_{N+1}(δ)  (3.26)

+ (3 − \frac{2}{δ})A_{N+1}(δ) + \frac{2}{δ}A_{N+2}(δ);

δ^{3} ∑_{n=1}^{N} C_{n}(δ) = C_{N+3}(δ) + (δ − 2)C_{N+2}(δ) + (1 − δ − 2δ^{2})C_{N+1}(δ) − δ^{2} =

= C_{N+2}(δ) + (δ − 2)C_{N+1}(δ) + (δ^{3} − 2δ^{2} − δ + 1)C_{N}(δ) − δ^{2}; (3.27)

δ ∑_{n=1}^{N} B_{n}(δ) = −1 + A_{N+1}(δ) − C_{N+1}(δ)  (3.28)

and

δ^{3} ∑_{n=1}^{N} B_{n}(δ) = C_{N+3}(δ) + (δ − 2)C_{N+2}(δ) + (1 − δ − δ^{2})C_{N+1}(δ) − δ^{2}.  (3.29)

Definition 3.13. We define A_{n}(δ) := 3A_{n}(δ) − B_{n}(δ) − C_{n}(δ), δ ∈ ℂ, n ∈ ℕ. Moreover, we set A_{n} := A_{n}(1) (see Table 7 and \underline{A033304} in [7]).

Lemma 3.14. From identities (3.17–3.19) the following special identities can be deduced: a)

A_{n}^{2} = A_{2n} + 2(ξ + ξ^{6})^{n} + 2(ξ^{2} + ξ^{5})^{n} + 2(ξ^{3} + ξ^{4})^{n}

= A_{2n} + 2(2 cos \frac{2π}{7} )^{n} + 2(2 cos \frac{4π}{7} )^{n} + 2(2 cos \frac{6π}{7} )^{n}

or

(2 cos \frac{2π}{7} )^{n} + (2 cos \frac{4π}{7} )^{n} + (2 cos \frac{6π}{7} )^{n} = \frac{1}{2}(A_{n}^{2} − A_{2n}),

and

(4 cos \frac{2π}{7} cos \frac{4π}{7} )^{n} + (4 cos \frac{2π}{7} cos \frac{6π}{7} )^{n} + (4 cos \frac{4π}{7} cos \frac{6π}{7} )^{n} =

= (−1)^{n}[(1 + ξ + ξ^{6})^{n} + (1 + ξ^{2} + ξ^{5})^{n} + (1 + ξ^{3} + ξ^{4})^{n}] = (−1)^{n}A_{n}.

Consequently, we get the following decomposition:

(𝕏 − (2 cos \frac{2π}{7} )^{n})(𝕏 − (2 cos \frac{4π}{7} )^{n})(𝕏 − (2 cos \frac{6π}{7} )^{n}) =

= 𝕏^{3} − \frac{1}{2}(A_{n}^{2} − A_{2n})𝕏^{2} + (−1)^{n}A_{n}𝕏 − 1;

b)

2^{−n}A_{n}(\frac{1}{2}) = ( cos \frac{π}{7} )^{2n} + ( cos \frac{2π}{7} )^{2n} + ( cos \frac{3π}{7} )^{2n};

c)

A_{n}^{3} = A_{3n} + 6(−1)^{n} + 3A_{n}(−1) + 3(ξ + ξ^{6} − ξ^{2} − ξ^{5} − 1)^{n}
+ 3(ξ^{3} + ξ^{4} − ξ − ξ^{6} − 1)^{n} + 3(ξ^{2} + ξ^{5} − ξ^{3} − ξ^{4} − 1)^{n};

d)

7C_{n+1}A_{n} − 7C_{n}A_{n+1} = (−1 + ξ^{3} + ξ^{4} − 4(ξ^{2} + ξ^{5}))(ξ^{3} + ξ^{4})^{n}+

+ (−1 + ξ^{2} + ξ^{5} − 4(ξ + ξ^{6}))(ξ^{2} + ξ^{5})^{n} + (−1 + ξ + ξ^{6} − 4(ξ^{3} + ξ^{4}))(ξ + ξ^{6})^{n} =

= ( − 1 + 2 cos \frac{6π}{7} − 8 cos \frac{4π}{7} )(2 cos \frac{6π}{7} )^{n}+

+ ( − 1 + 2 cos \frac{4π}{7} − 8 cos \frac{2π}{7} )(2 cos \frac{4π}{7} )^{n}+

+( − 1 + 2 cos \frac{2π}{7} − 8 cos \frac{6π}{7} )(2 cos \frac{2π}{7} )^{n} = \{ \begin{matrix}7; & for & n = 2, 4; \\ −7; & for & n = 6; \\ −70; & for & n = 8; \\ 14; & for & n = 1; \\ 7 · A_{(n+1)/2}; & for & n = 3, 5, 7; \end{matrix}

e)

7(B_{n+1} + C_{n+1} − 2A_{n+1})(A_{n} + B_{n} − C_{n}) − 7(B_{n} + C_{n} − 2A_{n})(A_{n+1} + B_{n+1} − C_{n+1}) = = (8 + 3(ξ^{2} + ξ^{5}))(ξ + ξ^{6})^{n} + (8 + 3(ξ + ξ^{6}))(ξ^{3} + ξ^{4})^{n} + (8 + 3(ξ^{3} + ξ^{4}))(ξ^{2} + ξ^{5})^{n} =

= (8 + 6 cos \frac{4π}{7} )(2 cos \frac{2π}{7} )^{n} + (8 + 6 cos \frac{2π}{7} )(2 cos \frac{6π}{7} )^{n}+

+ (8 + 6 cos \frac{6π}{7} )(2 cos \frac{4π}{7} )^{n} = \{ \begin{matrix}−14; & for & n = 1; \\ 49; & for & n = 2; \\ −56; & for & n = 3, \end{matrix}

and a more general identity f)

(α7A_{n+1} + β7B_{n+1} + γ7C_{n+1})(ε7A_{n} + ω7B_{n} + φ7C_{n})−

− (α7A_{n} + β7B_{n} + γ7C_{n})(ε7A_{n+1} + ω7B_{n+1} + φ7C_{n+1}) = = ((α_{0} + β_{0}(ξ + ξ^{6}) + γ_{0}(ξ^{2} + ξ^{5}))(1 + ξ + ξ^{6})^{n+1}+ + (α_{0} + β_{0}(ξ^{2} + ξ^{5}) + γ_{0}(ξ^{3} + ξ^{4}))(1 + ξ^{2} + ξ^{5})^{n+1}+ + (α_{0} + β_{0}(ξ^{3} + ξ^{4}) + γ_{0}(ξ + ξ^{6}))(1 + ξ^{3} + ξ^{4})^{n+1})× × ((ε_{0} + ω_{0}(ξ + ξ^{6}) + φ_{0}(ξ^{2} + ξ^{5}))(1 + ξ + ξ^{6})^{n}+

+ (ε_{0} + ω_{0}(ξ^{2} + ξ^{5}) + φ_{0}(ξ^{3} + ξ^{4}))(1 + ξ^{2} + ξ^{5})^{n}+ + (ε_{0} + ω_{0}(ξ^{3} + ξ^{4}) + φ_{0}(ξ + ξ^{6}))(1 + ξ^{3} + ξ^{4})^{n})− − ((α_{0} + β_{0}(ξ + ξ^{6}) + γ_{0}(ξ^{2} + ξ^{5}))(1 + ξ + ξ^{6})^{n}+ + (α_{0} + β_{0}(ξ^{2} + ξ^{5}) + γ_{0}(ξ^{3} + ξ^{4}))(1 + ξ^{2} + ξ^{5})^{n}+ + (α_{0} + β_{0}(ξ^{3} + ξ^{4}) + γ_{0}(ξ + ξ^{6}))(1 + ξ^{3} + ξ^{4})^{n})× × ((ε_{0} + ω_{0}(ξ + ξ^{6}) + φ_{0}(ξ^{2} + ξ^{5}))(1 + ξ + ξ^{6})^{n+1}+ + (ε_{0} + ω_{0}(ξ^{2} + ξ^{5}) + φ_{0}(ξ^{3} + ξ^{4}))(1 + ξ^{2} + ξ^{5})^{n+1}+ + (ε_{0} + ω_{0}(ξ^{3} + ξ^{4}) + φ_{0}(ξ + ξ^{6}))(1 + ξ^{3} + ξ^{4})^{n+1}) = = (A + (ξ + ξ^{6})B + (ξ^{2} + ξ^{5})C)(ξ + ξ^{6})^{n}+ + (A + (ξ + ξ^{6})C + (ξ^{3} + ξ^{4})B)(ξ^{3} + ξ^{4})^{n}+ + (A + (ξ^{2} + ξ^{5})B + (ξ^{3} + ξ^{4})C)(ξ^{3} + ξ^{5})^{n} =

= (A + 2 cos \frac{2π}{7} B + 2 cos \frac{4π}{7} C)(2 cos \frac{2π}{7} )^{n}+

+ (A − 2 cos \frac{π}{7} B + 2 cos \frac{2π}{7} C)( − 2 cos \frac{π}{7} )^{n}+

+ (A + 2 cos \frac{4π}{7} B − 2 cos \frac{π}{7} C)(2 cos \frac{4π}{7} )^{n},

where

A := 5ε_{0}β_{0} − 5α_{0}ω_{0} + α_{0}φ_{0} − ε_{0}γ_{0} + 2ω_{0}γ_{0} − 2β_{0}φ_{0},
B := 5α_{0}ω_{0} − ε_{0}β_{0} + 3ε_{0}γ_{0} − 3α_{0}φ_{0} + γ_{0}ω_{0} − β_{0}φ_{0},
C := 2ε_{0}β_{0} − 2α_{0}ω_{0} + 2β_{0}φ_{0} − 2γ_{0}ω_{0} + ε_{0}γ_{0} − α_{0}φ_{0},

α_{0} := 3α + β + γ,  β_{0} := α + 2β + γ,  γ_{0} := α + β + 2γ,
ε_{0} := 3ε + ω + φ,  ω_{0} := ε + 2ω + φ,  φ_{0} := ε + ω + 2φ.

Proof. The identity b) follows from (3.6).

Remark 3.15. The sequence \{\frac{1}{2}(A_{n}^{2} − A_{2n})\}_{n=1}^{∞} (see Table 7) is an accelerator sequence for Catalan’s constant (see [1]). The generating function of these numbers has the form

\frac{3 + 2 x − 2 x^{2}}{1 + x − 2 x^{2} − x^{3}}

(see also \underline{A094648} in [7]).

Some applications of identities (3.1–3.3) will be presented now.

3.1 Reduction formulas for indices

Lemma 3.16. The following identities hold:

A_{m+n}(δ) = A_{m}(δ)A_{n}(δ) + 2B_{m}(δ)B_{n}(δ) + C_{m}(δ)C_{n}(δ)  (3.30)
− B_{m}(δ)C_{n}(δ) − B_{n}(δ)C_{m}(δ),
B_{m+n}(δ) = −C_{m}(δ)C_{n}(δ) + A_{m}(δ)B_{n}(δ) + A_{n}(δ)B_{m}(δ),  (3.31)
C_{m+n}(δ) = B_{m}(δ)B_{n}(δ) − C_{m}(δ)C_{n}(δ) + A_{m}(δ)C_{n}(δ)  (3.32)
+ A_{n}(δ)C_{m}(δ) − B_{m}(δ)C_{n}(δ) − B_{n}(δ)C_{m}(δ).

Proof. First, we note that

(1 + δ(ξ + ξ^{6}))^{m+n} = A_{m+n}(δ) + B_{m+n}(δ)(ξ + ξ^{6}) + C_{m+n}(δ)(ξ^{2} + ξ^{5}).

On the other hand, we have

(1+δ(ξ + ξ^{6}))^{m+n} = (1 + δ(ξ + ξ^{6}))^{m}(1 + δ(ξ + ξ^{6}))^{n} =
= (A_{m}(δ) + B_{m}(δ)(ξ + ξ^{6}) + C_{m}(δ)(ξ^{2} + ξ^{5}))×
× (A_{n}(δ) + B_{n}(δ)(ξ + ξ^{6}) + C_{n}(δ)(ξ^{2} + ξ^{5}))

= A_{m}(δ)A_{n}(δ) + B_{m}(δ)B_{n}(δ)(ξ^{2} + ξ^{5} + 2) + C_{m}(δ)C_{n}(δ)(ξ^{4} + ξ^{3} + 2)
+ A_{m}(δ)B_{n}(δ)(ξ + ξ^{6}) + A_{m}(δ)C_{n}(δ)(ξ^{2} + ξ^{5}) + A_{n}(δ)B_{m}(δ)(ξ + ξ^{6})
+ A_{n}(δ)C_{m}(δ)(ξ^{2} + ξ^{5}) + B_{m}(δ)C_{n}(δ)(ξ + ξ^{6})(ξ^{2} + ξ^{5})
+ B_{n}(δ)C_{m}(δ)(ξ + ξ^{6})(ξ^{2} + ξ^{5})
(we have (ξ + ξ^{6})(ξ^{2} + ξ^{5}) = ξ^{3} + ξ^{6} + ξ + ξ^{4} = −1 − ξ^{2} − ξ^{5})
= A_{m}(δ)A_{n}(δ) + 2B_{m}(δ)B_{n}(δ) + C_{m}(δ)C_{n}(δ) − B_{m}(δ)C_{n}(δ) − B_{n}(δ)C_{m}(δ)
+ (ξ + ξ^{6})(A_{m}(δ)B_{n}(δ) + A_{n}(δ)B_{m}(δ) − C_{m}(δ)C_{n}(δ))
+ (ξ^{2} + ξ^{5})(B_{m}(δ)B_{n}(δ) − C_{m}(δ)C_{n}(δ) + A_{m}(δ)C_{n}(δ)
+ A_{n}(δ)C_{m}(δ) − B_{n}(δ)C_{m}(δ) − B_{m}(δ)C_{n}(δ))

hence by linear independence of 1, ξ + ξ^{6} = 2 cos \frac{2π}{7}, ξ^{2} + ξ^{5} = 2 cos \frac{4π}{7} over ℚ and by Lemma 2.8 the reduction formulas follow.

Corollary 3.17. We have

A_{2n}(δ) = A_{n}^{2}(δ) + B_{n}^{2}(δ) + (B_{n}(δ) − C_{n}(δ))^{2},  (3.33)
B_{2n}(δ) = 2A_{n}(δ)B_{n}(δ) − C_{n}^{2}(δ),  (3.34)
C_{2n}(δ) = B_{n}^{2}(δ) − C_{n}^{2}(δ) + 2C_{n}(δ)(A_{n}(δ) − B_{n}(δ)).  (3.35)

Remark 3.18. (Which comes from [7], sequence \underline{A006356}) Let u(k), v(k), w(k) be defined by u(1) = 1, v(1) = 0, w(1) = 0 and

u(k + 1) = u(k) + v(k) + w(k),
v(k + 1) = u(k) + v(k),
w(k + 1) = u(k).

Then we have

u(n + 1) = A_{n}, v(n + 1) = B_{n}, w(n + 1) = B_{n} − C_{n};

for every n ∈ ℕ ∪ {0}. So, according to Table 2 we get

{u(n)} = 1, 1, 3, 6, 14, 31, …,
{v(n)} = 0, 1, 2, 5, 11, 25, …,

and

{w(n)} = {u(n)}

(Benoit Cloitre (abcloitre@wanadoo.fr), Apr 05 2002). Moreover

(u(k))^{2} + (v(k))^{2} + (w(k))^{2} = u(2k) (Gary Adamson, Dec 23 2003).

The n-th term of the sequence {u(n)} is the number of paths for a ray of light that enters two layers of glass, and then is reflected exactly n times before leaving the layers of glass. One such path (with 2 plates of glass and 3 reflections) might be

                             ...\........./...
                             -----------------
                             ....\/\..../.....
                             -----------------
                             ........\/.......
                             -----------------

A_{3n}(δ) = A_{n}(δ)A_{2n}(δ) − B_{n}(δ)C_{2n}(δ) − B_{2n}(δ)C_{n}(δ)  (3.36)

+ 2B_{n}(δ)B_{2n}(δ) + C_{n}(δ)C_{2n}(δ)

= A_{n}^{3}(δ) − B_{n}^{3}(δ) + 6A_{n}(δ)B_{n}^{2}(δ) + 3A_{n}(δ)C_{n}^{2}(δ) − 3B_{n}(δ)C_{n}^{2}(δ)

+ 3B_{n}^{2}(δ)C_{n}(δ) − 6A_{n}(δ)B_{n}(δ)C_{n}(δ),

B_{3n}(δ) = A_{2n}(δ)B_{n}(δ) + A_{n}(δ)B_{2n}(δ) − C_{n}(δ)C_{2n}(δ)  (3.37)

= 2B_{n}^{3}(δ) + C_{n}^{3}(δ) + 3A_{n}^{2}(δ)B_{n}(δ) + 3A_{n}(δ)C_{n}^{2}(δ)

+ 3B_{n}(δ)C_{n}^{2}(δ) − 3B_{n}^{2}(δ)C_{n}(δ),

C_{3n}(δ) = B_{n}vB_{2n}(δ) + A_{n}(δ)C_{2n}(δ) + A_{2n}(δ)C_{n}(δ) − C_{n}(δ)C_{2n}(δ)  (3.38)
− B_{n}(δ)C_{2n}(δ) − B_{2n}(δ)C_{n}(δ)

= 3C_{n}^{3}(δ) − B_{n}^{3}(δ) + 3A_{n}^{2}(δ)C_{n}(δ) + 3A_{n}(δ)B_{n}^{2}(δ) + 3B_{n}^{2}(δ)C_{n}(δ)

− 6A_{n}(δ)B_{n}(δ)C_{n}(δ).

Lemma 3.20. The following identities are satisfied:

A_{n−k}(δ) = \frac{1}{Δ} | \begin{matrix}A_{n}(δ) & 2B_{k}(δ) − C_{k}(δ) & C_{k}(δ) − B_{k}(δ) \\ B_{n}(δ) & A_{k}(δ) & −C_{k}(δ) \\ C_{n}(δ) & B_{k}(δ) − C_{k}(δ) & A_{k}(δ) − B_{k}(δ) − C_{k}(δ) \end{matrix} | ,  (3.39)

B_{n−k}(δ) = \frac{1}{Δ} | \begin{matrix}A_{k}(δ) & A_{n}(δ) & C_{k}(δ) − B_{k}(δ) \\ B_{k}(δ) & B_{n}(δ) & −C_{k}(δ) \\ C_{k}(δ) & C_{n}(δ) & A_{k}(δ) − B_{k}(δ) − C_{k}(δ) \end{matrix} | ,  (3.40)

and

C_{n−k}(δ) = \frac{1}{Δ} | \begin{matrix}A_{k}(δ) & 2B_{k}(δ) − C_{k}(δ) & A_{n}(δ) \\ B_{k}(δ) & A_{k}(δ) & B_{n}(δ) \\ C_{k}(δ) & B_{k}(δ) − C_{k}(δ) & C_{n}(δ) \end{matrix} | ,  (3.41)

where

Δ := | \begin{matrix}A_{k}(δ) & 2B_{k}(δ) − C_{k}(δ) & C_{k}(δ) − B_{k}(δ) \\ B_{k}(δ) & A_{k}(δ) & −C_{k}(δ) \\ C_{k}(δ) & B_{k}(δ) − C_{k}(δ) & A_{k}(δ) − B_{k}(δ) − C_{k}(δ) \end{matrix} | .

Proof. First we note that

(1 + δ(ξ + ξ^{6}))^{n−k} = A_{n−k}(δ) + B_{n−k}(δ)(ξ + ξ^{6}) + C_{n−k}(δ)(ξ^{2} + ξ^{5}).

On the other hand we obtain

(1 + δ(ξ + ξ^{6}))^{n−k} = \frac{(1 + δ(ξ + ξ^{6}))^{n}}{(1 + δ(ξ + ξ^{6}))^{k}} = \frac{A_{n}(δ) + B_{n}(δ)(ξ + ξ^{6}) + C_{n}(δ)(ξ^{2} + ξ^{5})}{A_{k}(δ) + B_{k}(δ)(ξ + ξ^{6}) + C_{k}(δ)(ξ^{2} + ξ^{5})}.

The final form of formulas (3.39), (3.40) and (3.41) from the following identity could be derived:

\frac{a + b(ξ + ξ^{6}) + c(ξ^{2} + ξ^{5})}{d + e(ξ + ξ^{6}) + f(ξ^{2} + ξ^{5})} = α + β(ξ + ξ^{6}) + γ(ξ^{2} + ξ^{5}),

where

α := \frac{1}{D} | \begin{matrix}a & 2e − f & f − e \\ b & d & −f \\ c & e − f & d − e − f \end{matrix} | ,  β := \frac{1}{D} | \begin{matrix}d & a & f − e \\ e & b & −f \\ f & c & d − e − f \end{matrix} | ,
γ := \frac{1}{D} | \begin{matrix}d & 2e − f & a \\ e & d & b \\ f & e − f & c \end{matrix} |  and  D := | \begin{matrix}d & 2e − f & f − e \\ e & d & −f \\ f & e − f & d − e − f \end{matrix} | .

Lemma 3.21. The following identity for the determinant Δ from Lemma 3.20 holds:

Δ = | \begin{matrix}A_{k}(δ) & 2B_{k}(δ) − C_{k}(δ) & C_{k}(δ) − B_{k}(δ) \\ B_{k}(δ) & A_{k}(δ) & −C_{k}(δ) \\ C_{k}(δ) & B_{k}(δ) − C_{k}(δ) & A_{k}(δ) − B_{k}(δ) − C_{k}(δ) \end{matrix} | =

= (A_{k}(δ))^{3} + (B_{k}(δ))^{3} + (C_{k}(δ))^{3} + 3 A_{k}(δ) B_{k}(δ) C_{k}(δ) − (A_{k}(δ))^{2}(B_{k}(δ) + C_{k}(δ))−

− 2 A_{k}(δ)((B_{k}(δ))^{2} + (C_{k}(δ))^{2}) + 3 (B_{k}(δ))^{2} C_{k}(δ) − 4 B_{k}(δ) (C_{k}(δ))^{2} =
= (δ^{3} − 2 δ^{2} − δ + 1)^{k}. (3.42)

Proof. By (3.16) we obtain

[(1 + δ(ξ + ξ^{6}))(1 + δ(ξ^{2} + ξ^{5}))(1 + δ(ξ^{3} + ξ^{4}))]^{k} = (δ^{3} − 2 δ^{2} − δ + 1)^{k}. (3.43) On the other hand, by (3.1)–(3.3) we get

((1 + δ(ξ + ξ^{6}))(1 + δ(ξ^{2} + ξ^{5}))(1 + δ(ξ^{3} + ξ^{4})))^{k} =

= (1 + δ(ξ + ξ^{6}))^{k}(1 + δ(ξ^{2} + ξ^{5}))^{k}(1 + δ(ξ^{3} + ξ^{4}))^{k} =

= (A_{k}(δ) + B_{k}(δ)(ξ + ξ^{6}) + C_{k}(δ)(ξ^{2} + ξ^{5}))(A_{k}(δ) + B_{k}(δ)(ξ^{2} + ξ^{5}) + C_{k}(δ)(ξ^{3} + ξ^{4}))×

× (A_{k}(δ) + B_{k}(δ)(ξ^{3} + ξ^{4}) + C_{k}(δ)(ξ + ξ^{6})) =

= (A_{k}(δ))^{3} + a (B_{k}(δ))^{3} + a (C_{k}(δ))^{3} + b A_{k}(δ) B_{k}(δ) C_{k}(δ) +
+ c (A_{k}(δ))^{2}(B_{k}(δ) + C_{k}(δ)) + d A_{k}(δ)((B_{k}(δ))^{2} + (C_{k}(δ))^{2})+
+ e (B_{k}(δ))^{2} C_{k}(δ) + f B_{k}(δ) (C_{k}(δ))^{2}, (3.44)

where (see Lemma 2.3)

a = (ξ + ξ^{6})(ξ^{2} + ξ^{5})(ξ^{3} + ξ^{4}) = 1,
b = (ξ + ξ^{6})(ξ^{2} + ξ^{5}) + (ξ + ξ^{6})(ξ^{3} + ξ^{4}) + (ξ^{2} + ξ^{5})(ξ^{3} + ξ^{4})+
+ (ξ + ξ^{6})^{2} + (ξ^{2} + ξ^{5})^{2} + (ξ^{3} + ξ^{4})^{2} = 3,
c = ξ + ξ^{2} + ξ^{3} + ξ^{4} + ξ^{5} + ξ^{6} = −1,
d = (ξ + ξ^{6})(ξ^{2} + ξ^{5}) + (ξ + ξ^{6})(ξ^{3} + ξ^{4}) + (ξ^{2} + ξ^{5})(ξ^{3} + ξ^{4}) = −2,
e = (ξ + ξ^{6})^{2}(ξ^{2} + ξ^{5}) + (ξ + ξ^{6})(ξ^{3} + ξ^{4})^{2} + (ξ^{2} + ξ^{5})^{2}(ξ^{3} + ξ^{4}) = 3,
f = (ξ + ξ^{6})(ξ^{2} + ξ^{5})^{2} + (ξ + ξ^{6})^{2}(ξ^{3} + ξ^{4}) + (ξ^{2} + ξ^{5})(ξ^{3} + ξ^{4})^{2} = −4.

Hence (3.42) follows from (3.43) and (3.44).

3.2 Reduction formulas for δ-arguments

Next, our aim will be to prove some more general identities connecting the quasi-Fibonacci numbers with different δ’s (δ ≠ 2). Let us start with the following formula:

(2 + δ(ξ^{2} + ξ^{5}) + δ(ξ^{3} + ξ^{4}))^{n} = ((2 − δ) − δ(ξ + ξ^{6}))^{n} =

= (2 − δ)^{n}(1 + \frac{δ}{δ − 2}(ξ + ξ^{6}))^{n} = (2 − δ)^{n}A_{n}(\frac{δ}{δ − 2})+

+ (2 − δ)^{n}B_{n}(\frac{δ}{δ − 2})(ξ + ξ^{6}) + (2 − δ)^{n}C_{n}(\frac{δ}{δ − 2})(ξ^{2} + ξ^{5}). (3.45)

But, we can deduce another equivalent formula

(2 + δ(ξ^{2} + ξ^{5}) + δ(ξ^{3} + ξ^{4}))^{n} = ((1 + δ(ξ^{2} + ξ^{5})) + (1 + δ(ξ^{3} + ξ^{4})))^{n}  (3.46)

= ∑_{k=0}^{n} \binom{n}{k}(1 + δ(ξ^{2} + ξ^{5}))^{k}(1 + δ(ξ^{3} + ξ^{4}))^{n−k}

= ∑_{k=0}^{n} \binom{n}{k}(A_{k}(δ) + B_{k}(δ)(ξ^{2} + ξ^{5}) + C_{k}(δ)(ξ^{3} + ξ^{4}))×

× (A_{n−k}(δ) + B_{n−k}(δ)(ξ^{3} + ξ^{4}) + C_{n−k}(δ)(ξ + ξ^{6}))

= ∑_{k=0}^{n} \binom{n}{k}(A_{k}(δ)A_{n−k}(δ) + A_{k}(δ)B_{n−k}(δ)(ξ^{3} + ξ^{4}) + A_{k}(δ)C_{n−k}(δ)(ξ + ξ^{6})

+ A_{n−k}(δ)B_{k}(δ)(ξ^{2} + ξ^{5}) + B_{k}(δ)B_{n−k}(δ)(ξ^{5} + ξ^{6} + ξ + ξ^{2}) + B_{k}(δ)C_{n−k}(δ)(ξ^{3} + ξ + ξ^{6} + ξ^{4}) + A_{n−k}(δ)C_{k}(δ)(ξ^{3} + ξ^{4})

+ B_{n−k}(δ)C_{k}(δ)(ξ + ξ^{6} + 2) + C_{k}(δ)C_{n−k}(δ)(ξ^{4} + ξ^{5} + ξ^{2} + ξ^{3})) =

= ∑_{k=0}^{n} \binom{n}{k}(A_{k}(δ)A_{n−k}(δ) − A_{k}(δ)B_{n−k}(δ) − B_{k}(δ)C_{n−k}(δ) − A_{n−k}(δ)C_{k}(δ)

+ 2B_{n−k}(δ)C_{k}(δ) − C_{k}(δ)C_{n−k}(δ))

+ (ξ + ξ^{6}) ∑_{k=0}^{n} \binom{n}{k}(−A_{k}(δ)B_{n−k}(δ) + A_{k}(δ)C_{n−k}(δ) + B_{k}(δ)B_{n−k}(δ)

− A_{n−k}(δ)C_{k}(δ) + B_{n−k}(δ)C_{k}(δ) − C_{k}(δ)C_{n−k}(δ))

+ (ξ^{2} + ξ^{5}) ∑_{k=0}^{n} \binom{n}{k}(−A_{k}(δ)B_{n−k}(δ) + A_{n−k}(δ)B_{k}(δ)

+ B_{k}(δ)B_{n−k}(δ) − B_{k}(δ)C_{n−k}(δ) − A_{n−k}(δ)C_{k}(δ)).

From (3.45) and (3.46), again by linear independence of 1, ξ + ξ^{6} = 2 cos \frac{2}{7}π and ξ^{2} + ξ^{5} = 2 cos \frac{4}{7}π over ℚ and by Lemma 2.8 the identities below can be generated:

Lemma 3.22. We have

(2 − δ)^{n}A_{n}(\frac{δ}{δ − 2}) = ∑_{k=0}^{n} \binom{n}{k}(A_{k}(δ)A_{n−k}(δ) − A_{k}(δ)B_{n−k}(δ)  (3.47)
− B_{k}(δ)C_{n−k}(δ) − A_{n−k}(δ)C_{k}(δ) + 2B_{n−k}(δ)C_{k}(δ) − C_{k}(δ)C_{n−k}(δ)),

(2 − δ)^{n}B_{n}(\frac{δ}{δ − 2}) = ∑_{k=0}^{n} \binom{n}{k}( − A_{k}(δ)B_{n−k}(δ) + A_{k}(δ)C_{n−k}(δ)  (3.48)
+ B_{k}(δ)B_{n−k}(δ) − A_{n−k}(δ)C_{k}(δ) + B_{n−k}(δ)C_{k}(δ) − C_{k}(δ)C_{n−k}(δ))

and

(2 − δ)^{n}C_{n}(\frac{δ}{δ − 2}) = ∑_{k=0}^{n} \binom{n}{k}( − A_{k}(δ)B_{n−k}(δ)  (3.49)
+ A_{n−k}(δ)B_{k}(δ) + B_{k}(δ)B_{n−k}(δ) − B_{k}(δ)C_{n−k}(δ) − A_{n−k}(δ)C_{k}(δ)).

In the sequel, from the above formulas the following special identities can be obtained: the case δ = 1

A_{n}(−1) = ∑_{k=0}^{n} \binom{n}{k}(A_{k}A_{n−k} − A_{k}B_{n−k} − B_{k}C_{n−k} − A_{n−k}C_{k}  (3.50)
+ 2B_{n−k}C_{k} − C_{k}C_{n−k}),
B_{n}(−1) = ∑_{k=0}^{n} \binom{n}{k}( − A_{k}B_{n−k} + A_{k}C_{n−k} + B_{k}B_{n−k} − A_{n−k}C_{k}  (3.51)
+ B_{n−k}C_{k} − C_{k}C_{n−k}),
C_{n}(−1) = ∑_{k=0}^{n} \binom{n}{k}( − A_{k}B_{n−k} + A_{n−k}B_{k} + B_{k}B_{n−k}  (3.52)
− B_{k}C_{n−k} − A_{n−k}C_{k});

the case δ = 3

( − 1)^{n}A_{n}(3) = ∑_{k=0}^{n} \binom{n}{k}(A_{k}(3)A_{n−k}(3) − A_{k}(3)B_{n−k}(3)−  (3.53)

− B_{k}(3)C_{n−k}(3) − A_{n−k}(3)C_{k}(3) + 2B_{n−k}(3)C_{k}(3) − C_{k}(3)C_{n−k}(3)),

( − 1)^{n}B_{n}(3) = ∑_{k=0}^{n} \binom{n}{k}( − A_{k}(3)B_{n−k}(3) + A_{k}(3)C_{n−k}(3)+  (3.54)

+ B_{k}(3)B_{n−k}(3) − A_{n−k}(3)C_{k}(3) + B_{n−k}(3)C_{k}(3) − C_{k}(3)C_{n−k}(3)),

( − 1)^{n}C_{n}(3) = ∑_{k=0}^{n} \binom{n}{k}( − A_{k}(3)B_{n−k}(3) + A_{n−k}(3)B_{k}(3)+  (3.55)

+ B_{k}(3)B_{n−k}(3) − B_{k}(3)C_{n−k}(3) − A_{n−k}(3)C_{k}(3)).

The case δ = 2 must be treated in another way

(2 + 2(ξ^{2} + ξ^{5}) + 2(ξ^{3} + ξ^{4}))^{2n} = ( − 2(ξ + ξ^{6}))^{2n} =

= 2^{2n}(2 + (ξ^{2} + ξ^{5}))^{n} = 2^{3n}(1 + \frac{1}{2}(ξ^{2} + ξ^{5}))^{n} =

= 2^{3n}(A_{n}(\frac{1}{2}) + B_{n}(\frac{1}{2})(ξ^{2} + ξ^{5}) + C_{n}(\frac{1}{2})(ξ^{3} + ξ^{4})) =

= 2^{3n}(A_{n}(\frac{1}{2}) + B_{n}(\frac{1}{2})(ξ^{2} + ξ^{5}) + C_{n}(\frac{1}{2})(−1 − ξ − ξ^{6} − ξ^{2} − ξ^{5})) =

= 2^{3n}(A_{n}(\frac{1}{2}) − C_{n}(\frac{1}{2}) + (B_{n}(\frac{1}{2}) − C_{n}(\frac{1}{2}))(ξ^{2} + ξ^{5}) − C_{n}(\frac{1}{2})(ξ + ξ^{6})),

which, by (3.46), implies the next five identities

2^{3n}(A_{n}(\frac{1}{2}) − C_{n}(\frac{1}{2})) = ∑_{k=0}^{2n} \binom{2n}{k}(A_{k}(2)A_{2n−k}(2) − A_{k}(2)B_{2n−k}(2)−  (3.56)

− B_{k}(2)C_{2n−k}(2) − A_{2n−k}(2)C_{k}(2) + 2B_{2n−k}(2)C_{k}(2) − C_{k}(2)C_{2n−k}(2)),

2^{3n}(B_{n}(\frac{1}{2}) − C_{n}(\frac{1}{2})) = ∑_{k=0}^{2n} \binom{2n}{k}( − A_{k}(2)B_{2n−k}(2) − A_{2n−k}(2)B_{k}(2)+  (3.57)

+ B_{k}(2)B_{2n−k}(2) − B_{k}(2)C_{2n−k}(2) − A_{2n−k}(2)C_{k}(2)),

−2^{3n}C_{n}(\frac{1}{2}) = ∑_{k=0}^{2n} \binom{2n}{k}( − A_{k}(2)B_{2n−k}(2) + A_{k}(2)C_{2n−k}(2)+  (3.58)

+ B_{k}(2)B_{2n−k}(2) − A_{2n−k}(2)C_{k}(2) − B_{2n−k}(2)C_{k}(2) − C_{k}(2)C_{2n−k}(k)),

hence, after some manipulations, we obtain

2^{3n}A_{n}(\frac{1}{2}) = ∑_{k=0}^{2n} \binom{2n}{k}(A_{k}(2)A_{2n−k}(2) − A_{k}(2)C_{2n−k}(2)−  (3.59)

− B_{k}(2)B_{2n−k}(2) − B_{k}(2)C_{2n−k}(2) + B_{2n−k}(2)C_{k}(2)),

2^{3n}B_{n}(\frac{1}{2}) = ∑_{k=0}^{2n} \binom{2n}{k}( − A_{k}(2)C_{2n−k}(2) + A_{2n−k}(2)B_{k}(2)−  (3.60)

− B_{k}(2)C_{2n−k}(2) − B_{2n−k}(2)C_{k}(2) + C_{k}(2)C_{2n−k}(2)).

We have also the identity

(1 + ξ + ξ^{6})(1 + δ(ξ^{2} + ξ^{5})) = (1 + ξ + ξ^{6}) − δ,

i.e.,

(1 + ξ + ξ^{6})^{n}(1 + δ(ξ^{2} + ξ^{5}))^{n} = ((1 + ξ + ξ^{6}) − δ)^{n},

which, by (3.1) and (3.2), yields

(A_{n} + B_{n}(ξ + ξ^{6}) + C_{n}(ξ^{2} + ξ^{5}))(A_{n}(δ) + B_{n}(δ)(ξ^{2} + ξ^{5}) + C_{n}(δ)(ξ^{3} + ξ^{4})) =

= ∑_{k=0}^{n}(−δ)^{n−k}\binom{n}{k}(1 + ξ + ξ^{6})^{k} = ∑_{k=0}^{n}(−δ)^{n−k}\binom{n}{k}A_{k}+

+ (ξ + ξ^{6}) ∑_{k=0}^{n}(−δ)^{n−k}\binom{n}{k}B_{k} + (ξ^{2} + ξ^{5}) ∑_{k=0}^{n}(−δ)^{n−k}\binom{n}{k}C_{k}.

Hence, by Lemma 2.8, the following three identities can be discovered:

A_{n}A_{n}(δ) − A_{n}C_{n}(δ) − B_{n}B_{n}(δ) − B_{n}C_{n}(δ) + C_{n}B_{n}(δ) =  (3.61)

= ∑_{k=0}^{n}(−δ)^{n−k}\binom{n}{k}A_{k},

−A_{n}C_{n}(δ) + B_{n}A_{n}(δ) − B_{n}C_{n}(δ) − C_{n}B_{n}(δ) + C_{n}C_{n}(δ) =  (3.62)

= ∑_{k=0}^{n}(−δ)^{n−k}\binom{n}{k}B_{k},

A_{n}B_{n}(δ) − A_{n}C_{n}(δ) − B_{n}B_{n}(δ) + C_{n}A_{n}(δ) − C_{n}B_{n}(δ) + C_{n}C_{n}(δ) =  (3.63)

= ∑_{k=0}^{n}(−δ)^{n−k}\binom{n}{k}C_{k}.

3.3 Polynomials associated with quasi-Fibonacci numbers

of order 7

Directly from equation (3.1) we obtain (for x_{k} := 1 + 2δ cos \frac{2kπ}{7}, k = 1, 2, 3)

x_{k}^{n} = A_{n}(δ) + B_{n}(δ)(2 cos \frac{2kπ}{7} ) + C_{n}(δ)(2 cos \frac{4kπ}{7} )

= A_{n}(δ) + \frac{B_{n}(δ)}{δ} x_{k} − \frac{B_{n}(δ)}{δ} + \frac{C_{n}(δ)}{δ^{2}}(2δ^{2}(2 cos^{2} \frac{2kπ}{7} − 1))

= A_{n}(δ) + \frac{B_{n}(δ)}{δ} x_{k} − \frac{B_{n}(δ)}{δ} + \frac{C_{n}(δ)}{δ^{2}}(x_{k}^{2} − 2 x_{k} + 1 − 2δ^{2})

= A_{n}(δ) − \frac{B_{n}(δ)}{δ} − \frac{C_{n}(δ)}{δ^{2}}(2δ^{2} − 1) + \frac{B_{n}(δ)}{δ}x_{k} − 2\frac{C_{n}(δ)}{δ^{2}}x_{k} + \frac{C_{n}(δ)}{δ^{2}}x_{k}^{2},

i.e.,

W_{n,7}(x; δ) := δ^{2}x^{n} − C_{n}(δ)x^{2} + (2C_{n}(δ) − δB_{n}(δ))x+

+ δB_{n}(δ) + (2δ^{2} − 1)C_{n}(δ) − δ^{2}A_{n}(δ) = 0

for x = x_{k}, k = 1, 2, 3. So, by identity (3.16) the polynomial p_{7}(𝕏; δ) is a divisor of the polynomial W_{n,7}(𝕏; δ) (for every n ⩾ 3). In the sequel we obtain (for δ = 1)

(x^{3} − 2x^{2} − x + 1)|(x^{n} − C_{n}x^{2} + (2C_{n} − B_{n})x + B_{n} + C_{n} − A_{n})  (3.64)

for every n ∈ ℕ, n ⩾ 3. More precisely, the following decomposition holds:

p_{7}(𝕏; δ)( ∑_{k=1}^{n−2} C_{k+1}(δ)𝕏^{n−2−k}) = W_{n,7}(𝕏; δ),  (3.65)

hence, for δ = 1, we obtain

(x^{3} − 2x^{2} − x + 1)( ∑_{k=1}^{n−2} B_{k}x^{n−2−k}) = x^{n} − C_{n}x^{2} + (2C_{n} − B_{n})x + B_{n} + C_{n} − A_{n}. (3.66)

Remark 3.23. Equation (3.65) after differentiating and some manipulating enables us to generate the sums formulas of the following form:

(p_{7}(𝕏; δ))^{r+1}( ∑_{k=r}^{n} k^{r}C_{k}(δ)𝕏^{n−k−r}) = polynomial depending on W_{n,7}(𝕏; δ), p_{7}(𝕏; δ) and their derivatives

For example, the identity (3.65) implies the identity (3.25) (for 𝕏 = 1).

Remark 3.24. There exist other ways to obtain the relation (3.64). For example, by (3.5) we have

αA_{n+1}(δ) + βB_{n+1}(δ) + γC_{n+1}(δ) = = (α + β δ)A_{n}(δ) + (2 α δ + β + γ δ)B_{n}(δ) + ( − δ α + (1 − δ)γ)C_{n}(δ). (3.67)

We are interested when the following system of equations holds:

\frac{α + β δ}{α} = \frac{2 α δ + β + γ δ}{β} = \frac{−δ α + (1 − δ)γ}{γ} ,

which is equivalent to the following one:

\{ \begin{matrix}γ = \frac{1}{α}(β^{2} − 2 α^{2}), \\ (\frac{β}{α})^{3} + (\frac{β}{α})^{2} − 2(\frac{β}{α}) − 1 = 0. \end{matrix}

Hence, by Remark 3.11, it follows that

\frac{β}{α} ∈ \{1 + 2 cos \frac{2kπ}{7} : k = 1, 2, 3\}.

So the equation (3.67) then has the following form:

A_{n+1}(δ) + 2 cos \frac{2kπ}{7} B_{n+1}(δ) − (1 + 2 cos \frac{2kπ}{7} )^{−1}C_{n+1}(δ) =

= (1 + 2 δ cos \frac{2kπ}{7} )(A_{n}(δ) + 2 cos \frac{2kπ}{7} B_{n}(δ) − (1 + 2 cos \frac{2kπ}{7} )^{−1}C_{n}(δ)) =

= (1 + 2 δ cos \frac{2kπ}{7} )^{n}(A_{1}(δ) + 2 cos \frac{2kπ}{7} B_{1}(δ) − (1 + 2 cos \frac{2kπ}{7} )^{−1}C_{1}(δ)) =

(by (3.5))

= (1 + 2 δ cos \frac{2kπ}{7} )^{n+1}

or

δ^{2} x^{n+1} = δ(x + δ − 1) A_{n}(δ) + (x + δ − 1)(x − 1) B_{n}(δ) − δ^{2}C_{n}(δ),

i.e.,

δ^{2} x^{n+1} − x^{2} B_{n}(δ) + x((2 − δ) B_{n}(δ) − δ A_{n}(δ))+

+ δ (1 − δ) A_{n}(δ) + (δ − 1) B_{n}(δ) + δ^{2} C_{n}(δ) = 0

for x := 1 + 2δ cos \frac{2kπ}{7}, k = 1, 2, 3, which is equivalent to

W_{n,7}(𝕏; δ) = 0.

Lemma 3.25. Immediately from (3.66) we obtain the following identities:

(x^{3} − 2x^{2} − x + 1)^{2} · ( ∑_{k=1}^{n−2} B_{k}x^{n−2−k})′ =

= (x^{3} − 2x^{2} − x + 1)^{2} · ∑_{k=1}^{n−3}(n − 2 − k)B_{k}x^{n−3−k} =

= (n − 3)x^{n+2} − 2(n − 2)x^{n+1} − (n − 1)x^{n} + nx^{n−1} + C_{n}x^{4}+ + 2(B_{n} − 2C_{n})x^{3} + (3A_{n} − 5B_{n} + 2C_{n})x^{2} + (−4A_{n} + 4B_{n} + 2C_{n})x − A_{n} + 3C_{n}, (3.68)

and

(x^{3} − 2x^{2} − x + 1)^{3} · ( ∑_{k=1}^{n−2} B_{k}x^{n−2−k})′′ = (n^{2} − 7n + 12)x^{n+4}+

+ (−4n^{2} + 24n − 32)x^{n+3} + (2n^{2} − 10n + 18)x^{n+2} + (6n^{2} − 24n + 6)x^{n+1}+ + (−3n^{2} + 9n + 6)x^{n} + (2n − 2n^{2})x^{n−1} + (n^{2} − n)x^{n−2} − 2C_{n}x^{6}+ + (−6B_{n} + 12C_{n})x^{5} + (−12A_{n} + 24B_{n} − 18C_{n})x^{4} + (32A_{n} − 42B_{n} + 6C_{n})x^{3}+ + (−18A_{n} + 30B_{n} − 18C_{n})x^{2} + (−6A_{n} − 6B_{n} + 30C_{n})x − 6A_{n} + 4B_{N} + 8C_{n}. (3.69)

Corollary 3.26. We have

∑_{k=1}^{n−2}B_{k} = A_{n} − 2C_{n} − 1,
∑_{k=1}^{n−2}(−1)^{k}B_{k} = −1 + (−1)^{n}(A_{n} − 2B_{n} + 2C_{n}),
∑_{k=1}^{n−3}(n − 2 − k)B_{k} = −2(A_{n} − 2C_{n} − 1) − n + B_{n},
∑_{k=1}^{n−2}kB_{k} = nA_{n} − B_{n} − 2nC_{n},
∑_{k=1}^{n−3}(n − 2 − k)(−1)^{k}B_{k} = 6 − n + (−1)^{n−1}(6A_{n} − 11B_{n} + 8C_{n}),

and

∑_{k=1}^{n−3}(−1)^{k}kB_{k} = −4 + (−1)^{n}(6A_{n} − 11B_{n} + 8C_{n}+
+ (n − 2)(A_{n} − 2B_{n} + 2C_{n} − B_{n−2})).

3.4 Numerical remarks about the zeros of polynomials B_{n}(x)

Let us set

B_{n}(x) := ∑_{k=1}^{n} B_{k}x^{n−k}, n ∈ ℕ.

Then,

B_{1}(x) ≡ 1, B_{2}(x) = x + 2, B_{3}(x) = x^{2} + 2x + 5,

and, thus, we have the recurrence relations (see (3.20))

B_{n+3}(x) = x^{n+2} + 2B_{n+2}(x) + B_{n+1}(x) − B_{n}(x), n ⩾ 1.

All polynomials B_{n}(x) for n = 2k + 1, k ∈ ℕ, have an even degree, for example

B_{5}(x) = x^{4} + 2x^{3} + 5x^{2} + 11x + 25,
B_{7}(x) = x^{6} + 2x^{5} + 5x^{4} + 11x^{3} + 25x^{2} + 56x + 126.

As follows from calculations, these polynomials have no real zeros for n ⩽ 200. However, all polynomials B_{n}(x) for n = 2k, k ∈ ℕ, have an odd degree, for example

B_{4}(x) = x^{3} + 2x^{2} + 5x + 11,
B_{6}(x) = x^{5} + 2x^{4} + 5x^{3} + 11x^{2} + 25x + 56.

These polynomials have exactly one real zero, which is less than zero (for n ⩽ 200). The zeros s_{n} of the consecutive polynomials B_{n} make a decreasing sequence from s_{2} = −2 to s_{200} = −2.2442546 (by numerical calculations).

Remark 3.27. All numerical calculations were made in Mathematica.^{1}

Acknowledgment

The authors wish to express their gratitude to the referee for several helpful comments and suggestions concerning the first version of our paper.

1Mathematica is registered trademark of Wolfram Research Inc.

Table 1: ; \begin{aligned}& A_{n}(δ) \\ n = 0 & 1 \\ n = 1 & 1 \\ n = 2 & 2δ^{2} + 1 \\ n = 3 & −δ^{3} + 6δ^{2} + 1 \\ n = 4 & 5δ^{4} − 4δ^{3} + 12δ^{2} + 1 \\ n = 5 & −5δ^{5} + 25δ^{4} − 10δ^{3} + 20δ^{2} + 1 \\ n = 6 & 14δ^{6} − 30δ^{5} + 75δ^{4} − 20δ^{3} + 30δ^{2} + 1 \\ n = 7 & −19δ^{7} + 98δ^{6} − 105δ^{5} + 175δ^{4} − 35δ^{3} + 42δ^{2} + 1 \\ & B_{n}(δ) \\ n = 0 & 0 \\ n = 1 & δ \\ n = 2 & 2δ \\ n = 3 & 2δ^{3} + 3δ \\ n = 4 & −δ^{4} + 8δ^{3} + 4δ \\ n = 5 & 5δ^{5} − 5δ^{4} + 20δ^{3} + 5δ \\ n = 6 & −5δ^{6} + 30δ^{5} − 15δ^{4} + 40δ^{3} + 6δ \\ n = 7 & 14δ^{7} − 35δ^{6} + 105δ^{5} − 35δ^{4} + 70δ^{3} + 7δ \\ & C_{n}(δ) \\ n = 0 & 0 \\ n = 1 & 0 \\ n = 2 & δ^{2} \\ n = 3 & −δ^{3} + 3δ^{2} \\ n = 4 & 3δ^{4} − 4δ^{3} + 6δ^{2} \\ n = 5 & −4δ^{5} + 15δ^{4} − 10δ^{3} + 10δ^{2} \\ n = 6 & 9δ^{6} − 24δ^{5} + 45δ^{4} − 20δ^{3} + 15δ^{2} \\ n = 7 & −14δ^{7} + 63δ^{6} − 84δ^{5} + 105δ^{4} − 35δ^{3} + 21δ^{2} \end{aligned}

25

Table 2: ; \begin{matrix}n & A_{n} & B_{n} & C_{n} \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 2 & 3 & 2 & 1 \\ 3 & 6 & 5 & 2 \\ 4 & 14 & 11 & 5 \\ 5 & 31 & 25 & 11 \\ 6 & 70 & 56 & 25 \\ 7 & 157 & 126 & 56 \\ 8 & 353 & 283 & 126 \\ 9 & 793 & 636 & 283 \\ 10 & 1782 & 1429 & 636 \\ 11 & 4004 & 3211 & 1429 \end{matrix}  Table 3: ; \begin{matrix}n & A_{n}(−1) & B_{n}(−1) & C_{n}(−1) \\ 0 & 1 & 0 & 0 \\ 1 & 1 & −1 & 0 \\ 2 & 3 & −2 & 1 \\ 3 & 8 & −5 & 4 \\ 4 & 22 & −13 & 13 \\ 5 & 61 & −35 & 39 \\ 6 & 170 & −96 & 113 \\ 7 & 475 & −266 & 322 \\ 8 & 1329 & −741 & 910 \\ 9 & 3721 & −2070 & 2561 \\ 10 & 10422 & −5791 & 7192 \\ 11 & 29196 & −16213 & 20175 \end{matrix}

Table 4: ; \begin{matrix}n & A_{n}(\frac{1}{2}) & B_{n}(\frac{1}{2}) & C_{n}(\frac{1}{2}) \\ 0 & 1 & 0 & 0 \\ 1 & 1 & \frac{1}{2} & 0 \\ 2 & \frac{3}{2} & 1 & \frac{1}{4} \\ 3 & \frac{19}{8} & \frac{7}{4} & \frac{5}{8} \\ 4 & \frac{61}{16} & \frac{47}{16} & \frac{19}{16} \\ 5 & \frac{197}{32} & \frac{155}{32} & \frac{33}{16} \\ 6 & \frac{319}{32} & \frac{507}{64} & \frac{221}{64} \\ 7 & \frac{2069}{128} & \frac{413}{32} & \frac{91}{16} \\ 8 & \frac{3357}{128} & \frac{5373}{256} & \frac{595}{64} \\ 9 & \frac{10897}{256} & \frac{4365}{128} & \frac{7753}{512} \\ 10 & \frac{70755}{1024} & \frac{28357}{512} & \frac{25213}{1024} \\ 11 & \frac{229725}{2048} & \frac{184183}{2048} & \frac{81927}{2048} \end{matrix}  Table 5: ; \begin{matrix}n & A_{n}(2) & B_{n}(2) & C_{n}(2) \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 2 & 0 \\ 2 & 9 & 4 & 4 \\ 3 & 17 & 22 & 4 \\ 4 & 97 & 56 & 40 \\ 5 & 241 & 250 & 72 \\ 6 & 1097 & 732 & 428 \\ 7 & 3169 & 2926 & 1036 \\ 8 & 12801 & 9264 & 4816 \\ 9 & 40225 & 34866 & 13712 \\ 10 & 152265 & 115316 & 56020 \\ 11 & 501489 & 419846 & 174612 \end{matrix}

Table 6: ; \begin{matrix}n & A_{n}(3) & B_{n}(3) & C_{n}(3) \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 3 & 0 \\ 2 & 19 & 6 & 9 \\ 3 & 28 & 63 & 0 \\ 4 & 406 & 147 & 189 \\ 5 & 721 & 1365 & 63 \\ 6 & 8722 & 3528 & 3969 \\ 7 & 17983 & 29694 & 2646 \\ 8 & 188209 & 83643 & 83790 \\ 9 & 438697 & 648270 & 83349 \\ 10 & 4078270 & 1964361 & 1778112 \\ 11 & 10530100 & 14199171 & 2336859 \end{matrix}  Table 7: ; \begin{matrix}n & A_{n} & \frac{1}{2}(A_{n}^{2} − A_{2n}) & A_{n}(\frac{1}{2}) \\ 0 & 3 & 3 & 3 \\ 1 & 2 & −1 & \frac{5}{2} \\ 2 & 6 & 5 & \frac{13}{4} \\ 3 & 11 & −4 & \frac{19}{4} \\ 4 & 26 & 13 & \frac{117}{16} \\ 5 & 57 & −16 & \frac{185}{16} \\ 6 & 129 & 38 & \frac{593}{32} \\ 7 & 289 & −57 & \frac{3827}{128} \\ 8 & 650 & 117 & \frac{12389}{256} \\ 9 & 1460 & −193 & \frac{40169}{512} \\ 10 & 3281 & 370 & \frac{65169}{512} \\ 11 & 7372 & −639 & \frac{423065}{2048} \end{matrix}

References

[1] D. M. Bradley, A class of series acceleration formulae for Catalan’s constant, Ramanujan J. 3 (1999), 149–173.

[2] R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. A Foundation for Computer Science, Addison-Wesley, 1994.

[3] R. Grzymkowski and R. Wituła, Calculus Methods in Algebra, Part One, WPKJS, 2000 (in Polish).

[4] J. Kappraff and G. W. Adamson, A unified theory of proportion, Vis. Math. 5 (2003), electronic.

[5] T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley, 2001.

[6] I. Niven, Irrational Numbers, MAA, 1956 (appendix to the russian edition written by I. M. Yaglom).

[7] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, 2006.

[8] P. Steinbach, Golden fields: a case for the heptagon, Math. Magazine 70 (1997), 22–31.

[9] D. Surowski and P. McCombs, Homogeneous polynomials and the minimal polynomial of cos(2π/n), Missouri J. Math. Sci. 15 (2003), 4–14.

[10] W. Watkins and J. Zeitlin, The minimal polynomials of cos(2π/n), Amer. Math. Monthly 100 (1993), 471–474.

2000 Mathematics Subject Classification: Primary 11B83, 11A07; Secondary 39A10. Keywords: Fibonacci numbers, primitive roots of unity, recurrence relation.

(Concerned with sequences \underline{A006054}, \underline{A006356}, \underline{A033304}, \underline{A085810}, and \underline{A094648}.)

Received January 25 2006; revised version received August 21 2006; September 5 2006. Published in Journal of Integer Sequences, September 6 2006. Revised version, correcting some errors, April 19 2007.

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