New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7

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

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

Abstract

We give applications of the quasi-Fibonacci numbers of order 7 and the so-called sine-Fibonacci numbers of order 7 and many other new kinds of recurrent sequences to the decompositions of some polynomials. We also present the characteristic equations, generating functions and some properties of all these sequences. Finally, some new Ramanujan-type formulas are generated.

1 Introduction

The scope of the paper is the generalization of the following decompositions of polynomials [2, 3, 4, 8, 9]:

(𝕏−2 sin(\frac{2 π}{7}))(𝕏 − 2 sin(\frac{4 π}{7}))(𝕏 − 2 sin(\frac{8 π}{7})) = 𝕏^{3} − \sqrt{7} 𝕏^{2} + \sqrt{7},  (1.1)
(𝕏−4 sin^{2}(\frac{2 π}{7}))(𝕏 − 4 sin^{2}(\frac{4 π}{7}))(𝕏 − 4 sin^{2}(\frac{8 π}{7})) = 𝕏^{3} − 7 𝕏^{2} + 14 𝕏 − 7,  (1.2)
(𝕏−8 sin^{3}(\frac{2 π}{7}))(𝕏 − 8 sin^{3}(\frac{4 π}{7}))(𝕏 − 8 sin^{3}(\frac{8 π}{7})) = 𝕏^{3} − 4 \sqrt{7} 𝕏^{2} + 21 𝕏 + 7 \sqrt{7}, (1.3)
(𝕏−2 cos(\frac{2 π}{7}))(𝕏 − 2 cos(\frac{4 π}{7}))(𝕏 − 2 cos(\frac{8 π}{7})) = 𝕏^{3} + 𝕏^{2} − 2 𝕏 − 1,  (1.4)
(𝕏−4 cos^{2}(\frac{2 π}{7}))(𝕏 − 4 cos^{2}(\frac{4 π}{7}))(𝕏 − 4 cos^{2}(\frac{8 π}{7})) = 𝕏^{3} − 5 𝕏^{2} + 6 𝕏 − 1,  (1.5)
(𝕏−8 cos^{3}(\frac{2 π}{7}))(𝕏 − 8 cos^{3}(\frac{4 π}{7}))(𝕏 − 8 cos^{3}(\frac{8 π}{7})) = 𝕏^{3} + 4 𝕏^{2} − 11 𝕏 − 1,  (1.6)
(𝕏−8 sin(\frac{2 π}{7}) cos^{2}(\frac{8 π}{7}))(𝕏 − 8 sin(\frac{4 π}{7}) cos^{2}(\frac{2 π}{7}))(𝕏 − 8 sin(\frac{8 π}{7}) cos^{2}(\frac{4 π}{7})) =
= 𝕏^{3} − 3 \sqrt{7} 𝕏^{2} + 14 𝕏 + \sqrt{7},  (1.7)
etc.

The main incentive for generating the decompositions of these polynomials is provided by the properties of the so-called quasi-Fibonacci numbers of order 7, A_{n}(δ), B_{n}(δ) and C_{n}(δ), n ∈ ℕ, described in [10] by means of the relations

(1 + δ (ξ^{k} + ξ^{6k}))^{n} = A_{n}(δ) + B_{n}(δ) (ξ^{k} + ξ^{6k}) + C_{n}(δ) (ξ^{2k} + ξ^{5k})  (1.8)

for k = 1, 2, 3, where ξ ∈ ℂ is a primitive root of unity of order 7 (i.e., ξ^{7} = 1 and ξ ≠ 1), δ ∈ ℂ, δ ≠ 0. Besides, an essential rôle in the decompositions of polynomials discussed in the paper is played by related numbers (δ ∈ ℂ, n ∈ ℕ):

A_{n}(δ) := 3 A_{n}(δ) − B_{n}(δ) − C_{n}(δ) = = (1 + δ (ξ + ξ^{6}))^{n} + (1 + δ (ξ^{2} + ξ^{5}))^{n} + (1 + δ (ξ^{3} + ξ^{4}))^{n} (1.9)

and

B_{n}(δ) := \frac{1}{2}((A_{n}(δ))^{2} − A_{2n}(δ)) =
= ((1 + 2 δ cos(\frac{2 π}{7}))(1 + 2 δ cos(\frac{4 π}{7})))^{n} +
+ ((1 + 2 δ cos(\frac{2 π}{7}))(1 + 2 δ cos(\frac{8 π}{7})))^{n} +
+ ((1 + 2 δ cos(\frac{4 π}{7}))(1 + 2 δ cos(\frac{8 π}{7})))^{n} =
= 3 (A_{n}(δ))^{2} − 2 A_{n}(δ) B_{n}(δ) − 2 A_{n}(δ) C_{n}(δ) +
+ 3 B_{n}(δ) C_{n}(δ) − 2 (B_{n}(δ))^{2} − 2 (C_{n}(δ))^{2} =
= A_{n}(δ) (2 A_{n}(δ) − 3A_{n}(δ)) − 2 (B_{n}(δ) − C_{n}(δ))^{2} − B_{n}(δ) C_{n}(δ).  (1.10)

Furthermore, to simplify the formulas, we will write

A_{n} = A_{n}(1), B_{n} = B_{n}(1), A_{n} = A_{n}(1), B_{n} = B_{n}(1) and C_{n} = C_{n}(1),  (1.11)

for every n ∈ ℕ. We note that the tables of values of these numbers can be found in the article [10].

2 Basic decompositions

Wituła et al. [10] determined the following two formulas:

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

= 𝕏^{3} − B_{n} 𝕏^{2} + (−1)^{n} A_{n} 𝕏 − 1 (2.1)

and

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

= 𝕏^{3} − A_{n}(δ) 𝕏^{2} + B_{n}(δ) 𝕏 − (1 − δ − 2 δ^{2} + δ^{3})^{n}. (2.2)

From (2.2) three special formulas follow:

(𝕏 − (1 + cos^{2}(\frac{2 π}{7}))^{n})(𝕏 − (1 + cos^{2}(\frac{4 π}{7}))^{n})(𝕏 − (1 + cos^{2}(\frac{8 π}{7}))^{n}) = = 𝕏^{3} − (\frac{3}{2})^{n} A_{n}(\frac{1}{6}) 𝕏^{2} + (\frac{3}{2})^{2n} B_{n}(\frac{1}{6}) 𝕏 − (\frac{13}{8})^{2n}, (2.3)

(𝕏 − (2 sin(\frac{2 π}{7}))^{2n})(𝕏 − (2 sin(\frac{4 π}{7}))^{2n})(𝕏 − (2 sin(\frac{8 π}{7}))^{2n}) =

= 𝕏^{3} − 2^{n} A_{n}( − \frac{1}{2}) 𝕏^{2} + 2^{2n} B_{n}( − \frac{1}{2}) 𝕏 − 7^{n} (2.4)

and

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

= 𝕏^{3} − 2^{n} A_{n}(\frac{1}{2}) 𝕏^{2} + 2^{2n} B_{n}(\frac{1}{2}) 𝕏 − 1. (2.5)

Comparing formulas (2.1) and (2.5) two new identities are generated

A_{2n} = 2^{2n} B_{n}(\frac{1}{2}) and B_{2n} = 2^{n} A_{n}(\frac{1}{2}).  (2.6)

We also have the decomposition (𝕏 − ((1 + δ (ξ + ξ^{6})) (1 + δ (ξ^{2} + ξ^{5})))^{n}) (𝕏 − ((1 + δ (ξ + ξ^{6})) (1 + δ (ξ^{3} + ξ^{4})))^{n}) ×

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

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

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

= 𝕏^{3} − B_{n}(δ) 𝕏^{2} + (1 − δ − 2 δ^{2} + δ^{3}) A_{n}(δ) 𝕏 − (1 − δ − 2 δ^{2} + δ^{3})^{2n} =
:= r_{n}(𝕏; δ). (2.7)

Now let us set

Ξ_{n} := Ξ_{n}(δ, ε, η) = 2^{n} (δ cos^{n} (\frac{2 π}{7}) + ε cos^{n} (\frac{4 π}{7}) + η cos^{n} (\frac{8 π}{7})),  (2.8)
Υ_{n} := Υ_{n}(δ, ε, η) = 2^{n} (ε cos^{n} (\frac{2 π}{7}) + η cos^{n} (\frac{4 π}{7}) + δ cos^{n} (\frac{8 π}{7})),  (2.9)
Θ_{n} := Θ_{n}(δ, ε, η) = 2^{n} (η cos^{n} (\frac{2 π}{7}) + δ cos^{n} (\frac{4 π}{7}) + ε cos^{n} (\frac{8 π}{7}))  (2.10)

for any δ, ε, η ∈ ℂ and n ∈ ℕ_{0}. Lemma 2.1 The following general decomposition formula holds

(𝕏 − Ξ_{n})(𝕏 − Υ_{n})(𝕏 − Θ_{n}) =

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

− [δ ε η (B_{3n} + 3) + δ^{3} + ε^{3} + η^{3} + (−1)^{n} (δ ε^{2} + ε η^{2} + η δ^{2}) A_{n}(−1) +

+ 2^{2n} (δ^{2} ε + ε^{2} η + η^{2} δ) ∑_{k=0}^{n}(−1)^{k} \binom{n}{k} A_{2n−k}(\frac{1}{2})]. (2.11)

Below an illustrative example connected with Lemma 2.1 will be presented. Examle 2.2 A. M. Yaglom and I. M. Yaglom [11] (see also [5, problem 230] and [6, problem 329]) considered the following polynomial:

w(x) = \binom{2m + 1}{1} x^{m} − \binom{2m + 1}{3} x^{m−1} + \binom{2m + 1}{5} x^{m−2} − …

and proved that it has the roots

x_{k} = cot^{2} (\frac{k π}{2m + 1}), k = 1, 2, …, m.

In particular, for m = 3 taking into account that

cot^{2} (\frac{π}{7}) = cot^{2} (\frac{8 π}{7}), cot^{2} (\frac{3 π}{7}) = cot^{2} (\frac{4 π}{7})

we have the decomposition (see also formula (6.14) below):

(x − cot^{2} (\frac{2 π}{7})) (x − cot^{2} (\frac{4 π}{7})) (x − cot^{2} (\frac{8 π}{7})) = x^{3} − 5 x^{2} + 3 x − \frac{1}{7}

and as a corollary – the decomposition

(x + 7 cot^{2} (\frac{2 π}{7})) (x + 7 cot^{2} (\frac{4 π}{7})) (x + 7 cot^{2} (\frac{8 π}{7})) = x^{3} + 35 x^{2} + 147 x + 49. (2.12)

According to (2.8) for n = 1, let us try to find a linear combination −7 cot^{2} (\frac{2 π}{7}) = 2 (δ cos (\frac{2 π}{7}) + ε cos (\frac{4 π}{7}) + η cos (\frac{8 π}{7})) or, the same,

− 7 cos^{2} (\frac{2 π}{7}) = 2 (δ cos (\frac{2 π}{7}) + ε cos (\frac{4 π}{7}) + η cos (\frac{8 π}{7})) −

− δ cos^{3} (\frac{2 π}{7}) − ε cos (\frac{4 π}{7}) cos^{2} (\frac{2 π}{7}) − η cos (\frac{8 π}{7}) cos^{2} (\frac{2 π}{7}).

Decreasing powers and taking into account the identity

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

we find

(2 δ + ε − 3 η) cos (\frac{2 π}{7}) + (δ + 3 ε − 3 η + 7) cos (\frac{4 π}{7}) = −\frac{δ}{2} + \frac{ε}{2} + η − 7.

Thus, for finding δ, ε, η we have the linear system (see Corollary 2.5 in [10]):

\{ \begin{matrix}2 δ + ε − 3 η = 0 \\ δ + 3 ε − 3 η = −7 \\ δ − ε − 2 η = −14 \end{matrix}

with the solution

δ = 17,  ε = 5,  η = 13.

Hence,

−7 cot^{2} (\frac{2 π}{7}) = 2 (17 cos (\frac{2 π}{7}) + 5 cos (\frac{4 π}{7}) + 13 cos (\frac{8 π}{7})).

Analogously, we obtain

−7 cot^{2} (\frac{4 π}{7}) = 2 (13 cos (\frac{2 π}{7}) + 17 cos (\frac{4 π}{7}) + 5 cos (\frac{8 π}{7})),
−7 cot^{2} (\frac{8 π}{7}) = 2 (5 cos (\frac{2 π}{7}) + 13 cos (\frac{4 π}{7}) + 17 cos (\frac{8 π}{7})).

Thus the decomposition (2.12) corresponds to Lemma 2.1 with

Ξ_{1}(17, 5, 13), Υ_{1}(17, 5, 13), Θ_{1}(17, 5, 13).

Remark 2.3 The identity (2.12) was found earlier by Shevelev [4].

Remark 2.4 The formula (2.11), in some cases which are subject of our interest, especially when coefficients δ, ε, η are the corresponding values of trigonometric functions, becomes rather complicated. Thus, in the next two sections we attempt to designate the relevant coefficients of decomposition (2.11), including the recurrent coefficients, on the grounds of new sequences that are easier to analyze.

3 The first group of special cases of (2.11)

Let us set

a_{n} = 2^{2n+1} [ sin(\frac{2 π}{7})( cos(\frac{8 π}{7}))^{2n} + sin(\frac{4 π}{7})( cos(\frac{2 π}{7}))^{2n} + sin(\frac{8 π}{7})( cos(\frac{4 π}{7}))^{2n}],  (3.1)
b_{n} = 2^{2n+1} [ sin(\frac{4 π}{7})( cos(\frac{8 π}{7}))^{2n} + sin(\frac{8 π}{7})( cos(\frac{2 π}{7}))^{2n} + sin(\frac{2 π}{7})( cos(\frac{4 π}{7}))^{2n}],  (3.2)
c_{n} = 2^{2n+1} [ sin(\frac{8 π}{7})( cos(\frac{8 π}{7}))^{2n} + sin(\frac{2 π}{7})( cos(\frac{2 π}{7}))^{2n} + sin(\frac{4 π}{7})( cos(\frac{4 π}{7}))^{2n}],  (3.3)

for n = 0, 1, 2, …

Lemma 3.1 The following recurrence relations hold:

\{ \begin{matrix}a_{n+1} = 2 a_{n} + b_{n}, \\ b_{n+1} = a_{n} + 2 b_{n} − c_{n}, \\ c_{n+1} = c_{n} − b_{n}, \end{matrix}  (3.4)

for n = 0, 1, 2, … and a_{0} = b_{0} = c_{0} = \sqrt{7}. Moreover, elements of each sequences \{a_{n}\}_{n=0}^{∞}, {b_{n}}_{n=0}^{∞} and {c_{n}}_{n=0}^{∞} satisfy the recurrence equation

x_{n+2} − 5 x_{n+1} + 6 x_{n} − x_{n−1} = 0,  n = 0, 1, 2, …  (3.5)

(in view of decomposition (1.5) an appropriate characteristic polynomial is compatible with the definition of numbers a_{n}, b_{n} and c_{n}).

The first twelve values of numbers a_{n}^{∗} = a_{n}/\sqrt{7}, b_{n}^{∗} = b_{n}/\sqrt{7} and c_{n}^{∗} = c_{n}/\sqrt{7} are presented in Table 1.

Now, let us set

α_{n} = 2^{2n} [ sin(\frac{2 π}{7})( cos(\frac{8 π}{7}))^{2n−1} + sin(\frac{4 π}{7})( cos(\frac{2 π}{7}))^{2n−1} + sin(\frac{8 π}{7})( cos(\frac{4 π}{7}))^{2n−1}], (3.6)
β_{n} = 2^{2n} [ sin(\frac{4 π}{7})( cos(\frac{8 π}{7}))^{2n−1} + sin(\frac{8 π}{7})( cos(\frac{2 π}{7}))^{2n−1} + sin(\frac{2 π}{7})( cos(\frac{4 π}{7}))^{2n−1}], (3.7)
γ_{n} = 2^{2n} [ sin(\frac{8 π}{7})( cos(\frac{8 π}{7}))^{2n−1} + sin(\frac{2 π}{7})( cos(\frac{2 π}{7}))^{2n−1} + sin(\frac{4 π}{7})( cos(\frac{4 π}{7}))^{2n−1}], (3.8)

for n = 1, 2, …

Lemma 3.2 We have

α_{1} = 0,  β_{1} = −2 \sqrt{7},  and  γ_{1} = \sqrt{7}.

The elements of sequences {α_{n}}_{n=1}^{∞}, {β_{n}}_{n=1}^{∞} and {γ_{n}}_{n=1}^{∞} satisfy the system of recurrence relations (3.4) and, selectively, recurrence relation (3.5).

The first twelve values of numbers α_{n}^{∗} = α_{n}/\sqrt{7}, β_{n}^{∗} = β_{n}/\sqrt{7} and γ_{n}^{∗} = γ_{n}/\sqrt{7} are presented in Table 1.

Remark 3.3 There exists a simple relationships between numbers α_{n}, β_{n}, γ_{n}, n ∈ ℕ, and a_{n}, b_{n}, c_{n}, n ∈ ℕ. We have

α_{n} ≡ c_{n},  β_{n} ≡ −a_{n−1} − b_{n−1},  γ_{n} ≡ a_{n−1}.

These relations are easily derived from the definitions of respective numbers, for example

β_{n} = 2^{2(n−1)+1}[2 sin(\frac{4 π}{7}) cos(\frac{8 π}{7})( cos(\frac{8 π}{7}))^{2(n−1)} +
+ 2 sin(\frac{8 π}{7}) cos(\frac{2 π}{7})( cos(\frac{2 π}{7}))^{2(n−1)} + 2 sin(\frac{2 π}{7}) cos(\frac{4 π}{7})( cos(\frac{4 π}{7}))^{2(n−1)}] =

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

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

= −a_{n−1} − b_{n−1}.

Let us also set

f_{n} = 2^{n+1} [ cos(\frac{2 π}{7})( cos(\frac{4 π}{7}))^{n} + cos(\frac{4 π}{7})( cos(\frac{8 π}{7}))^{n} + cos(\frac{8 π}{7})( cos(\frac{2 π}{7}))^{n}],  (3.9)
g_{n} = 2^{n+1} [ cos(\frac{8 π}{7})( cos(\frac{4 π}{7}))^{n} + cos(\frac{2 π}{7})( cos(\frac{8 π}{7}))^{n} + cos(\frac{4 π}{7})( cos(\frac{2 π}{7}))^{n}],  (3.10)
h_{n} = 2^{n+1} [( cos(\frac{2 π}{7}))^{n+1} + ( cos(\frac{4 π}{7}))^{n+1} + ( cos(\frac{8 π}{7}))^{n+1}],  (3.11)

for n = 0, 1, 2, …

Lemma 3.4 We have

f_{0} = g_{0} = h_{0} = −1,  and  h_{1} = 5

and

\{ \begin{matrix}f_{n+1} = f_{n} + g_{n}, & n ≥ 0, \\ g_{n+1} = f_{n} + h_{n}, & n ≥ 0, \\ h_{n+1} = g_{n} + 2 h_{n−1}, & n ≥ 1. \end{matrix}  (3.12)

The elements of sequences {f_{n}}_{n=0}^{∞}, {g_{n}}_{n=0}^{∞} and {h_{n}}_{n=0}^{∞} satisfy the following recurrence relation (see formula (1.4)):

x_{n+3} + x_{n+2} − 2 x_{n+1} − x_{n} = 0,  n = 0, 1, …  (3.13)

Proof: By (3.12) we obtain

g_{n} = f_{n+1} − f_{n},  (3.14)
h_{n} = g_{n+1} − f_{n} = f_{n+2} − f_{n+1} − f_{n},  (3.15)

and, finally, the following identity

f_{n+3} − f_{n+2} − f_{n+1} = 2 (f_{n+1} − f_{n} − f_{n−1}) + f_{n+1} − f_{n},

i.e.,

f_{n+3} − f_{n+2} − 4 f_{n+1} + 3 f_{n} + 2 f_{n−1} = 0.  (3.16)

But we also have the following decomposition of respective characteristic polynomial

x^{4} − x^{3} − 4 x^{2} + 3 x + 2 = (x − 2)(x^{3} + x^{2} − 2 x − 1),

which implies the following form of (3.16):

f_{n+3} + f_{n+2} − 2 f_{n+1} − f_{n} = 2 (f_{n+2} + f_{n+1} − 2 f_{n} − f_{n−1}).  (3.17)

Since

f_{3} + f_{2} − 2 f_{1} − f_{0} = 0

so, we obtain the required identity (3.13) from (3.17), (3.14) and (3.15). □

The first twelve elements of the sequences {f_{n}}_{n=0}^{∞}, {g_{n}}_{n=0}^{∞} and {h_{n}}_{n=0}^{∞} are presented in Table 1. Remark 3.5 We note, that

h_{n−1} = B_{n} = \frac{1}{2}((A_{n})^{2} − A_{2n}), n = 1, 2, … (3.18)

is an accelerator sequence for Catalan’s constant (see [1] and \underline{A094648} [7]).

The next lemma contains a sequence of eight identities and simultanously six newly defined auxiliary sequences of real numbers {Ã_{n}}, {B̃_{n}}, {C̃_{n}}, {F_{n}}, {G_{n}} and {H_{n}}. Lemma 3.6 The following identities hold

4 cos(\frac{2 π}{7}) (4 cos(\frac{2 π}{7}) cos(\frac{8 π}{7}))^{n} + 4 cos(\frac{4 π}{7})(4 cos(\frac{2 π}{7}) cos(\frac{4 π}{7}))^{n} + + 4 cos(\frac{8 π}{7})(4 cos(\frac{4 π}{7}) cos(\frac{8 π}{7}))^{n} = = f_{n}^{2} + g_{n}^{2} − h_{n}^{2} + h_{2n+1} − 2 h_{2n} − 4 h_{2n−1} − f_{2n} := 2 F_{n}, (3.19)

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

+ 2 cos(\frac{8 π}{7})(4 cos(\frac{2 π}{7}) cos(\frac{8 π}{7}))^{n} = = F_{n} + h_{n}^{2} − f_{n}^{2} − h_{2n+1} + 2 h_{2n} + 2 h_{2n−1} := G_{n}, (3.20)

2 cos(\frac{2 π}{7}) (4 cos(\frac{4 π}{7}) cos(\frac{8 π}{7}))^{n} + 2 cos(\frac{4 π}{7})(4 cos(\frac{2 π}{7}) cos(\frac{8 π}{7}))^{n} + + 2 cos(\frac{8 π}{7})(4 cos(\frac{2 π}{7}) cos(\frac{4 π}{7}))^{n} = = f_{n}^{2} − 2 (h_{2n} + h_{2n−1}) − F_{n} = h_{n}^{2} − h_{2n+1} − G_{n} := H_{n}, (3.21)

g_{n}^{2} − 2 h_{2n−1} − f_{2n} = (−1)^{n}(A_{n} + A_{n+1} − 7 A_{n}) = G_{n} + F_{n},  (3.22)
f_{n}^{2} − 2 h_{2n} − 2 h_{2n−1} = (−1)^{n}(A_{n} − A_{n−2}) = H_{n} + F_{n},  (3.23)

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

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

= a_{2n} − a_{n} h_{2n−1} + \frac{\sqrt{7}}{2} (h_{2n−1}^{2} − h_{4n−1}) := Ã_{n}, (3.24)

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

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

= b_{2n} − b_{n} h_{2n−1} + \frac{\sqrt{7}}{2} (h_{2n−1}^{2} − h_{4n−1}) := B̃_{n}, (3.25)

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

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

= c_{2n} − c_{n} h_{2n−1} + \frac{\sqrt{7}}{2} (h_{2n−1}^{2} − h_{4n−1}) := C̃_{n}, (3.26)

Now we are ready to present the final result of this section. All recurrent sequences defined in this section are applied below to the description of the coefficients of certain polynomials.

Theorem 3.7 The following decompositions of polynomials hold:

(𝕏 − 2 sin(\frac{2 π}{7})(2 cos(\frac{8 π}{7}))^{n})(𝕏 − 2 sin(\frac{4 π}{7})(2 cos(\frac{2 π}{7}))^{n})(𝕏 − 2 sin(\frac{8 π}{7})(2 cos(\frac{4 π}{7}))^{n}) =

= \{ \begin{matrix}𝕏^{3} − a_{k} 𝕏^{2} + 7 B_{2k} 𝕏 + \sqrt{7}, & for n = 2 k, \\ 𝕏^{3} − α_{k−1} 𝕏^{2} − 7 B_{2k−1} 𝕏 + \sqrt{7}, & for n = 2 k − 1, \end{matrix} (3.27)

(𝕏 − 2 sin(\frac{4 π}{7})(2 cos(\frac{8 π}{7}))^{n})(𝕏 − 2 sin(\frac{8 π}{7})(2 cos(\frac{2 π}{7}))^{n})(𝕏 − 2 sin(\frac{2 π}{7})(2 cos(\frac{4 π}{7}))^{n}) =

= \{ \begin{matrix}𝕏^{3} − b_{k} 𝕏^{2} + 7 (C_{2k} − B_{2k}) 𝕏 + \sqrt{7}, & for n = 2 k, \\ 𝕏^{3} − β_{k−1} 𝕏^{2} + 7 (B_{2k−1} − C_{2k−1}) 𝕏 + \sqrt{7}, & for n = 2 k − 1, \end{matrix} (3.28)

(𝕏 − 2 sin(\frac{2 π}{7})(2 cos(\frac{2 π}{7}))^{n})(𝕏 − 2 sin(\frac{4 π}{7})(2 cos(\frac{4 π}{7}))^{n})(𝕏 − 2 sin(\frac{8 π}{7})(2 cos(\frac{8 π}{7}))^{n}) =

= \{ \begin{matrix}𝕏^{3} − c_{k} 𝕏^{2} − 7 C_{2k} 𝕏 + \sqrt{7}, & for n = 2 k, \\ 𝕏^{3} − γ_{k−1} 𝕏^{2} + 7 C_{2k−1} 𝕏 + \sqrt{7}, & for n = 2 k − 1, \end{matrix} (3.29)

(𝕏 − 2 cos(\frac{2 π}{7})(2 cos(\frac{4 π}{7}))^{n})(𝕏 − 2 cos(\frac{4 π}{7})(2 cos(\frac{8 π}{7}))^{n})(𝕏 − 2 cos(\frac{8 π}{7})(2 cos(\frac{2 π}{7}))^{n}) = = 𝕏^{3} − f_{n} 𝕏^{2} + (−1)^{n} (7 A_{n} − 3 A_{n}) 𝕏 − 1, (3.30)

(𝕏 − 2 cos(\frac{8 π}{7})(2 cos(\frac{4 π}{7}))^{n})(𝕏 − 2 cos(\frac{2 π}{7})(2 cos(\frac{8 π}{7}))^{n})(𝕏 − 2 cos(\frac{4 π}{7})(2 cos(\frac{2 π}{7}))^{n}) = = 𝕏^{3} − g_{n} 𝕏^{2} + (−1)^{n} (A_{n} + A_{n+1} − 7 A_{n}) 𝕏 − 1, (3.31)

(𝕏 − 2 cos(\frac{2 π}{7})(4 cos(\frac{2 π}{7}) cos(\frac{8 π}{7}))^{n})(𝕏 − 2 cos(\frac{4 π}{7})(4 cos(\frac{2 π}{7}) cos(\frac{4 π}{7}))^{n}) ×

× (𝕏 − 2 cos(\frac{8 π}{7})(4 cos(\frac{4 π}{7}) cos(\frac{8 π}{7}))^{n}) = = 𝕏^{3} − F_{n} 𝕏^{2} + (f_{n} + h_{n}) 𝕏 − 1, (3.32)

(𝕏 − 2 cos(\frac{2 π}{7})(4 cos(\frac{2 π}{7}) cos(\frac{4 π}{7}))^{n})(𝕏 − 2 cos(\frac{4 π}{7})(4 cos(\frac{4 π}{7}) cos(\frac{8 π}{7}))^{n}) ×

× (𝕏 − 2 cos(\frac{8 π}{7})(4 cos(\frac{2 π}{7}) cos(\frac{8 π}{7}))^{n}) = = 𝕏^{3} − G_{n} 𝕏^{2} + (f_{n} + g_{n}) 𝕏 − 1, (3.33)

(𝕏 − 2 cos(\frac{2 π}{7})(4 cos(\frac{4 π}{7}) cos(\frac{8 π}{7}))^{n})(𝕏 − 2 cos(\frac{4 π}{7})(4 cos(\frac{2 π}{7}) cos(\frac{8 π}{7}))^{n}) ×

× (𝕏 − 2 cos(\frac{8 π}{7})(4 cos(\frac{2 π}{7}) cos(\frac{4 π}{7}))^{n}) =

= 𝕏^{3} − H_{n} 𝕏^{2} + (g_{n} + h_{n}) 𝕏 − 1. (3.34)

Proof: Ad (3.30) We have

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

hence, by (1.4) we obtain

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

= [(2 cos(\frac{2 π}{7}) + 2 cos(\frac{4 π}{7}) + 2 cos(\frac{8 π}{7})) − (2 cos(\frac{2 π}{7}) + 2 cos(\frac{8 π}{7})) + 2 cos(\frac{2 π}{7})] × ×(4 cos(\frac{4 π}{7}) cos(\frac{8 π}{7}))^{n} = −(4 cos(\frac{4 π}{7}) cos(\frac{8 π}{7}))^{n} −4 cos(\frac{2 π}{7}) cos(\frac{4 π}{7})(4 cos(\frac{4 π}{7}) cos(\frac{8 π}{7}))^{n} +

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

Treating, in a similar way, the following products:

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

and adding all three received decompositions, by (2.1), we obtain

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

+ 4 cos(\frac{2 π}{7}) cos(\frac{8 π}{7})(4 cos(\frac{2 π}{7}) cos(\frac{4 π}{7}))^{n} + 4 cos(\frac{4 π}{7}) cos(\frac{8 π}{7})(4 cos(\frac{2 π}{7}) cos(\frac{8 π}{7}))^{n}. But, by [10], we have

A_{n+3} − 2 A_{n+2} − A_{n+1} + A_{n} ≡ 0,

hence

A_{n+1} − A_{n} − A_{n−1} = A_{n} − A_{n−2},

which implies decomposition (3.30). □ Remark 3.8 It should also be noted that the following interesting identity holds:

4 A_{n} − A_{n−2} = 7 A_{n}.  (3.35)

4 The second group of special cases of (2.11)

Let us set

u_{n} = 2 cos(\frac{2 π}{7})(2 sin(\frac{2 π}{7}))^{n} + 2 cos(\frac{4 π}{7})(2 sin(\frac{4 π}{7}))^{n} + 2 cos(\frac{8 π}{7})(2 sin(\frac{8 π}{7}))^{n},  (4.1)
v_{n} = 2 cos(\frac{4 π}{7})(2 sin(\frac{2 π}{7}))^{n} + 2 cos(\frac{8 π}{7})(2 sin(\frac{4 π}{7}))^{n} + 2 cos(\frac{2 π}{7})(2 sin(\frac{8 π}{7}))^{n},  (4.2)
w_{n} = 2 cos(\frac{8 π}{7})(2 sin(\frac{2 π}{7}))^{n} + 2 cos(\frac{2 π}{7})(2 sin(\frac{4 π}{7}))^{n} + 2 cos(\frac{4 π}{7})(2 sin(\frac{8 π}{7}))^{n},  (4.3)
x_{n} = 2 sin(\frac{4 π}{7})(2 sin(\frac{2 π}{7}))^{n} + 2 sin(\frac{8 π}{7})(2 sin(\frac{4 π}{7}))^{n} + 2 sin(\frac{2 π}{7})(2 sin(\frac{8 π}{7}))^{n},  (4.4)
y_{n} = 2 sin(\frac{8 π}{7})(2 sin(\frac{2 π}{7}))^{n} + 2 sin(\frac{2 π}{7})(2 sin(\frac{4 π}{7}))^{n} + 2 sin(\frac{4 π}{7})(2 sin(\frac{8 π}{7}))^{n},  (4.5)
z_{n} = (2 sin(\frac{2 π}{7}))^{n+1} + (2 sin(\frac{4 π}{7}))^{n+1} + (2 sin(\frac{8 π}{7}))^{n+1},  (4.6)

for n ∈ ℕ and u_{0} = v_{0} = w_{0} = −1 and z_{0} = y_{0} = z_{0} = \sqrt{7}, z_{1} = 7 (see Table 3, and for sequence \{−w_{2n+1}/\sqrt{7}\} see \underline{A115146} [7], and see \underline{A079309} [7] for sequence \{z_{2n}/\sqrt{7}\}).

Then, as may by verified without difficulty, the following recurrence relations hold

\{ \begin{matrix}u_{n+1} = x_{n}, \\ v_{n+1} = −y_{n} − z_{n} = x_{n} − \sqrt{7} z_{n−1}, \\ w_{n+1} = y_{n} − x_{n}, \\ x_{n+1} = u_{n} − w_{n}, \\ y_{n+1} = w_{n} − v_{n}, \\ z_{n+1} = 2 z_{n−1} − v_{n}, \end{matrix}  (4.7)

for every n ∈ ℕ. Hence, we easily obtain

\{ \begin{matrix}x_{n+2} = x_{n} − w_{n+1} = 2 x_{n} − y_{n}, \\ y_{n+2} = y_{n} − v_{n+1} − x_{n} = 2 y_{n} − x_{n} + z_{n}, \\ z_{n+2} = y_{n} + 3 z_{n}, \end{matrix}  (4.8)

for every n ∈ ℕ and finally the recurrence relation (see also equation (1.2)):

z_{n+6} − 7 z_{n+4} + 14 z_{n+2} − 7 z_{n} = 0,  (4.9)

which also satisfies the remaining sequences discussed in this section: {x_{n}}, {y_{n}}, {u_{n}}, {v_{n}} and {w_{n}}.

Remark 4.1 The characteristic polynomial of equation (4.9) (after rescaling) has the form of (1.2) and was recognized by Johannes Kepler (1571–1630). The roots of this polynomial are equal to |A_{1}A_{2}|^{2}, |A_{1}A_{3}|^{2} and |A_{1}A_{4}|^{2}, where A_{1}A_{2} … A_{7} is a regular convex heptagon inscribed in the unit circle [2].

Theorem 4.2 The following decompositions of polynomials hold:

(𝕏 − 2 cos(\frac{2 π}{7})(2 sin(\frac{2 π}{7}))^{n})(𝕏 − 2 cos(\frac{4 π}{7})(2 sin(\frac{4 π}{7}))^{n})(𝕏 − 2 cos(\frac{8 π}{7})(2 sin(\frac{8 π}{7}))^{n}) =

= 𝕏^{3} − u_{n} 𝕏^{2} + \frac{1}{2}(u_{n}^{2} − v_{2n} − 2 z_{2n−1}) 𝕏 − (−\sqrt{7} )^{n}, (4.10)

(𝕏 − 2 cos(\frac{4 π}{7})(2 sin(\frac{2 π}{7}))^{n})(𝕏 − 2 cos(\frac{8 π}{7})(2 sin(\frac{4 π}{7}))^{n})(𝕏 − 2 cos(\frac{2 π}{7})(2 sin(\frac{8 π}{7}))^{n}) =

= 𝕏^{3} − v_{n} 𝕏^{2} + \frac{1}{2}(v_{n}^{2} − w_{2n} − 2 z_{2n−1}) 𝕏 − (−\sqrt{7} )^{n}, (4.11)

(𝕏 − 2 cos(\frac{8 π}{7})(2 sin(\frac{2 π}{7}))^{n})(𝕏 − 2 cos(\frac{2 π}{7})(2 sin(\frac{4 π}{7}))^{n})(𝕏 − 2 cos(\frac{4 π}{7})(2 sin(\frac{8 π}{7}))^{n}) =

= 𝕏^{3} − w_{n} 𝕏^{2} + \frac{1}{2}(w_{n}^{2} − u_{2n} − 2 z_{2n−1}) 𝕏 − (−\sqrt{7} )^{n}, (4.12)

(𝕏 − 2 sin(\frac{4 π}{7})(2 sin(\frac{2 π}{7}))^{n})(𝕏 − 2 sin(\frac{8 π}{7})(2 sin(\frac{4 π}{7}))^{n})(𝕏 − 2 sin(\frac{2 π}{7})(2 sin(\frac{8 π}{7}))^{n}) =

= 𝕏^{3} − x_{n} 𝕏^{2} + \frac{1}{2}(x_{n}^{2} + w_{2n} − 2 z_{2n−1}) 𝕏 − (−\sqrt{7} )^{n+1}, (4.13)

(𝕏 − 2 sin(\frac{8 π}{7})(2 sin(\frac{2 π}{7}))^{n})(𝕏 − 2 sin(\frac{2 π}{7})(2 sin(\frac{4 π}{7}))^{n})(𝕏 − 2 sin(\frac{4 π}{7})(2 sin(\frac{8 π}{7}))^{n}) =

= 𝕏^{3} − y_{n} 𝕏^{2} + \frac{1}{2}(y_{n}^{2} + u_{2n} − 2 z_{2n−1}) 𝕏 − (−\sqrt{7} )^{n+1}, (4.14)

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

= 𝕏^{3} − z_{n} 𝕏^{2} + \frac{1}{2}(z_{n}^{2} − z_{2n+1}) 𝕏 − (−\sqrt{7} )^{n+1}. (4.15)

Additionally, primarily to describe the generating functions of the sequences {u_{n}}, {v_{n}}, {w_{n}}, {x_{n}}, {y_{n}} and {z_{n}} (see Section 7) six new recurrence sequences were introduced, which, from a certain point of view, are conjugated with sequences {u_{n}}, …, {z_{n}}. Let us set

u_{n}^{∗} = 2 cos(\frac{2 π}{7})(4 sin(\frac{4 π}{7}) sin(\frac{8 π}{7}))^{n} + 2 cos(\frac{4 π}{7})(4 sin(\frac{2 π}{7}) sin(\frac{8 π}{7}))^{n} +
+ 2 cos(\frac{8 π}{7})(4 sin(\frac{2 π}{7}) sin(\frac{4 π}{7}))^{n},  (4.16)
v_{n}^{∗} = 2 cos(\frac{2 π}{7})(4 sin(\frac{2 π}{7}) sin(\frac{4 π}{7}))^{n} + 2 cos(\frac{4 π}{7})(4 sin(\frac{4 π}{7}) sin(\frac{8 π}{7}))^{n} +
+ 2 cos(\frac{8 π}{7})(4 sin(\frac{2 π}{7}) sin(\frac{8 π}{7}))^{n},  (4.17)
w_{n}^{∗} = 2 cos(\frac{2 π}{7})(4 sin(\frac{2 π}{7}) sin(\frac{8 π}{7}))^{n} + 2 cos(\frac{4 π}{7})(4 sin(\frac{2 π}{7}) sin(\frac{4 π}{7}))^{n} +
+ 2 cos(\frac{8 π}{7})(4 sin(\frac{4 π}{7}) sin(\frac{8 π}{7}))^{n},  (4.18)
x_{n}^{∗} = 2 sin(\frac{2 π}{7})(4 sin(\frac{2 π}{7}) sin(\frac{4 π}{7}))^{n} + 2 sin(\frac{4 π}{7})(4 sin(\frac{4 π}{7}) sin(\frac{8 π}{7}))^{n} +
+ 2 sin(\frac{8 π}{7})(4 sin(\frac{2 π}{7}) sin(\frac{8 π}{7}))^{n},  (4.19)
y_{n}^{∗} = 2 sin(\frac{2 π}{7})(4 sin(\frac{2 π}{7}) sin(\frac{8 π}{7}))^{n} + 2 sin(\frac{4 π}{7})(4 sin(\frac{2 π}{7}) sin(\frac{4 π}{7}))^{n} +
+ 2 sin(\frac{8 π}{7})(4 sin(\frac{4 π}{7}) sin(\frac{8 π}{7}))^{n},  (4.20)
z_{n}^{∗} = 2 sin(\frac{2 π}{7})(4 sin(\frac{4 π}{7}) sin(\frac{8 π}{7}))^{n} + 2 sin(\frac{4 π}{7})(4 sin(\frac{2 π}{7}) sin(\frac{8 π}{7}))^{n} +
+ 2 sin(\frac{8 π}{7})(4 sin(\frac{2 π}{7}) sin(\frac{4 π}{7}))^{n},  (4.21)

for n ∈ ℕ and u_{0}^{∗} = v_{0}^{∗} = w_{0}^{∗} = −1 and x_{0}^{∗} = y_{0}^{∗} = z_{0}^{∗} = \sqrt{7} (see Table 4).

An easy computation shows that the following recurrence relations hold

\{ \begin{matrix}u_{n+1}^{∗} = u_{n}^{∗} + w_{n}^{∗} − v_{n}^{∗} − z_{n−1}^{2} + z_{2n−1}, \\ v_{n+1}^{∗} = z_{n−1}^{2} − z_{2n−1} − u_{n}^{∗}, \\ w_{n+1}^{∗} = u_{n}^{∗} − w_{n}^{∗}, \end{matrix}  (4.22)

for every n ≥ 1. Hence, we obtain

u_{n+1}^{∗} + v_{n+1}^{∗} = w_{n}^{∗} − v_{n}^{∗},  (4.23)
u_{n}^{∗} = w_{n+1}^{∗} + w_{n}^{∗},  (4.24)
w_{n+2}^{∗} + w_{n+1}^{∗} − w_{n}^{∗} = −v_{n+1}^{∗} − v_{n}^{∗},  (4.25)
w_{n+3}^{∗} − 3 w_{n+1}^{∗} − w_{n}^{∗} = z_{2n+1} + z_{2n−1} − z_{n}^{2} − z_{n−1}^{2}.  (4.26)

for every n ≥ 1.

Also the following recurrence relations hold

\{ \begin{matrix}x_{n+1}^{∗} = 2 y_{n}^{∗} − z_{n}^{∗}, \\ y_{n+1}^{∗} = 2 x_{n}^{∗} + y_{n}^{∗} − z_{n}^{∗}, \\ z_{n+1}^{∗} = −x_{n}^{∗} − y_{n}^{∗} − z_{n}^{∗}, \end{matrix}  (4.27)

for every n ∈ ℕ, which implies the identities

y_{n+1}^{∗} = −x_{n+2}^{∗} + 2 x_{n+1}^{∗} + 7 x_{n}^{∗},  (4.28)
x_{n+2}^{∗} + 7 x_{n}^{∗} + 7 x_{n−1}^{∗} = 0  (4.29)

for every n ≥ 1. Theorem 4.3 The following decompositions of polynomials hold:

(𝕏 − 2 cos(\frac{2 π}{7})(4 sin(\frac{4 π}{7}) sin(\frac{8 π}{7}))^{n})(𝕏 − 2 cos(\frac{4 π}{7})(4 sin(\frac{2 π}{7}) sin(\frac{8 π}{7}))^{n}) ×

× (𝕏 − 2 cos(\frac{8 π}{7})(4 sin(\frac{2 π}{7}) sin(\frac{4 π}{7}))^{n}) =

= 𝕏^{3} − u_{n}^{∗} 𝕏^{2} + (−\sqrt{7} )^{n} (u_{n} + v_{n}) 𝕏 − 7^{n}, (4.30)

(𝕏 − 2 cos(\frac{2 π}{7})(4 sin(\frac{2 π}{7}) sin(\frac{4 π}{7}))^{n})(𝕏 − 2 cos(\frac{4 π}{7})(4 sin(\frac{4 π}{7}) sin(\frac{8 π}{7}))^{n}) ×

× (𝕏 − 2 cos(\frac{8 π}{7})(4 sin(\frac{2 π}{7}) sin(\frac{8 π}{7}))^{n}) =

= 𝕏^{3} − v_{n}^{∗} 𝕏^{2} + (−\sqrt{7} )^{n} (v_{n} + w_{n}) 𝕏 − 7^{n}, (4.31)

(𝕏 − 2 cos(\frac{2 π}{7})(4 sin(\frac{2 π}{7}) sin(\frac{8 π}{7}))^{n})(𝕏 − 2 cos(\frac{4 π}{7})(4 sin(\frac{2 π}{7}) sin(\frac{4 π}{7}))^{n}) ×

× (𝕏 − 2 cos(\frac{8 π}{7})(4 sin(\frac{4 π}{7}) sin(\frac{8 π}{7}))^{n}) =

= 𝕏^{3} − w_{n}^{∗} 𝕏^{2} + (−\sqrt{7} )^{n} (u_{n} + w_{n}) 𝕏 − 7^{n}, (4.32)

(𝕏 − 2 sin(\frac{2 π}{7})(4 sin(\frac{2 π}{7}) sin(\frac{4 π}{7}))^{n})(𝕏 − 2 sin(\frac{4 π}{7})(4 sin(\frac{4 π}{7}) sin(\frac{8 π}{7}))^{n}) ×

× (𝕏 − 2 sin(\frac{8 π}{7})(4 sin(\frac{2 π}{7}) sin(\frac{8 π}{7}))^{n}) = = 𝕏^{3} − x_{n}^{∗} 𝕏^{2} + (−\sqrt{7} )^{n} (w_{n} − v_{n}) 𝕏 − 7^{n}, (4.33)

(𝕏 − 2 sin(\frac{2 π}{7})(4 sin(\frac{2 π}{7}) sin(\frac{8 π}{7}))^{n})(𝕏 − 2 sin(\frac{4 π}{7})(4 sin(\frac{2 π}{7}) sin(\frac{4 π}{7}))^{n}) ×

× (𝕏 − 2 sin(\frac{8 π}{7})(4 sin(\frac{4 π}{7}) sin(\frac{8 π}{7}))^{n}) = = 𝕏^{3} − y_{n}^{∗} 𝕏^{2} + (−\sqrt{7} )^{n} (u_{n} − w_{n}) 𝕏 − 7^{n}, (4.34)

(𝕏 − 2 sin(\frac{2 π}{7})(4 sin(\frac{4 π}{7}) sin(\frac{8 π}{7}))^{n})(𝕏 − 2 sin(\frac{4 π}{7})(4 sin(\frac{2 π}{7}) sin(\frac{8 π}{7}))^{n}) ×

× (𝕏 − 2 sin(\frac{8 π}{7})(4 sin(\frac{2 π}{7}) sin(\frac{4 π}{7}))^{n}) =

= 𝕏^{3} − z_{n}^{∗} 𝕏^{2} + (−\sqrt{7} )^{n} (v_{n} − w_{n}) 𝕏 − 7^{n}. (4.35)

Lemma 4.4 The following two groups of identities hold:

\begin{aligned}u_{n}^{2} & = 2 z_{2n−1} + v_{2n} + v_{n}^{∗} + u_{n}^{∗}, \\ v_{n}^{2} & = 2 z_{2n−1} + w_{2n} + v_{n}^{∗} + w_{n}^{∗}, \\ w_{n}^{2} & = 2 z_{2n−1} + u_{2n} + u_{n}^{∗} + w_{n}^{∗}, \\ x_{n}^{2} & = 2 z_{2n−1} − w_{2n} + w_{n}^{∗} − v_{n}^{∗}, \\ y_{n}^{2} & = 2 z_{2n−1} − u_{2n} + u_{n}^{∗} − w_{n}^{∗}, \\ z_{n}^{2} & = 2 z_{2n−1} + 2 vn^{′} − 2 u_{n}^{∗}, \end{aligned}  (4.36)

and

\begin{aligned}(u_{n}^{∗})^{2} & = z_{2n−1}^{2} − z_{4n−1} + v_{n}^{∗} + 2 (−\sqrt{7} )^{n}(u_{n} + v_{n}), \\ (v_{n}^{∗})^{2} & = z_{2n−1}^{2} − z_{4n−1} + w_{n}^{∗} + 2 (−\sqrt{7} )^{n}(v_{n} + w_{n}), \\ (w_{n}^{∗})^{2} & = z_{2n−1}^{2} − z_{4n−1} + u_{n}^{∗} + 2 (−\sqrt{7} )^{n}(u_{n} + w_{n}), \\ (x_{n}^{∗})^{2} & = z_{2n−1}^{2} − z_{4n−1} − w_{n}^{∗} + 2 (−\sqrt{7} )^{n}(w_{n} − v_{n}), \\ (y_{n}^{∗})^{2} & = z_{2n−1}^{2} − z_{4n−1} − u_{n}^{∗} + 2 (−\sqrt{7} )^{n}(u_{n} − w_{n}), \\ (z_{n}^{∗})^{2} & = z_{2n−1}^{2} − z_{4n−1} − v_{n}^{∗} + 2 (−\sqrt{7} )^{n}(v_{n} − u_{n}). \end{aligned}  (4.37)

5 Some Ramanujan-type formulas

Let p, q, r ∈ ℝ. Shevelev [4] (see also [2]) proved that if z_{1}, z_{2}, z_{3} are roots of the polynomial

w(z) = z^{3} + p z^{2} + q z + r

and z_{1}, z_{2}, z_{3} are all reals and at least the following condition holds

p \sqrt[3]{r} + 3 (\sqrt[3]{r})^{2} + q = 0,  (5.1)

then the following formula holds

\sqrt[3]{z_{1}} + \sqrt[3]{z_{2}} + \sqrt[3]{z_{3}} = \sqrt[3]{−p − 6 \sqrt[3]{r} + \sqrt[3]{(p + 6 \sqrt[3]{r})^{3} − (p − 3 \sqrt[3]{r})^{3}}} = = \sqrt[3]{−p − 6 \sqrt[3]{r} + 3 \sqrt[3]{9 r + 3 p (\sqrt[3]{r})^{2} + p^{2} \sqrt[3]{r}}} (5.2)

(all radicals are determined to be real).

It was verified that only following polynomials (among all discussed in this paper) satisfy the condition (5.1); simultanously below the respective form of identity (5.2) is presented:

polynomial (1.4) and polynomial (2.1) for n = 1 (it is the classical Ramanujan formula):

\sqrt[3]{2 cos(\frac{2 π}{7})} + \sqrt[3]{2 cos(\frac{4 π}{7})} + \sqrt[3]{2 cos(\frac{8 π}{7})} = \sqrt[3]{5 − 3 \sqrt[3]{7}};  (5.3)

polynomial (3.31) for n = 4:

cos(\frac{2 π}{7}) \sqrt[3]{sec(\frac{8 π}{7})} + cos(\frac{4 π}{7}) \sqrt[3]{sec(\frac{2 π}{7})} + cos(\frac{8 π}{7}) \sqrt[3]{sec(\frac{4 π}{7})} = \frac{1}{2} \sqrt[3]{18 (2 − \sqrt[3]{7})}; (5.4)

polynomial (3.31) for n = 3:

2 cos(\frac{2 π}{7}) \sqrt[3]{2 cos(\frac{4 π}{7})} + 2 cos(\frac{4 π}{7}) \sqrt[3]{2 cos(\frac{8 π}{7})} + 2 cos(\frac{8 π}{7}) \sqrt[3]{2 cos(\frac{2 π}{7})} =

= \sqrt[3]{−2 − 3 \sqrt[3]{49}}; (5.5)

polynomial (3.32) for n = 2, 3, respectively:

\sqrt[3]{\frac{cos(\frac{2 π}{7})}{1 + cos(\frac{8 π}{7})}} + \sqrt[3]{\frac{cos(\frac{4 π}{7})}{1 + cos(\frac{2 π}{7})}} + \sqrt[3]{\frac{cos(\frac{8 π}{7})}{1 + cos(\frac{4 π}{7})}} = \sqrt[3]{11 − 3 \sqrt[3]{49}},  (5.6)

sec(\frac{4 π}{7}) \sqrt[3]{2 cos(\frac{2 π}{7})} + sec(\frac{8 π}{7}) \sqrt[3]{2 cos(\frac{4 π}{7})} + sec(\frac{2 π}{7}) \sqrt[3]{2 cos(\frac{8 π}{7})} = −2 \sqrt[3]{9 (1 + \sqrt[3]{7})}; (5.7)

polynomial (3.30) for n = 1, polynomial (3.31) for n = 1 and polynomial (3.34) for n = 2:

\sqrt[3]{sec(\frac{2 π}{7})} + \sqrt[3]{sec(\frac{4 π}{7})} + \sqrt[3]{sec(\frac{8 π}{7})} = \sqrt[3]{8 − 6 \sqrt[3]{7}};  (5.8)

polynomial (4.11) for n = 6:

sin^{2}(\frac{2 π}{7}) \sqrt[3]{2 cos(\frac{4 π}{7})} + sin^{2}(\frac{4 π}{7}) \sqrt[3]{2 cos(\frac{8 π}{7})} + sin^{2}(\frac{8 π}{7}) \sqrt[3]{2 cos(\frac{2 π}{7})} =

= −\frac{1}{4} \sqrt[3]{63 (1 + \sqrt[3]{7})}; (5.9)

polynomial (4.13) for n = 2:

\sqrt[3]{\frac{sin(\frac{2 π}{7})}{sin(\frac{8 π}{7})}} + \sqrt[3]{\frac{sin(\frac{4 π}{7})}{sin(\frac{2 π}{7})}} + \sqrt[3]{\frac{sin(\frac{8 π}{7})}{sin(\frac{4 π}{7})}} = \sqrt[3]{5 − 3 \sqrt[3]{7}};  (5.10)

polynomial (4.14) for n = 2:

\sqrt[3]{\frac{sin(\frac{2 π}{7})}{sin(\frac{4 π}{7})}} + \sqrt[3]{\frac{sin(\frac{4 π}{7})}{sin(\frac{8 π}{7})}} + \sqrt[3]{\frac{sin(\frac{8 π}{7})}{sin(\frac{2 π}{7})}} = \sqrt[3]{4 − 3 \sqrt[3]{7}};  (5.11)

polynomial (4.14) for n = 5:

sin^{2}(\frac{2 π}{7}) \sqrt[3]{\frac{sin(\frac{8 π}{7})}{sin(\frac{2 π}{7})}} + sin^{2}(\frac{4 π}{7}) \sqrt[3]{\frac{sin(\frac{2 π}{7})}{sin(\frac{4 π}{7})}} + sin^{2}(\frac{8 π}{7}) \sqrt[3]{\frac{sin(\frac{4 π}{7})}{sin(\frac{8 π}{7})}} = \frac{1}{4} \sqrt[3]{77 − 21 \sqrt[3]{49}}; (5.12)

polynomial (4.30) for n = 3:

\sqrt[6]{7} ( csc(\frac{2 π}{7}) \sqrt[3]{2 cos(\frac{2 π}{7})} + csc(\frac{4 π}{7}) \sqrt[3]{2 cos(\frac{4 π}{7})} + csc(\frac{8 π}{7}) \sqrt[3]{2 cos(\frac{8 π}{7})}) =

= 2 \sqrt[3]{2 + 3 \sqrt[3]{49}}; (5.13)

polynomial (4.32) for n = 3:

\sqrt[6]{7} ( csc(\frac{4 π}{7}) \sqrt[3]{2 cos(\frac{2 π}{7})} + csc(\frac{8 π}{7}) \sqrt[3]{2 cos(\frac{4 π}{7})} + csc(\frac{2 π}{7}) \sqrt[3]{2 cos(\frac{8 π}{7})}) =

= 2 \sqrt[3]{3 \sqrt[3]{7} − 5}; (5.14)

polynomial (4.32) for n = 6:

csc^{2}(\frac{4 π}{7}) \sqrt[3]{2 cos(\frac{2 π}{7})} + csc^{2}(\frac{8 π}{7}) \sqrt[3]{2 cos(\frac{4 π}{7})} + csc^{2}(\frac{2 π}{7}) \sqrt[3]{2 cos(\frac{8 π}{7})} =

= −4 \sqrt[3]{\frac{2 + 3 \sqrt[3]{49}}{7}}. (5.15)

Remark 5.1 Shevelev [4] presented the identities (5.10) and (5.11) in the following alternative form:

\sqrt[3]{2 + sec(\frac{2 π}{7})} + \sqrt[3]{2 + sec(\frac{4 π}{7})} + \sqrt[3]{2 + sec(\frac{8 π}{7})} = \sqrt[3]{6 \sqrt[3]{7} − 10}  (5.16)

and

\sqrt[3]{1 + 2 cos(\frac{2 π}{7})} + \sqrt[3]{1 + 2 cos(\frac{4 π}{7})} + \sqrt[3]{1 + 2 cos(\frac{8 π}{7})} =

= \sqrt[3]{\frac{2 cos(\frac{2 π}{7})}{2 cos(\frac{2 π}{7}) + 1}} + \sqrt[3]{\frac{2 cos(\frac{4 π}{7})}{2 cos(\frac{4 π}{7}) + 1}} + \sqrt[3]{\frac{2 cos(\frac{8 π}{7})}{2 cos(\frac{8 π}{7}) + 1}} = \sqrt[3]{3 \sqrt[3]{7} − 4}, (5.17)

respectively. Remark 5.2 By (5.9), (5.3) and by applied the formula 2 sin^{2} α = 1 − cos(2α) we obtain

2 cos (\frac{2 π}{7}) \sqrt[3]{2 cos (\frac{2 π}{7})} + 2 cos (\frac{4 π}{7}) \sqrt[3]{2 cos (\frac{4 π}{7})} + 2 cos (\frac{8 π}{7}) \sqrt[3]{2 cos (\frac{8 π}{7})} =

= \sqrt[3]{63 (1 + \sqrt[3]{7})} − 2 \sqrt[3]{5 − 3 \sqrt[3]{7}} . (5.18)

Moreover, from (5.3), (5.8) and the equality 8 cos(\frac{2π}{7}) cos(\frac{4π}{7}) cos(\frac{8π}{7}) = 1 we get \sqrt[3]{(2 cos (\frac{2 π}{7}))^{2}} + \sqrt[3]{(2 cos (\frac{4 π}{7}))^{2}} + \sqrt[3]{(2 cos (\frac{8 π}{7}))^{2}} =

= \sqrt[3]{(5 − 3 \sqrt[3]{7})^{2}} − 2 \sqrt[3]{4 − 3 \sqrt[3]{7}} . (5.19)

6 The sine-Fibonacci numbers of order 7

The quasi-Fibonacci numbers of order 7 introduced by Wituła et al. in [10] and further developed in Section 1 constitute the simplest, only one-parameter type of the so-called cosine-Fibonacci numbers of order 7. The name “cosine-Fibonacci numbers” is derived from the form of the decomposition of formulas: (1 + δ (ξ + ξ^{6}))^{n} = (1 + 2 δ cos(\frac{2π}{7}))^{n}, … In this Section we shall introduce and analyze a sine variety of these numbers, created in the course of decomposing formulas: (1 + δ (ξ − ξ^{6}))^{n} = (1 + 2 i δ sin(\frac{2π}{7}))^{n}, …

Theorem 6.1 Let ξ, δ ∈ ℂ, ξ^{7} = 1 and ξ ≠ 1. Then, for every n ∈ ℕ, there exist polynomials p_{n}, r_{n}, s_{n}, k_{n}, l_{n} ∈ ℤ[δ], called here the sine-Fibonacci numbers of order 7, so that

(1 + δ (ξ − ξ^{6}))^{n} = p_{n}(δ) + r_{n}(δ) (ξ − ξ^{6}) + s_{n}(δ) (ξ^{2} − ξ^{5}) +
+ k_{n}(δ) (ξ + ξ^{6}) + l_{n}(δ) (ξ^{2} + ξ^{5}),  (6.1)
(1 + δ (ξ^{2} − ξ^{5}))^{n} = p_{n}(δ) + r_{n}(δ) (ξ^{2} − ξ^{5}) + s_{n}(δ) (ξ^{4} − ξ^{3}) +
+ k_{n}(δ) (ξ^{2} + ξ^{5}) + l_{n}(δ) (ξ^{4} + ξ^{3}),  (6.2)
(1 + δ (ξ^{4} − ξ^{3}))^{n} = p_{n}(δ) + r_{n}(δ) (ξ^{4} − ξ^{3}) + s_{n}(δ) (ξ − ξ^{6}) +
+ k_{n}(δ) (ξ^{4} + ξ^{3}) + l_{n}(δ) (ξ + ξ^{6}).  (6.3)

We have p_{1}(δ) = 1, r_{1}(δ) = δ and s_{1}(δ) = k_{1}(δ) = l_{1}(δ) = 0 (see Table 7, where the initial values of these polynomial are presented). These polynomials are connected by recurrence relations

\{ \begin{matrix}p_{n+1}(δ) = p_{n}(δ) − δ s_{n}(δ) − 2 δ r_{n}(δ) − i \sqrt{7} δl_{n}(δ), \\ r_{n+1}(δ) = r_{n}(δ) + δ p_{n}(δ), \\ s_{n+1}(δ) = s_{n}(δ) + δ k_{n}(δ) + δ l_{n}(δ), \\ k_{n+1}(δ) = k_{n}(δ) − 2 δ s_{n}(δ), \\ l_{n+1}(δ) = l_{n}(δ) + δ r_{n}(δ) − δ s_{n}(δ), \end{matrix}  (6.4)

for n ∈ ℕ. Hence, the following relationships can be deduced:

δ p_{n}(δ) = r_{n+1}(δ) − r_{n}(δ),  (6.5)
2 δ s_{n}(δ) = k_{n+1}(δ) − k_{n}(δ),  (6.6)
2 δ^{2} l_{n}(δ) = −k_{n+2}(δ) + 2 k_{n+1}(δ) − (1 + 2 δ^{2}) k_{n}(δ),  (6.7)
2 δ^{3} r_{n}(δ) = −k_{n+3}(δ) + 3 k_{n+2}(δ) − 3 (1 + δ^{2}) k_{n+1}(δ) + (1 + 3 δ^{2}) k_{n}(δ)  (6.8)

and, at last, the main identity

k_{n+5}(δ) − 5 k_{n+4} + (10 + 5 δ^{2}) k_{n+3}(δ) + (i \sqrt{7} δ^{3} − 15 δ^{2} − 10) k_{n+2}(δ)+

+ (7 δ^{4} − i 2 \sqrt{7} δ^{3} + 15 δ^{2} + 5) k_{n+1}(δ)+

+ (i 2 \sqrt{7} δ^{5} − 7 δ^{4} + i \sqrt{7} δ^{3} − 5 δ^{2} − 1) k_{n}(δ) = 0. (6.9)

This identity satisfies, also by (6.5)–(6.8), the remaining polynomials: p_{n}(δ), r_{n}(δ), s_{n}(δ), and l_{n}(δ), n ∈ ℕ. The characteristic polynomial corresponding to identity (6.9) has the following decomposition:

𝕏^{5} − 5 𝕏^{4} + (10 + 5 δ^{2}) 𝕏^{3} + (i \sqrt{7} δ^{3} − 15 δ^{2} − 10) 𝕏^{2} + (7 δ^{4} − i 2 \sqrt{7} δ^{3} + 15 δ^{2} + 5) 𝕏 +

+ i 2 \sqrt{7} δ^{5} − 7 δ^{4} + i \sqrt{7} δ^{3} − 5 δ^{2} − 1 = (𝕏 − 1 − δ (ξ − ξ^{6}))(𝕏 − 1 − δ (ξ^{2} − ξ^{5})) ×

× (𝕏 − 1 − δ (ξ^{4} − ξ^{3}))(𝕏 − 1 + δ (ξ + ξ^{2} + ξ^{4}))(𝕏 − 1 + δ (ξ^{3} + ξ^{5} + ξ^{6})) =
= (𝕏^{3} + (−3 − i \sqrt{7} δ) 𝕏^{2} + (3 + i 2 \sqrt{7} δ) 𝕏 − i \sqrt{7} δ^{3} − i \sqrt{7} δ − 1) ×

× (𝕏^{2} + (−2 + i \sqrt{7} δ) 𝕏 + 1 − 2 δ^{2} − i \sqrt{7} δ). (6.10)

Hence, we obtain, for example, the following explicit form of k_{n}(δ):

k_{n}(δ) = a (1 + δ (ξ − ξ^{6}))^{n} + b (1 + δ (ξ^{2} − ξ^{5}))^{n} +

+ c (1 + δ (ξ^{4} − ξ^{3}))^{n} + d (1 − δ (ξ + ξ^{2} + ξ^{4}))^{n} + e (1 + δ (ξ^{3} + ξ^{5} + ξ^{6}))^{n}, (6.11)

where

a = −2(5 + (ξ^{2} + ξ^{5}) + 7 (ξ^{3} + ξ^{4}))^{−1} = \frac{2}{301}(46 (ξ + ξ^{6}) + 5 (ξ^{3} + ξ^{4}) − 11),
b = 2(2 + 7 (ξ^{2} + ξ^{5}) + 6 (ξ^{3} + ξ^{4}))^{−1} = \frac{2}{187}(19 (ξ + ξ^{6}) + 46 (ξ^{2} + ξ^{5}) − 5),
c = −2(4 + 6 (ξ^{2} + ξ^{5}) − (ξ^{3} + ξ^{4}))^{−1} = \frac{2}{301}(5 (ξ^{2} + ξ^{5}) + 46 (ξ^{3} + ξ^{4}) − 11),
d = −2(−5 + 2 (ξ + ξ^{2} + ξ^{4}))^{−1} = \frac{2}{43}(5 − 2 (ξ^{3} + ξ^{5} + ξ^{6})),
e = 2(7 + 2 (ξ + ξ^{2} + ξ^{4}))^{−1} = \frac{2}{43}(7 + 2 (ξ^{3} + ξ^{5} + ξ^{6})).

Corollary 6.2 The following identity holds:

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

= 3 p_{n}(δ) − k_{n}(δ) − l_{n}(δ) + i \sqrt{7}(r_{n}(δ) + s_{n}(δ)). (6.12)

Definition 6.3 To simplify the notation, we shall write Ω_{n} instead of Ω_{n}(1). Theorem 6.4 The following decompositions of the polynomials hold:

q_{n}(𝕏; δ) :=

= (𝕏 − (1 + δ (2 i sin(\frac{2 π}{7}))^{n}))(𝕏 − (1 + δ (2 i sin(\frac{4 π}{7}))^{n}))(𝕏 − (1 + δ (2 i sin(\frac{8 π}{7}))^{n})) =

= 𝕏^{3} − Ω_{n}(δ) 𝕏^{2} + \frac{1}{2}((Ω_{n}(δ))^{2} − Ω_{2n}(δ)) 𝕏 − (1 + i \sqrt{7} δ + i \sqrt{7} δ^{3})^{n}, (6.13)

(x − ( cot(\frac{2 π}{7}))^{n})(x − ( cot(\frac{4 π}{7}))^{n})(x − ( cot(\frac{8 π}{7}))^{n}) =

= x^{3} − (\frac{3}{\sqrt{7}})^{n} A_{n}(\frac{2}{3}) x^{2} + Ω_{n}(\frac{2 i}{\sqrt{7}}) x + \frac{(−1)^{n−1}}{(\sqrt{7} )^{n}} . (6.14)

Moreover, the following identity holds:

Ω_{n}^{2}(δ) = Ω_{2n}(δ)+2 (3 p_{n}^{2}(δ)−2 l_{n}^{2}(δ)−2 k_{n}^{2}(δ)−2 p_{n}(δ) k_{n}(δ)−2 p_{n}(δ) l_{n}(δ)−7 r_{n}(δ) s_{n}(δ)+

+ 3 k_{n}(δ) l_{n}(δ) + i 2 \sqrt{7} (p_{n}(δ) r_{n}(δ) + p_{n}(δ) s_{n}(δ) − r_{n}(δ) k_{n}(δ) − s_{n}(δ) l_{n}(δ)) +

+ i \sqrt{7}(r_{n}(δ) l_{n}(δ) − k_{n}(δ) s_{n}(δ))) =

= Ω_{2n}(δ) + 4 p_{n}(δ) Ω_{n}(δ) + 2 ( − 3 p_{n}^{2}(δ) − 2 l_{n}^{2}(δ) − 2 k_{n}^{2}(δ) − 7 r_{n}(δ) s_{n}(δ) + 3 k_{n}(δ) l_{n}(δ) + + i \sqrt{7}(r_{n}(δ) l_{n}(δ) − 2 r_{n}(δ) k_{n}(δ) − k_{n}(δ) s_{n}(δ) − 2 s_{n}(δ) l_{n}(δ))). (6.15)

7 Generating functions

The generating functions of almost all sequences discussed and defined in this paper are presented in this section.

By (1.9) and (2.2) we obtain (for k ∈ ℕ):

∑_{n=0}^{∞} A_{kn}(δ) 𝕏^{n} = (1 − (1 + δ (ξ + ξ^{6}))^{k} 𝕏)^{−1} + (1 − (1 + δ (ξ^{2} + ξ^{5}))^{k} 𝕏)^{−1} +

+ (1 − (1 + δ (ξ^{3} + ξ^{4}))^{k} 𝕏)^{−1} = \frac{3 − 2 A_{k}(δ) 𝕏 + B_{k}(δ) 𝕏^{2}}{1 − A_{k}(δ) 𝕏 + B_{k}(δ) 𝕏^{2} − (1 − δ − 2 δ^{2} + δ^{3})^{k} 𝕏^{3}} , (7.1)

and, in the sequel, for k = 1 we get

∑_{n=0}^{∞} A_{n}(δ) 𝕏^{n} = (1 − (1 + δ (ξ + ξ^{6})) 𝕏)^{−1} + (1 − (1 + δ (ξ^{2} + ξ^{5})) 𝕏)^{−1} +

+ (1 − (1 + δ (ξ^{3} + ξ^{4})) 𝕏)^{−1} = \frac{𝕏^{2} p7^{′}(1/𝕏; δ)}{𝕏^{3} p_{7}(1/𝕏; δ)} =

= \frac{3 + 2 (δ − 3) 𝕏 + (3 − 2 δ − 2 δ^{2}) 𝕏^{2}}{1 + (δ − 3) 𝕏 + (3 − 2 δ − 2 δ^{2}) 𝕏^{2} + (−1 + δ + 2 δ^{2} − δ^{3}) 𝕏^{3}} , (7.2)

where p_{7}(𝕏, δ) = 𝕏^{3} + (δ − 3)𝕏^{2} + (3 − 2δ − 2δ^{2})𝕏 + (−1 + δ + 2δ^{2} − δ^{3}) (see [10]). By (1.10) and (2.7) we get (for k ∈ ℕ):

∑_{n=0}^{∞} B_{kn}(δ) 𝕏^{n} = \frac{𝕏^{2} rk^{′}(1/𝕏; δ)}{𝕏^{3} r_{k}(1/𝕏; δ)} =

= \frac{3 − 2 B_{k}(δ) 𝕏 + (1 − δ − 2 δ^{2} + δ^{3}) A_{k}(δ) 𝕏^{2}}{1 − B_{k}(δ) 𝕏 + (1 − δ − 2 δ^{2} + δ^{3}) A_{k}(δ) 𝕏^{2} − (1 − δ − 2 δ^{2} + δ^{3})^{2k} 𝕏^{3}} , (7.3)

where r_{n}(𝕏, δ) is defined in (2.7). By (3.19) from [10] we obtain

∑_{n=0}^{∞} A_{n}(δ) 𝕏^{n} = \frac{2 − ξ^{3} − ξ^{4}}{1 − (1 + δ (ξ + ξ^{6})) 𝕏} + \frac{2 − ξ − ξ^{6}}{1 − (1 + δ (ξ^{2} + ξ^{5})) 𝕏} +

+ \frac{2 − ξ^{2} − ξ^{5}}{1 − (1 + δ (ξ^{3} + ξ^{4})) 𝕏} = \frac{1 + (−2 + δ) 𝕏 + (1 − δ) 𝕏^{2}}{𝕏^{3} p_{7}(1/𝕏; δ)}; (7.4)

by (3.18) and (3.17) from [10] we obtain respectively:

∑_{n=0}^{∞} B_{n}(δ) 𝕏^{n} = \frac{ξ + ξ^{6} − ξ^{3} − ξ^{4}}{1 − (1 + δ (ξ + ξ^{6})) 𝕏} + \frac{ξ^{2} + ξ^{5} − ξ − ξ^{6}}{1 − (1 + δ (ξ^{2} + ξ^{5})) 𝕏} +

+ \frac{ξ^{3} + ξ^{4} − ξ^{2} − ξ^{5}}{1 − (1 + δ (ξ^{3} + ξ^{4})) 𝕏} = \frac{δ 𝕏 + (δ^{2} − δ) 𝕏^{2}}{𝕏^{3} p_{7}(1/𝕏; δ)} (7.5)

and

∑_{n=0}^{∞} C_{n}(δ) 𝕏^{n} = ∑_{n=0}^{∞} (3A_{n}(δ) − B_{n}(δ) − A_{n}(δ)) 𝕏^{n} = \frac{δ^{2} 𝕏^{2}}{𝕏^{3} p_{7}(1/𝕏; δ)}.  (7.6)

Equally, on the grounds of (3.1)–(3.3) and (3.24)–(3.26), we obtain the following formulas:

∑_{n=0}^{∞} a_{kn} 𝕏^{n} = \frac{\sqrt{7} − (b_{k} + c_{k}) 𝕏 + Ã_{k} 𝕏^{2}}{1 − B_{2k} 𝕏 + A_{2k} 𝕏^{2} − 𝕏^{3}} ,  (7.7)
∑_{n=0}^{∞} b_{kn} 𝕏^{n} = \frac{\sqrt{7} − (a_{k} + c_{k}) 𝕏 + B̃_{k} 𝕏^{2}}{1 − B_{2k} 𝕏 + A_{2k} 𝕏^{2} − 𝕏^{3}} ,  (7.8)
∑_{n=0}^{∞} c_{kn} 𝕏^{n} = \frac{\sqrt{7} − (a_{k} + b_{k}) 𝕏 + C̃_{k} 𝕏^{2}}{1 − B_{2k} 𝕏 + A_{2k} 𝕏^{2} − 𝕏^{3}} .  (7.9)

The special cases (for k = 1) can also be generated in the following selective ways: by (3.1) and (1.5) we obtain

∑_{n=0}^{∞} a_{n} 𝕏^{n} = \frac{2 sin (\frac{2 π}{7})}{1 − 4 cos^{2} (\frac{8 π}{7}) 𝕏} + \frac{2 sin (\frac{4 π}{7})}{1 − 4 cos^{2} (\frac{2 π}{7}) 𝕏} + \frac{2 sin (\frac{8 π}{7})}{1 − 4 cos^{2} (\frac{4 π}{7}) 𝕏} =

= \frac{\sqrt{7} − 2 \sqrt{7} 𝕏 − \sqrt{7} 𝕏^{2}}{1 − 5 𝕏 + 6 𝕏^{2} − 𝕏^{3}} ; (7.10)

however, by (3.2) and (3.4) we obtain

∑_{n=0}^{∞} b_{n} 𝕏^{n} = ∑_{n=0}^{∞} (a_{n+1} + 2 a_{n}) 𝕏^{n} = \frac{1 + 2 𝕏}{𝕏}( ∑_{n=0}^{∞} a_{n} 𝕏^{n}) − \frac{\sqrt{7}}{𝕏} =

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

= \frac{\sqrt{7} − 3 \sqrt{7} 𝕏 + 3 \sqrt{7} 𝕏^{2}}{1 − 5 𝕏 + 6 𝕏^{2} − 𝕏^{3}} (7.11)

and

∑_{n=0}^{∞} c_{n} 𝕏^{n} = ∑_{n=0}^{∞} (a_{n} + 2 b_{n} − b_{n+1}) 𝕏^{n} = \frac{\sqrt{7} − 5 \sqrt{7} 𝕏 + 4 \sqrt{7} 𝕏^{2}}{1 − 5 𝕏 + 6 𝕏^{2} − 𝕏^{3}} .  (7.12)

By (6.12) and (6.13) we get

∑_{n=0}^{∞} Ω_{kn}(δ) 𝕏^{n} = (1 − (1 + δ (ξ − ξ^{6}))^{k} 𝕏)^{−1} + (1 − (1 + δ (ξ^{2} − ξ^{5}))^{k} 𝕏)^{−1} +

+ (1 − (1 + δ (ξ^{3} − ξ^{4}))^{k} 𝕏)^{−1} = \frac{𝕏^{2} qk^{′}(1/𝕏; δ)}{𝕏^{3} q_{k}(1/𝕏; δ)} =

= \frac{3 − 2 Ω_{k}(δ) 𝕏 + \frac{1}{2} ((Ω_{k}(δ))^{2} − Ω_{2k}(δ)) 𝕏^{2}}{1 − Ω_{k}(δ) 𝕏 + \frac{1}{2} ((Ω_{k}(δ))^{2} − Ω_{2k}(δ)) 𝕏^{2} − (1 + i \sqrt{7} δ + i \sqrt{7} δ^{3})^{k} 𝕏^{3}} , (7.13)

and, the special case, for k = 1 (by (6.12) and (6.10)):

∑_{n=0}^{∞} Ω_{n}(δ) 𝕏^{n} = (1 − (1 + δ (ξ − ξ^{6})) 𝕏)^{−1} + (1 − (1 + δ (ξ^{2} − ξ^{5})) 𝕏)^{−1} +

+ (1 − (1 + δ (ξ^{4} − ξ^{3})) 𝕏)^{−1} = \frac{𝕏^{2} q1^{′}(1/𝕏; δ)}{𝕏^{3} q_{1}(1/𝕏; δ)} =

= \frac{3 − 2 (3 + i \sqrt{7} δ) 𝕏 + (3 + i 2 \sqrt{7} δ) 𝕏^{2}}{1 − (3 + i \sqrt{7} δ) 𝕏 + (3 + i 2 \sqrt{7} δ) 𝕏^{2} − (1 + i \sqrt{7} δ + i \sqrt{7} δ^{3}) 𝕏^{3}} , (7.14)

where q_{k}(𝕏; δ) is defined in (6.13).

By (1.4), (3.9)–(3.11) and (3.19)–(3.21) the following formulas may be obtained:

∑_{n=0}^{∞} f_{kn} 𝕏^{n} = \frac{2 cos (\frac{2 π}{7})}{1 − (2 cos(\frac{4 π}{7}))^{k} 𝕏} + \frac{2 cos (\frac{4 π}{7})}{1 − (2 cos(\frac{8 π}{7}))^{k} 𝕏} + \frac{2 cos (\frac{8 π}{7})}{1 − (2 cos(\frac{2 π}{7}))^{k} 𝕏} =

= \frac{−1 − (g_{k} + h_{k}) 𝕏 + F_{k} 𝕏^{2}}{1 − h_{k−1} 𝕏 + \frac{1}{2}(h_{k−1}^{2} − h_{2k−1}) 𝕏^{2} − 𝕏^{3}} , (7.15)

∑_{n=0}^{∞} g_{kn} 𝕏^{n} = \frac{2 cos (\frac{2 π}{7})}{1 − (2 cos(\frac{8 π}{7}))^{k} 𝕏} + \frac{2 cos (\frac{4 π}{7})}{1 − (2 cos(\frac{2 π}{7}))^{k} 𝕏} + \frac{2 cos (\frac{8 π}{7})}{1 − (2 cos(\frac{4 π}{7}))^{k} 𝕏} =

= \frac{−1 − (f_{k} + h_{k}) 𝕏 + G_{k} 𝕏^{2}}{1 − h_{k−1} 𝕏 + \frac{1}{2}(h_{k−1}^{2} − h_{2k−1}) 𝕏^{2} − 𝕏^{3}} , (7.16)

∑_{n=0}^{∞} h_{kn−1} 𝕏^{n} = \frac{1}{1 − (2 cos(\frac{2 π}{7}))^{k} 𝕏} + \frac{1}{1 − (2 cos(\frac{4 π}{7}))^{k} 𝕏} + \frac{1}{1 − (2 cos(\frac{8 π}{7}))^{k} 𝕏} =

= \frac{3 − 2 h_{k−1} 𝕏 + \frac{1}{2}(h_{k−1}^{2} − h_{2k−1}) 𝕏^{2}}{1 − h_{k−1} 𝕏 + \frac{1}{2}(h_{k−1}^{2} − h_{2k−1}) 𝕏^{2} − 𝕏^{3}} =^{by (2.1)} \frac{3 − 2 B_{k} 𝕏 + (−1)^{k} A_{k} 𝕏^{2}}{1 − B_{k} 𝕏 + (−1)^{k} A_{k} 𝕏^{2} − 𝕏^{3}} . (7.17)

(h_{−1} := 1).

We note that the special case of the formulas (7.15) and (7.16) for k = 1 can be treated in another way (polynomial (1.4) will be denoted here by p_{7}(𝕏)):

∑_{n=0}^{∞} h_{n} 𝕏^{n} = \frac{ξ + ξ^{6}}{1 − (ξ + ξ^{6}) 𝕏} + \frac{ξ^{2} + ξ^{5}}{1 − (ξ^{2} + ξ^{5}) 𝕏} + \frac{ξ^{3} + ξ^{4}}{1 − (ξ^{3} + ξ^{4}) 𝕏} =

= −\frac{(𝕏^{3} p_{7}(1/𝕏))′}{𝕏^{3} p_{7}(1/𝕏)} = \frac{−1 + 4 𝕏 + 3 𝕏^{2}}{1 + 𝕏 − 2 𝕏^{2} − 𝕏^{3}} , (7.18)

hence by (3.12):

∑_{n=0}^{∞} g_{n} 𝕏^{n} = −1 + ∑_{n=1}^{∞} (h_{n+1} − 2 h_{n−1}) 𝕏^{n} =

= −1 + 𝕏^{−1} ∑_{n=0}^{∞} h_{n} 𝕏^{n} − 𝕏^{−1} (h_{0} + h_{1} 𝕏) − 2 𝕏 ∑_{n=0}^{∞} h_{n} 𝕏^{n} =

= −6 + 𝕏^{−1} + \frac{1 − 2 𝕏^{2}}{𝕏} ∑_{n=0}^{∞} h_{n} 𝕏^{n} = \frac{−1 − 3 𝕏 + 3 𝕏^{2}}{1 + 𝕏 − 2 𝕏^{2} − 𝕏^{3}} (7.19)

and, finally, using (3.12) again we get

∑_{n=0}^{∞} f_{n} 𝕏^{n} = ∑_{n=0}^{∞} (g_{n+1} − h_{n}) 𝕏^{n} =

= 𝕏^{−1} ∑_{n=0}^{∞} g_{n} 𝕏^{n} + 𝕏^{−1} − ∑_{n=0}^{∞} h_{n} 𝕏^{n} = −\frac{1 + 3 𝕏 + 4 𝕏^{2}}{1 + 𝕏 − 2 𝕏^{2} − 𝕏^{3}} . (7.20)

By (1.1), (4.4)–(4.6), (4.15) and (4.19)–(4.21) we have

∑_{n=0}^{∞} x_{kn} 𝕏^{n} = \frac{2 sin (\frac{2 π}{7})}{1 − (2 sin(\frac{8 π}{7}))^{k} 𝕏} + \frac{2 sin (\frac{4 π}{7})}{1 − (2 sin(\frac{2 π}{7}))^{k} 𝕏} + \frac{2 sin (\frac{8 π}{7})}{1 − (2 sin(\frac{4 π}{7}))^{k} 𝕏} =

= \frac{\sqrt{7} − (y_{k} + z_{k}) 𝕏 + x_{k}^{∗} 𝕏^{2}}{1 − z_{k−1} 𝕏 + \frac{1}{2}(z_{k−1}^{2} − z_{2k−1}) 𝕏^{2} − (−\sqrt{7} )^{k} 𝕏^{3}} , (7.21)

∑_{n=0}^{∞} y_{kn} 𝕏^{n} = \frac{2 sin (\frac{2 π}{7})}{1 − (2 sin(\frac{4 π}{7}))^{k} 𝕏} + \frac{2 sin (\frac{4 π}{7})}{1 − (2 sin(\frac{8 π}{7}))^{k} 𝕏} + \frac{2 sin (\frac{8 π}{7})}{1 − (2 sin(\frac{2 π}{7}))^{k} 𝕏} =

= \frac{\sqrt{7} − (x_{k} + z_{k}) 𝕏 + y_{k}^{∗} 𝕏^{2}}{1 − z_{k−1} 𝕏 + \frac{1}{2}(z_{k−1}^{2} − z_{2k−1}) 𝕏^{2} − (−\sqrt{7} )^{k} 𝕏^{3}} , (7.22)

∑_{n=0}^{∞} z_{kn} 𝕏^{n} = \frac{2 sin (\frac{2 π}{7})}{1 − (2 sin(\frac{2 π}{7}))^{k} 𝕏} + \frac{2 sin (\frac{4 π}{7})}{1 − (2 sin(\frac{4 π}{7}))^{k} 𝕏} + \frac{2 sin (\frac{8 π}{7})}{1 − (2 sin(\frac{8 π}{7}))^{k} 𝕏} =

= \frac{\sqrt{7} − (x_{k} + y_{k}) 𝕏 + z_{k}^{∗} 𝕏^{2}}{1 − z_{k−1} 𝕏 + \frac{1}{2}(z_{k−1}^{2} − z_{2k−1}) 𝕏^{2} − (−\sqrt{7} )^{k} 𝕏^{3}} , (7.23)

and (z_{−1} := 1):

∑_{n=0}^{∞} z_{kn−1} 𝕏^{n} = (1 − (2 sin(\frac{2 π}{7}))^{k} 𝕏)^{−1} + (1 − (2 sin(\frac{4 π}{7}))^{k} 𝕏)^{−1} +

+ (1 − (2 sin(\frac{8 π}{7}))^{k} 𝕏)^{−1} = \frac{3 − 2 z_{k−1} 𝕏 + \frac{1}{2}(z_{k−1}^{2} − z_{2k−1}) 𝕏^{2}}{1 − z_{k−1} 𝕏 + \frac{1}{2}(z_{k−1}^{2} − z_{2k−1}) 𝕏^{2} − (−\sqrt{7} )^{k} 𝕏^{3}} . (7.24)

By (1.1), (1.4), (4.1)–(4.3), (4.15), (4.16)–(4.18), (4.30)–(4.32) and (4.6), respectively, we get

∑_{n=0}^{∞} u_{kn} 𝕏^{n} = \frac{2 cos (\frac{2 π}{7})}{1 − (2 sin(\frac{2 π}{7}))^{k} 𝕏} + \frac{2 cos (\frac{4 π}{7})}{1 − (2 sin(\frac{4 π}{7}))^{k} 𝕏} + \frac{2 cos (\frac{8 π}{7})}{1 − (2 sin(\frac{8 π}{7}))^{k} 𝕏} =

= \frac{−1 − (v_{k} + w_{k}) 𝕏 + u_{k}^{∗} 𝕏^{2}}{1 − z_{k−1} 𝕏 + \frac{1}{2}(z_{k−1}^{2} − z_{2k−1}) 𝕏^{2} − (−\sqrt{7} )^{k} 𝕏^{3}} , (7.25)

∑_{n=0}^{∞} v_{kn} 𝕏^{n} = \frac{2 cos (\frac{4 π}{7})}{1 − (2 sin(\frac{2 π}{7}))^{k} 𝕏} + \frac{2 cos (\frac{8 π}{7})}{1 − (2 sin(\frac{4 π}{7}))^{k} 𝕏} + \frac{2 cos (\frac{2 π}{7})}{1 − (2 sin(\frac{8 π}{7}))^{k} 𝕏} =

= \frac{−1 − (u_{k} + w_{k}) 𝕏 + v_{k}^{∗} 𝕏^{2}}{1 − z_{k−1} 𝕏 + \frac{1}{2}(z_{k−1}^{2} − z_{2k−1}) 𝕏^{2} − (−\sqrt{7} )^{k} 𝕏^{3}} , (7.26)

∑_{n=0}^{∞} w_{kn} 𝕏^{n} = \frac{2 cos (\frac{8 π}{7})}{1 − (2 sin(\frac{2 π}{7}))^{k} 𝕏} + \frac{2 cos (\frac{2 π}{7})}{1 − (2 sin(\frac{4 π}{7}))^{k} 𝕏} + \frac{2 cos (\frac{4 π}{7})}{1 − (2 sin(\frac{8 π}{7}))^{k} 𝕏} =

= \frac{−1 − (u_{k} + v_{k}) 𝕏 + w_{k}^{∗} 𝕏^{2}}{1 − z_{k−1} 𝕏 + \frac{1}{2}(z_{k−1}^{2} − z_{2k−1}) 𝕏^{2} − (−\sqrt{7} )^{k} 𝕏^{3}} , (7.27)

∑_{n=0}^{∞} u_{kn}^{∗} 𝕏^{n} = \frac{2 cos (\frac{2 π}{7})}{1 − (4 sin(\frac{4 π}{7}) sin(\frac{8 π}{7}))^{k} 𝕏} + \frac{2 cos (\frac{4 π}{7})}{1 − (4 sin(\frac{2 π}{7}) sin(\frac{8 π}{7}))^{k} 𝕏} +

+ \frac{2 cos (\frac{8 π}{7})}{1 − (4 sin(\frac{2 π}{7}) sin(\frac{4 π}{7}))^{k} 𝕏} = \frac{−1 − (v_{k}^{∗} + w_{k}^{∗}) 𝕏 + (−\sqrt{7} )^{k} u_{k} 𝕏^{2}}{1 − \frac{1}{2}(z_{k−1}^{2} − z_{2k−1}) 𝕏 + (−\sqrt{7} )^{k} z_{k−1} 𝕏^{2} − 7^{k} 𝕏^{3}} , (7.28)

∑_{n=0}^{∞} v_{kn}^{∗} 𝕏^{n} = \frac{2 cos (\frac{2 π}{7})}{1 − (4 sin(\frac{2 π}{7}) sin(\frac{4 π}{7}))^{k} 𝕏} + \frac{2 cos (\frac{4 π}{7})}{1 − (4 sin(\frac{4 π}{7}) sin(\frac{8 π}{7}))^{k} 𝕏} +

+ \frac{2 cos (\frac{8 π}{7})}{1 − (4 sin(\frac{2 π}{7}) sin(\frac{8 π}{7}))^{k} 𝕏} = \frac{−1 − (u_{k}^{∗} + w_{k}^{∗}) 𝕏 + (−\sqrt{7} )^{k} v_{k} 𝕏^{2}}{1 − \frac{1}{2}(z_{k−1}^{2} − z_{2k−1}) 𝕏 + (−\sqrt{7} )^{k} z_{k−1} 𝕏^{2} − 7^{k} 𝕏^{3}} , (7.29)

∑_{n=0}^{∞} w_{kn}^{∗} 𝕏^{n} = \frac{2 cos (\frac{2 π}{7})}{1 − (4 sin(\frac{2 π}{7}) sin(\frac{8 π}{7}))^{k} 𝕏} + \frac{2 cos (\frac{4 π}{7})}{1 − (4 sin(\frac{2 π}{7}) sin(\frac{4 π}{7}))^{k} 𝕏} +

+ \frac{2 cos (\frac{8 π}{7})}{1 − (4 sin(\frac{4 π}{7}) sin(\frac{8 π}{7}))^{k} 𝕏} = \frac{−1 − (u_{k}^{∗} + v_{k}^{∗}) 𝕏 + (−\sqrt{7} )^{k} w_{k} 𝕏^{2}}{1 − \frac{1}{2}(z_{k−1}^{2} − z_{2k−1}) 𝕏 + (−\sqrt{7} )^{k} z_{k−1} 𝕏^{2} − 7^{k} 𝕏^{3}} . (7.30)

By (1.1), (4.19)–(4.21), (4.33)–(4.35) and (4.6), we obtain

∑_{n=0}^{∞} x_{kn}^{∗} 𝕏^{n} = \frac{2 sin (\frac{2 π}{7})}{1 − (4 sin(\frac{2 π}{7}) sin(\frac{4 π}{7}))^{k} 𝕏} + \frac{2 sin (\frac{4 π}{7})}{1 − (4 sin(\frac{4 π}{7}) sin(\frac{8 π}{7}))^{k} 𝕏} +

+ \frac{2 sin (\frac{8 π}{7})}{1 − (4 sin(\frac{2 π}{7}) sin(\frac{8 π}{7}))^{k} 𝕏} = \frac{\sqrt{7} − (y_{k}^{∗} + z_{k}^{∗}) 𝕏 + (−\sqrt{7} )^{k} x_{k} 𝕏^{2}}{1 − \frac{1}{2}(z_{k−1}^{2} − z_{2k−1}) 𝕏 + (−\sqrt{7} )^{k} z_{k−1} 𝕏^{2} − 7^{k} 𝕏^{3}} , (7.31)

∑_{n=0}^{∞} y_{kn}^{∗} 𝕏^{n} = \frac{2 sin (\frac{2 π}{7})}{1 − (4 sin(\frac{2 π}{7}) sin(\frac{8 π}{7}))^{k} 𝕏} + \frac{2 sin (\frac{4 π}{7})}{1 − (4 sin(\frac{2 π}{7}) sin(\frac{4 π}{7}))^{k} 𝕏} +

+ \frac{2 sin (\frac{8 π}{7})}{1 − (4 sin(\frac{4 π}{7}) sin(\frac{8 π}{7}))^{k} 𝕏} = \frac{\sqrt{7} − (x_{k}^{∗} + z_{k}^{∗}) 𝕏 + (−\sqrt{7} )^{k} y_{k} 𝕏^{2}}{1 − \frac{1}{2}(z_{k−1}^{2} − z_{2k−1}) 𝕏 + (−\sqrt{7} )^{k} z_{k−1} 𝕏^{2} − 7^{k} 𝕏^{3}} , (7.32)

∑_{n=0}^{∞} z_{kn}^{∗} 𝕏^{n} = \frac{2 sin (\frac{2 π}{7})}{1 − (4 sin(\frac{4 π}{7}) sin(\frac{8 π}{7}))^{k} 𝕏} + \frac{2 sin (\frac{4 π}{7})}{1 − (4 sin(\frac{2 π}{7}) sin(\frac{8 π}{7}))^{k} 𝕏} +

+ \frac{2 sin (\frac{8 π}{7})}{1 − (4 sin(\frac{2 π}{7}) sin(\frac{4 π}{7}))^{k} 𝕏} = \frac{\sqrt{7} − (x_{k}^{∗} + y_{k}^{∗}) 𝕏 + (−\sqrt{7} )^{k} z_{k} 𝕏^{2}}{1 − \frac{1}{2}(z_{k−1}^{2} − z_{2k−1}) 𝕏 + (−\sqrt{7} )^{k} z_{k−1} 𝕏^{2} − 7^{k} 𝕏^{3}} . (7.33)

8 Jordan decomposition

For sequences A_{n}(δ), B_{n}(δ) and C_{n}(δ) we have the equality

[ \begin{matrix}A_{n+1}(δ) \\ B_{n+1}(δ) \\ C_{n+1}(δ) \end{matrix} ] = W(δ) [ \begin{matrix}A_{n}(δ) \\ B_{n}(δ) \\ C_{n}(δ) \end{matrix} ] , n ∈ ℕ, (8.1)

where

W(δ) = [ \begin{matrix}1 & 2 δ & −δ \\ δ & 1 & 0 \\ 0 & δ & 1 − δ \end{matrix} ] .  (8.2)

Matrix W(δ) is a diagonalized matrix, and the following decomposition can be obtained

W(δ) = A · [ \begin{matrix}1 + δ (ξ + ξ^{6}) & 0 & 0 \\ 0 & 1 + δ (ξ^{2} + ξ^{5}) & 0 \\ 0 & 0 & 1 + δ (ξ^{3} + ξ^{4}) \end{matrix} ] · A^{−1},  (8.3)

where

A = [ \begin{matrix}1 + (ξ + ξ^{6})^{−1} & 1 + (ξ^{2} + ξ^{5})^{−1} & 1 + (ξ^{3} + ξ^{4})^{−1} \\ (ξ + ξ^{6})^{−2} + (ξ + ξ^{6})^{−1} & (ξ^{2} + ξ^{5})^{−2} + (ξ^{2} + ξ^{5})^{−1} & (ξ^{3} + ξ^{4})^{−2} + (ξ^{3} + ξ^{4})^{−1} \\ (ξ + ξ^{6})^{−2} & (ξ^{2} + ξ^{5})^{−2} & (ξ^{3} + ξ^{4})^{−2} \end{matrix} ]

and

A^{−1} = \frac{1}{7} [ \begin{matrix}(ξ + ξ^{6})^{2} & (ξ + ξ^{6})^{3} & (ξ + ξ^{6})^{4} \\ (ξ^{2} + ξ^{5})^{2} & (ξ^{2} + ξ^{5})^{3} & (ξ^{2} + ξ^{5})^{4} \\ (ξ^{3} + ξ^{4})^{2} & (ξ^{3} + ξ^{4})^{3} & (ξ^{3} + ξ^{4})^{4} \end{matrix} ] · [ \begin{matrix}−3 & 2 & 5 \\ 1 & 1 & −2 \\ 2 & −1 & −2 \end{matrix} ] .

It should be noticed, that characteristic polynomial w(λ) of matrix W(δ) is equal to

w(λ) = −δ^{3} p_{7}(\frac{λ − 1}{δ}).

9 Tables

Table 1: ; \begin{matrix}n & a_{n}/\sqrt{7} & b_{n}/\sqrt{7} & c_{n}/\sqrt{7} & α_{n}/\sqrt{7} & β_{n}/\sqrt{7} & γ_{n}/\sqrt{7} & f_{n} & g_{n} & h_{n} \\ 0 & 1 & 1 & 1 & 0 & −2 & 1 & −1 & −1 & −1 \\ 1 & 3 & 2 & 0 & −2 & −5 & 3 & −2 & −2 & 5 \\ 2 & 8 & 7 & −2 & −9 & −15 & 8 & −4 & 3 & −4 \\ 3 & 23 & 24 & −9 & −33 & −47 & 23 & −1 & −8 & 13 \\ 4 & 70 & 80 & −33 & −113 & −150 & 70 & −9 & 12 & −16 \\ 5 & 220 & 263 & −113 & −376 & −483 & 220 & 3 & −25 & 38 \\ 6 & 703 & 859 & −376 & −1235 & −1562 & 703 & −22 & 41 & −57 \\ 7 & 2265 & 2797 & −1235 & −4032 & −5062 & 2265 & 19 & −79 & 117 \\ 8 & 7327 & 9094 & −4032 & −13126 & −16421 & 7327 & −60 & 136 & −193 \\ 9 & 23748 & 29547 & −13126 & −42673 & −53295 & 23748 & 76 & −253 & 370 \\ 10 & 77043 & 95968 & −42673 & −138641 & −173011 & 77043 & −177 & 446 & −639 \\ 11 & 250054 & 311652 & −138641 & −450293 & −561706 & 250054 & 269 & −816 & 1186 \end{matrix}

Table 2: ; \begin{matrix}n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ ε_{n} & 1 & 0 & −2 & 2 & 2 & −6 & 2 & 10 & −14 & −6 & 34 & −22 \\ ω_{n} & 0 & 1 & −1 & −1 & 3 & −1 & −5 & 7 & 3 & −17 & 11 & 23 \\ Ψ_{n} & 2 & −1 & −3 & 5 & 1 & −11 & 9 & 13 & −31 & 5 & 57 & −67 \end{matrix}

Table 3: ; \begin{matrix}n & u_{n} & v_{n} & w_{n} & x_{n} & y_{n} & z_{n} \\ 0 & −1 & −1 & −1 & \sqrt{7} & \sqrt{7} & \sqrt{7} \\ 1 & \sqrt{7} & −2\sqrt{7} & 0 & 0 & 0 & 7 \\ 2 & 0 & −7 & 0 & \sqrt{7} & 2\sqrt{7} & 4\sqrt{7} \\ 3 & \sqrt{7} & −6\sqrt{7} & \sqrt{7} & 0 & 7 & 21 \\ 4 & 0 & −28 & 7 & 0 & 7\sqrt{7} & 14\sqrt{7} \\ 5 & 0 & −21\sqrt{7} & 7\sqrt{7} & −7 & 35 & 70 \\ 6 & −7 & −105 & 42 & −7\sqrt{7} & 28\sqrt{7} & 49\sqrt{7} \\ 7 & −7\sqrt{7} & −77\sqrt{7} & 35\sqrt{7} & −49 & 147 & 245 \\ 8 & −49 & −392 & 196 & −42\sqrt{7} & 112\sqrt{7} & 175\sqrt{7} \\ 9 & −42\sqrt{7} & −287\sqrt{7} & 154\sqrt{7} & −245 & 588 & 882 \\ 10 & −245 & −1470 & 833 & −196\sqrt{7} & 441\sqrt{7} & 637\sqrt{7} \\ 11 & −196\sqrt{7} & −1078\sqrt{7} & 637\sqrt{7} & −1078 & 2303 & 3234 \end{matrix}

Table 4: ; \begin{matrix}n & u_{n}^{∗} & v_{n}^{∗} & w_{n}^{∗} & x_{n}^{∗}/\sqrt{7} & y_{n}^{∗}/\sqrt{7} & z_{n}^{∗}/\sqrt{7} \\ 0 & −1 & −1 & −1 & 1 & 1 & 1 \\ 1 & −7 & 7 & 0 & 1 & 2 & −3 \\ 2 & −14 & 7 & −7 & 7 & 7 & 0 \\ 3 & −56 & 42 & −7 & 14 & 21 & −14 \\ 4 & −147 & 98 & −49 & 56 & 63 & −21 \\ 5 & −490 & 343 & −98 & 147 & 196 & −98 \\ 6 & −1421 & 980 & −392 & 490 & 588 & −245 \\ 7 & −4459 & 3087 & −1029 & 1421 & 1813 & −833 \\ 8 & −13377 & 9261 & −3430 & 4459 & 5488 & −2401 \\ 9 & −41160 & 28469 & −9947 & 13377 & 16807 & −7546 \\ 10 & −124852 & 86436 & −31213 & 41160 & 51107 & −22638 \\ 11 & −381759 & 264110 & −93639 & 124852 & 156065 & −69629 \end{matrix}

Table 5: ; \begin{matrix}n & F_{n} & G_{n} & H_{n} & Ã_{n}/\sqrt{7} & B̃_{n}/\sqrt{7} & C̃_{n}/\sqrt{7} \\ 0 & −1 & −1 & −1 & 1 & 1 & 1 \\ 1 & −4 & 3 & 3 & −1 & 3 & 4 \\ 2 & 5 & −9 & −2 & −8 & 15 & 19 \\ 3 & −15 & 20 & 6 & −42 & 76 & 95 \\ 4 & 31 & −46 & −11 & −213 & 384 & 479 \\ 5 & −72 & 103 & 26 & −1076 & 1939 & 2418 \\ 6 & 160 & −232 & −57 & −5433 & 9790 & 12208 \\ 7 & −361 & 521 & 129 & −27431 & 49429 & 61637 \\ 8 & 810 & −1171 & −289 & −138497 & 249563 & 311200 \\ 9 & −1821 & 2631 & 650 & −699260 & 1260023 & 1571223 \\ 10 & 4091 & −5912 & −1460 & −3530506 & 6361752 & 7932975 \\ 11 & −9193 & 13284 & 3281 & −17825233 & 32119960 & 40052935 \end{matrix}

Table 6: ; \begin{matrix}n & Ω_{n}(δ) \\ 0 & 3 \\ 1 & i\sqrt{7}δ + 3 \\ 2 & −7δ^{2} + 2i\sqrt{7}δ + 3 \\ 3 & −4i\sqrt{7}δ^{3} − 21δ^{2} + 3i\sqrt{7}δ + 3 \\ 4 & 21δ^{4} − 16i\sqrt{7}δ^{3} − 42δ^{2} + 4i\sqrt{7}δ + 3 \\ 5 & 14i\sqrt{7}δ^{5} + 105δ^{4} − 40i\sqrt{7}δ^{3} − 70δ^{2} + 5i\sqrt{7}δ + 3 \\ 6 & −70δ^{6} + 84i\sqrt{7}δ^{5} + 315δ^{4} − 80i\sqrt{7}δ^{3} − 105δ^{2} + 6i\sqrt{7}δ + 3 \\ 7 & −49i\sqrt{7}δ^{7} − 490δ^{6} + 294i\sqrt{7}δ^{5} + 735δ^{4} − 140i\sqrt{7}δ^{3} − 147δ^{2} + 7i\sqrt{7}δ + 3 \end{matrix}

Table 7:_{\begin{matrix}n & p_{n}(δ) \\ 0 & 1 \\ 1 & 1 \\ 2 & −2δ^{2} + 1 \\ 3 & −i\sqrt{7}δ^{3} − 6δ^{2} + 1 \\ 4 & 3δ^{4} − 4i\sqrt{7}δ^{3} − 12δ^{2} + 1 \\ 5 & 5i\sqrt{7}δ^{5} + 15δ^{4} − 10i\sqrt{7}δ^{3} − 20δ^{2} + 1 \\ 6 & −8δ^{6} + 30i\sqrt{7}δ^{5} + 45δ^{4} − 20i\sqrt{7}δ^{3} − 30δ^{2} + 1 \\ 7 & −17i\sqrt{7}δ^{7} − 56δ^{6} + 105i\sqrt{7}δ^{5} + 105δ^{4} − 35i\sqrt{7}δ^{3} − 42δ^{2} + 1 \\ n & r_{n}(δ) \\ 0 & 0 \\ 1 & δ \\ 2 & 2δ \\ 3 & −2δ^{3} + 3δ \\ 4 & −i\sqrt{7}δ^{4} − 8δ^{3} + 4δ \\ 5 & 3δ^{5} − 5i\sqrt{7}δ^{4} − 20δ^{3} + 5δ \\ 6 & 5i\sqrt{7}δ^{6} + 18δ^{5} − 15i\sqrt{7}δ^{4} − 40δ^{3} + 6δ \\ 7 & −8δ^{7} + 35i\sqrt{7}δ^{6} + 63δ^{5} − 35i\sqrt{7}δ^{4} − 70δ^{3} + 7δ \\ n & s_{n}(δ) & k_{n}(δ) \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\ 2 & 0 & 0 \\ 3 & δ^{3} & 0 \\ 4 & 4δ^{3} & −2δ^{4} \\ 5 & −5δ^{5} + 10δ^{3} & −10δ^{4} \\ 6 & −i\sqrt{7}δ^{6} − 30δ^{5} + 20δ^{3} & 10δ^{6} − 30δ^{4} \\ 7 & 18δ^{7} − 7i\sqrt{7}δ^{6} − 105δ^{5} + 35δ^{3} & 2i\sqrt{7}δ^{7} + 70δ^{6} − 70δ^{4} \\ n & l_{n}(δ) \\ 0 & 0 \\ 1 & 0 \\ 2 & δ^{2} \\ 3 & 3δ^{2} \\ 4 & −3δ^{4} + 6δ^{2} \\ 5 & −i\sqrt{7}δ^{5} − 15δ^{4} + 10δ^{2} \\ 6 & 8δ^{6} − 6i\sqrt{7}δ^{5} − 45δ^{4} + 15δ^{2} \\ 7 & 6i\sqrt{7}δ^{7} + 56δ^{6} − 21i\sqrt{7}δ^{5} − 105δ^{4} + 21δ^{2} \end{matrix}}

10 Acknowledgments

The authors wish to express their gratitude to the referee for several helpful comments and suggestions concerning the first version of our paper, especially for preparing Example 2.2, which is added to the present version of the paper.

References

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

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

[3] K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Springer Verlag, 1982.

[4] V. S. Shevelev, Three Ramanujan’s formulas, Kvant 6 (1988), 52–55 (in Russian).

[5] D. O. Shklyarsky, N. N. Chentzov, I. M. Yaglom, The USSR Olympiad Problem Book, W. H. Freeman & Co., 1962.

[6] D. O. Shklyarsky, N. N. Chentsov, I. M. Yaglom, Selected Problems and Theorems in Elementary Mathematics, Arithmetic and Algebra, Mir, 1979.

[7] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, 2007.

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

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

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

[11] A. M. Yaglom and I. M. Yaglom, An elementary proof of the Wallis, Leibniz and Euler formulas for π, Uspekhi Matem. Nauk VIII (1953), 181–187 (in Russian).

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

(Concerned with sequences \underline{A001607}, \underline{A002249}, \underline{A079309}, \underline{A094648}, \underline{A110512}, and \underline{A115146}.)

Received April 19 2007; revised version received May 17 2007. Published in Journal of Integer Sequences, June 4 2007.

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