Relating Balancing Polynomials to Lucas-Balancing Polynomials via Bernoulli Numbers

Mouloud Goubi Department of Mathematics UMMTO University Laboratory of Algebra and Number Theory (USTHB) Tizi Ouzou Algeria

                    mouloud.goubi@ummto.dz

Abstract We derive relations for balancing and Lucas-balancing polynomials. The results

provide extensions of some results proved recently by Frontczak.

1 Introduction

The Fibonacci numbers F_{n} and Lucas numbers L_{n} [2, 6] are sequences satisfying the Fibonacci recursion relation

X_{n+1} = X_{n} + X_{n−1}.  (1)

The initial conditions are, respectively, F_{0} = 0, F_{1} = 1 and L_{0} = 2, L_{1} = 1. The first elements of the sequences are

F : 0, 1, 1, 2, 3, 5, 8, 13, 21, ⋯

and

L : 2, 1, 3, 4, 7, 11, 18, 29, 47, · · · .

Fibonacci numbers (F_{n})_{n≥0} and Lucas numbers (L_{n})_{n≥0} are regarded as the some of the most important sequences in mathematics. Relationships between them have been studied in the past. Hoggatt [6] states as exercises several identities relating Fibonacci numbers with Lucas numbers.

These numbers are also related to balancing and Lucas-balancing polynomials. Balancing polynomials B_{n}^{∗}(x) are defined by the recurrence (see [4, 5, 7])

B_{n}^{∗}(x) = 6xB_{n−1}^{∗}(x) − B_{n−2}^{∗}(x), n ≥ 2,  (2)

with initial terms B_{0}^{∗}(x) = 0 and B_{1}^{∗}(x) = 1. Lucas-balancing polynomials C_{n}(x) are defined by

C_{n}(x) = 6xC_{n−1}(x) − C_{n−2}(x), n ≥ 2,  (3)

with initial terms C_{0}(x) = 1 and C_{1}(x) = 3x. The numbers B_{n}^{∗} = B_{n}^{∗}(1) (or C_{n} = C_{n}(1)) are so-called balancing numbers (or Lucas-balancing numbers). The Binet formulas for B_{n}^{∗}(x) and C_{n}(x) are

B_{n}^{∗}(x) = \frac{λ^{n}(x) − λ^{−n}(x)}{λ(x) − λ^{−1}(x)} and C_{n}(x) = \frac{1}{2} (λ^{n}(x) + λ^{−n}(x)) ,  (4)

where λ(x) = 3x + \sqrt{9x^{2} − 1} and λ^{−1}(x) = 3x − \sqrt{9x^{2} − 1}. We refer to Frontczak [4] for more details. Accordingly, the exponential generating functions of polynomials B_{n}^{∗}(x) and C_{n}(x) are given by Frontczak [3, Lemma 2, p. 3] as follows:

F (x, t) = \frac{e^{3xt}}{\sqrt{9x^{2} − 1}} sinh (t\sqrt{9x^{2} − 1}) = ∑_{n≥0} B_{n}^{∗}(x)\frac{t^{n}}{n!},  (5)

and

G (x, t) = e^{3xt} cosh (t\sqrt{9x^{2} − 1}) = ∑_{n≥0} C_{n}(x)\frac{t^{n}}{n!},  (6)

where sinh and cosh are the hyperbolic sine and cosine functions. The relations that we spoke about previously are given in the papers [3, 4] as follows:

B_{n}^{∗} (\frac{L_{2m}}{6} ) = \frac{F_{2mn}}{F_{2m}} , C_{n} (\frac{L_{2m}}{6} ) = \frac{L_{2m}}{2} ,  (7)

and

B_{n}^{∗} (\frac{i}{6}L_{2m+1}) = i^{n−1} \frac{F_{(2m+1)n}}{F_{2m+1}} , C_{n} (\frac{i}{6}L_{2m+1}) = i^{n} \frac{L_{(2m+1)n}}{2} ,  (8)

where m is an integer and i = \sqrt{−1} is the imaginary unit.

The Cauchy product of exponential generating functions is defined by

(∑_{n≥0} a_{n}(x)\frac{t^{n}}{n!}) (∑_{n≥0} b_{n}(x)\frac{t^{n}}{n!}) = ∑_{n≥0} (∑_{k=0}^{n} \binom{n}{k}a_{k}(x)b_{n−k}(x)) \frac{t^{n}}{n!}.  (9)

This product allows us to provide extensions of Theorem 3 and Corollary 3 proved by Frontczak [3] in 2019.

2 Connection between balancing polynomials and Lucasbalancing polynomials

As usual, we use B_{n} for the n^{th} Bernoulli number and E_{n} for the n^{th} Euler number. These classical numbers (see Apostol [1]) are defined respectively by

B(t) = \frac{t}{e^{t} − 1} = ∑_{k≥0} B_{k} \frac{t^{k}}{k!}, |t| < 2π  (10)

and

E(t) = \frac{2}{e^{t} + 1} = ∑_{k≥0} E_{k} \frac{t^{k}}{k!}, |t| < π.  (11)

The first few Bernoulli numbers are B_{0} = 1, B_{1} = −\frac{1}{2}, B_{2} = \frac{1}{6}. Also B_{2n+1} = 0 for n ≥ 1. The first few Euler numbers are E_{0} = 1, E_{2} = −1, E_{4} = 5 and E_{2n+1} = 0 for n ≥ 0.

Frontczak [3, Theorem 3] proved the following identities

∑_{{k=0 \atop n≡k (mod 2)}}^{n} \binom{n}{k} (2\sqrt{9x^{2} − 1})^{n−k} B_{n−k}B_{k}^{∗}(x) = nC_{n−1}(x)  (12)

and

∑_{{k=0 \atop n≡k (mod 2)}}^{n} \binom{n}{k} (2\sqrt{9x^{2} − 1})^{n−k} (2^{n−k} − 1) B_{n−k}C_{k}(x) = n (9x^{2} − 1) B_{n−1}^{∗}(x),  (13)

where n ≡ k (mod 2) means that n − k is a multiple of 2. An improvement of the identities (12) and (13) is given in the following theorem:

Theorem 1. Let n ≥ 1. Then

∑_{k=0}^{n} \binom{n}{k} (2\sqrt{9x^{2} − 1})^{n−k} B_{n−k}B_{k}^{∗}(x) = n (C_{n−1}(x) − \sqrt{9x^{2} − 1}B_{n−1}^{∗}(x))  (14)

and

\begin{matrix}∑_{k=0}^{n} \binom{n}{k} (2\sqrt{9x^{2} − 1})^{n−k} (2^{n−k} − 1) B_{n−k}C_{k}(x) = n (9x^{2} − 1) B_{n−1}^{∗}(x) − n\sqrt{9x^{2} − 1}C_{n−1}(x). \\ (15) \end{matrix}

Proof. Let the polynomial

A_{n}(x) = ∑_{k=0}^{n} \binom{n}{k}B_{n−k}^{∗}(x) (2\sqrt{9x^{2} − 1})^{k} B_{k}.

Then A_{n}(x) is defined in means of the Cauchy product of the functions

\frac{e^{3xt}}{\sqrt{9x^{2} − 1}} sinh (t\sqrt{9x^{2} − 1}) and \frac{2t\sqrt{9x^{2} − 1}}{e^{2t\sqrt{9x^{2}−1}} − 1}.

But the product is

A(x, t) = \frac{2te^{3xt}}{e^{2t\sqrt{9x^{2}−1}} − 1} sinh (t\sqrt{9x^{2} − 1}) .

Since

coth t = 1 + \frac{2}{e^{2t} − 1} = 1 + \frac{1}{t} ∑_{n≥0} 2^{n}B_{n} \frac{t^{n}}{n!}

then

A(x, t) = te^{3xt} (coth (t\sqrt{9x^{2} − 1}) − 1) sinh (t\sqrt{9x^{2} − 1}) .

Furthermore

A(x, t) = te^{3xt} cosh (t\sqrt{9x^{2} − 1}) − te^{3xt} sinh (t\sqrt{9x^{2} − 1})

Finally

A(x, t) = tG(x, t) − t\sqrt{9x^{2} − 1}F (x, t)

and

\frac{2te^{3xt}}{e^{2t\sqrt{9x^{2}−1}} − 1} sinh (t\sqrt{9x^{2} − 1}) = ∑_{n≥0} n (C_{n−1}(x) − \sqrt{9x^{2} − 1}B_{n−1}^{∗}(x)) \frac{t^{n}}{n!}.

Thus A_{n}(x) is identical with the polynomial

n (C_{n−1}(x) − \sqrt{9x^{2} − 1}B_{n−1}^{∗}(x)) ,

and the result (14) follows. Let the polynomial

D_{n}(x) = ∑_{k=0}^{n} \binom{n}{k}C_{k}(x) (2\sqrt{9x^{2} − 1})^{n−k} (2^{n−k} − 1) B_{n−k}.

Then D_{n}(x) = J_{n}(x) − K_{n}(x) with

J_{n}(x) = ∑_{k=0}^{n} \binom{n}{k}C_{k}(x) (4\sqrt{9x^{2} − 1})^{n−k} B_{n−k}

and

K_{n}(x) = ∑_{k=0}^{n} \binom{n}{k}C_{k}(x) (2\sqrt{9x^{2} − 1})^{n−k} B_{n−k}.

But the polynomial J_{n}(x) is generated by the function

\frac{4t\sqrt{9x^{2} − 1}}{e^{4t\sqrt{9x^{2}−1}} − 1}e^{3xt} cosh (t\sqrt{9x^{2} − 1})

and the polynomial K_{n}(x) is generated by the function

\frac{2t\sqrt{9x^{2} − 1}}{e^{2t\sqrt{9x^{2}−1}} − 1}e^{3xt} cosh (t\sqrt{9x^{2} − 1}) .

Furthermore the generating function of the polynomial D_{n}(x) is

D(x, t) = \frac{4t\sqrt{9x^{2} − 1}}{e^{4t\sqrt{9x^{2}−1}} − 1}e^{3xt} cosh (t\sqrt{9x^{2} − 1}) − \frac{2t\sqrt{9x^{2} − 1}}{e^{2t\sqrt{9x^{2}−1}} − 1}e^{3xt} cosh (t\sqrt{9x^{2} − 1}) .

Then

D(x, t) = \frac{2t\sqrt{9x^{2} − 1}}{e^{2t\sqrt{9x^{2}−1}} − 1} (\frac{2}{e^{2t\sqrt{9x^{2}−1}} + 1} − 1) e^{3xt} cosh (t\sqrt{9x^{2} − 1})

and

D(x, t) = −\frac{2t\sqrt{9x^{2} − 1}}{e^{2t\sqrt{9x^{2}−1}} + 1}e^{3xt} cosh (t\sqrt{9x^{2} − 1})

Furthermore

D(x, t) = −\frac{t\sqrt{9x^{2} − 1}}{e^{t\sqrt{9x^{2}−1}}}e^{3xt}

but

\frac{1}{e^{t\sqrt{9x^{2}−1}}} = e^{t\sqrt{9x^{2}−1}} − 2 sinh (t\sqrt{9x^{2} − 1})

then

D(x, t) = t\sqrt{9x^{2} − 1} (2 sinh (t\sqrt{9x^{2} − 1}) − e^{t\sqrt{9x^{2}−1}}) e^{3xt}

and

D(x, t) = t\sqrt{9x^{2} − 1} (sinh (t\sqrt{9x^{2} − 1}) − cosh (t\sqrt{9x^{2} − 1})) e^{3xt}.

Which means that

D(x, t) = (9x^{2} − 1) ∑_{n≥0} nB_{n−1}^{∗}(x)\frac{t^{n}}{n!} − \sqrt{9x^{2} − 1} ∑_{n≥0} nC_{n−1}(x)\frac{t^{n}}{n!}.

Finally we obtain

D_{n}(x) = n (9x^{2} − 1) B_{n−1}^{∗}(x) − n\sqrt{9x^{2} − 1}C_{n−1}(x)

and the identity (15) follows.

Frontczak [3, Corollary 4] proved the following relations between balancing numbers and Bernoulli numbers

∑_{{k=0 \atop n≡k (mod 2)}}^{n} \binom{n}{k}32^{\frac{n−k}{2}} B_{k}^{∗}B_{n−k} = nC_{n−1}

and

∑_{{k=0 \atop n≡k (mod 2)}}^{n} \binom{n}{k}32^{\frac{n−k}{2}} (2^{n−k} − 1) C_{k}B_{n−k} = 8nB_{n−1}^{∗}.

An improvement of these identities is given in the following corollary. Corollary 2. Let n ≥ 1. Then

∑_{k=0}^{n} \binom{n}{k}32^{\frac{n−k}{2}} B_{k}^{∗}B_{n−k} = n (C_{n−1} − 2\sqrt{2}B_{n−1}^{∗})  (16)

and

∑_{k=0}^{n} \binom{n}{k}32^{\frac{n−k}{2}} (2^{n−k} − 1) B_{n−k}C_{k} = 8nB_{n−1}^{∗} − 2n\sqrt{2}C_{n−1}.  (17)

Proof. Evaluate (14) and (15) at the point x = 1 to get the desired results.

References

[1] T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976.

[2] P. F. Byrd, Relations between Euler and Lucas numbers, Fibonacci Quart. 13 (1975), 111–114.

[3] R. Frontczak, Relating Fibonacci numbers to Bernoulli numbers via balancing polynomials, J. Integer Sequences 22 (2019) Article 19.5.3.

[4] R. Frontczak, On balancing polynomials, Appl. Math. Sci. 13 (2019), 57–66.

[5] R. Frontczak, Powers of balancing polynomials and some consequences for Fibonacci sums, Inter. Jour. Math. Anal. 13 (2019), 109–115.

[6] V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers, Fibonacci Association, 1971.

[7] B. K. Patel, N. Irmak, and P. K. Ray, Incomplete balancing and Lucas-balancing numbers, Math. Rep. 20 (2018), 59–72.

2010 Mathematics Subject Classification: Primary 11B37; Secondary 11B65, 05A15. Keywords: Bernoulli number, balancing polynomial, balancing number, Fibonacci number, Lucas number, Cauchy product of generating functions.

(Concerned with sequences \underline{A000032}, \underline{A000045}, \underline{A001109}, \underline{A100615}, and \underline{A122045}.)

Received December 12 2019; revised versions received February 27 2010; March 1 2020. Published in Journal of Integer Sequences, March 18 2020.

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