Some identities of Genocchi polynomials arising from Genocchi basis

Taekyun Kim; Seog-Hoon Rim; Dmitry V Dolgy; Sang-Hun Lee

doi:10.1186/1029-242X-2013-43

Research
Open Access

Some identities of Genocchi polynomials arising from Genocchi basis

Taekyun Kim¹Email author,
Seog-Hoon Rim²,
Dmitry V Dolgy³ and
Sang-Hun Lee⁴

Journal of Inequalities and Applications20132013:43

https://doi.org/10.1186/1029-242X-2013-43

Received: 21 December 2012
Accepted: 13 January 2013
Published: 11 February 2013

Abstract

In this paper, we give some interesting identities which are derived from the basis of Genocchi. From our methods which are treated in this paper, we can derive some new identities associated with Bernoulli and Euler polynomials.

MSC:11B68, 11S80.

Keywords

Vector Space
Dimensional Vector
Good Basis
Natural Basis
Euler Number

1 Introduction

As is well known, the Genocchi polynomials are defined by the generating function to be

\frac{2 t}{e^{t} + 1} e^{x t} = e^{G (x) t} = \sum_{n = 0}^{\infty} G_{n} (x) \frac{t^{n}}{n!} (see [1–9])

(1)

with the usual convention about replacing $G^{n} (x)$ by $G_{n} (x)$ .

In the special case

x = 0

,

G_{n} (0) = G_{n}

are called the n th Genocchi numbers. From (1), we note that

G_{0} = 0, G_{n} (1) + G_{n} = 2 δ_{n, 1} (see [10–16]),

(2)

where

δ_{n, k}

is the Kronecker symbol.

G_{n} (x) = {(G + x)}^{n} = \sum_{l = 0}^{n} (\binom{n}{l}) G_{l} x^{n - l} (see [6–8, 17]) .

(3)

Thus, by (2) and (3), we see that

\frac{d}{d x} G_{n} (x) = n G_{n - 1} (x), deg G_{n} (x) = n - 1 .

(4)

The n th Bernoulli polynomials are also defined by the generating function to be

\frac{t}{e^{t} - 1} e^{x t} = e^{B (x) t} = \sum_{n = 0}^{\infty} B_{n} (x) \frac{t^{n}}{n!} (see [14–16])

(5)

with the usual convention about replacing $B^{n} (x)$ by $B_{n} (x)$ .

In the special case

x = 0

,

B_{n} (0) = B_{n}

are called the n th Bernoulli numbers. By (5), we get

B_{0} = 1, B_{n} (1) - B_{n} = δ_{1, n} (see [8, 9, 17])

(6)

and

B_{n} (x) = \sum_{l = 0}^{n} (\binom{n}{l}) B_{l} x^{n - l} = \sum_{l = 0}^{n} (\binom{n}{l}) B_{n - l} x^{l} .

(7)

The Euler numbers are defined by

E_{0} = 1, {(E + 1)}^{n} + E_{n} = 2 δ_{0, n} .

(8)

The Euler polynomials are defined by

E_{n} (x) = {(E + x)}^{n} = \sum_{l = 0}^{n} (\binom{n}{l}) E_{n - l} x^{l} (see [7–13, 17]) .

(9)

Let $P_{n} = {p (x) \in Q [x] ∣ deg p (x) \leq n}$ be the $(n + 1)$ -dimensional vector space over ℚ. Probably, ${1, x, \dots, x^{n}}$ is the most natural basis for $P_{n}$ . But ${G_{1} (x), G_{2} (x), \dots, G_{n + 1} (x)}$ is also a good basis for the space $P_{n}$ for our purpose of arithmetical applications of Genocchi polynomials. Let $p (x) \in P_{n}$ . Then $p (x)$ can be expressed by $p (x) = \sum_{1 \leq k \leq n + 1} b_{k} G_{k} (x)$ .

In this paper, we develop some new methods to obtain some new identities and properties of Genocchi polynomials which are derived from the Genocchi basis.

2 Genocchi basis and some identities of Genocchi polynomials

Let us take

p (x) \in P_{n}

. Then

p (x)

can be expressed as a ℚ-linear combination of

G_{1} (x), G_{2} (x), \dots, G_{n + 1} (x)

as follows:

p (x) = \sum_{1 \leq k \leq n + 1} b_{k} G_{k} (x) = b_{1} G_{1} (x) + b_{2} G_{2} (x) + \dots + b_{n + 1} G_{n + 1} (x) .

(10)

Now, let us define the operator

\tilde{△}

by

\tilde{△} p (x) = p (x + 1) + p (x) .

(11)

Then, by (10) and (11), we set

g (x) = \tilde{△} p (x) = \sum_{1 \leq k \leq n + 1} b_{k} (G_{k} (x + 1) + G_{k} (x)) .

(12)

From (1), we note that

\sum_{n = 0}^{\infty} {G_{n} (x + 1) + G_{n} (x)} \frac{t^{n}}{n!} = \frac{2 t}{e^{t} + 1} e^{(x + 1) t} + \frac{2 t}{e^{t} + 1} e^{x t} .

(13)

By (2), (3) and (13), we get

\frac{G_{n + 1} (x + 1) + G_{n + 1} (x)}{n + 1} = 2 x^{n} .

(14)

From (12) and (14), we get

g (x) = \tilde{△} p (x) = 2 \sum_{1 \leq k \leq n + 1} k b_{k} x^{k - 1} .

(15)

For

r \in N

, let us take the r th derivative of

g (x)

in (15) as follows:

g^{(r)} (x) = \frac{d^{r} g (x)}{d x^{r}} = 2 \sum_{1 \leq k \leq n + 1} k (k - 1) \dots (k - 1 - r + 1) b_{k} x^{k - 1 - r} .

(16)

Thus, by (16), we get

g^{(r)} (0) = \frac{d^{r} g (x)}{d x^{r}} |_{x = 0} = 2 (r + 1)! b_{r + 1} .

(17)

From (11) and (17), we have

b_{r + 1} = \frac{1}{2 (r + 1)!} {p^{(r)} (1) + p^{(r)} (0)},

(18)

where $p^{(r)} (a) = \frac{d^{r} g (x)}{d x^{r}} |_{x = a}$ .

Therefore, by (10) and (18), we obtain the following theorem.

Theorem 1 For $n \in N$ , let $p (x) \in P_{n}$ with $p (x) = \sum_{1 \leq k \leq n + 1} b_{k} G_{k} (x)$ .

Then we have

b_{k} = \frac{1}{2 k!} (p^{(k - 1)} (1) + p^{(k - 1)} (0)) .

Let us assume that

p (x) = B_{n} (x)

. Then by Theorem 1, we get

B_{n} (x) = \sum_{1 \leq k \leq n + 1} b_{k} G_{k} (x),

(19)

where

b_{k} = \frac{1}{2 k!} {p^{(k - 1)} (1) + p^{(k - 1)} (0)} = \frac{1}{2 k!} {(n)}_{k - 1} {B_{n - k + 1} (1) + B_{n - k + 1}} .

(20)

From (6) and (20), we have

b_{k} = \frac{1}{2 (n + 1)} (\binom{n + 1}{k}) {δ_{n, k} + 2 B_{n - k + 1}} .

(21)

By (19) and (21), we get

\begin{aligned} B_{n} (x) & = \frac{1}{n + 1} \sum_{1 \leq k \leq n - 1} (\binom{n + 1}{k}) B_{n - k + 1} G_{k} (x) + \frac{1}{2} (1 + 2 B_{1}) G_{n} (x) + \frac{1}{2 (n + 1)} 2 G_{n + 1} (x) \\ = \frac{1}{n + 1} \sum_{1 \leq k \leq n - 1} (\binom{n + 1}{k}) B_{n - k + 1} G_{k} (x) + \frac{1}{n + 1} G_{n + 1} (x) . \end{aligned}

(22)

Therefore, by (22), we obtain the following theorem.

Theorem 2 For

n \in N

, we have

B_{n} (x) = \frac{1}{n + 1} \sum_{1 \leq k \leq n - 1} (\binom{n + 1}{k}) B_{n - k + 1} G_{k} (x) + \frac{1}{n + 1} G_{n + 1} (x) .

In particular, if we take

p (x) = E_{n} (x) \in P_{n}

, then we have

E_{n} (x) = \sum_{1 \leq k \leq n + 1} b_{k} G_{k} (x),

(23)

where

b_{k} = \frac{1}{2 k!} {p^{(k - 1)} (1) + p^{(k - 1)} (0)} = \frac{1}{2 k!} {(n)}_{k - 1} {E_{n - k + 1} (1) + E_{n - k + 1}} .

(24)

By (8) and (24), we get

\begin{aligned} b_{k} & = \frac{1}{2 (n + 1)} (\binom{n + 1}{k}) {2 δ_{n - k + 1, 0} - E_{n - k + 1} + E_{n - k + 1}} \\ = \frac{1}{n + 1} (\binom{n + 1}{k}) δ_{n + 1, k} . \end{aligned}

(25)

From (23) and (25), we have

E_{n} (x) = \frac{1}{n + 1} G_{n + 1} (x) .

Let us take

p (x) \in P_{n}

with

p (x) = \sum_{0 \leq k \leq n} B_{k} (x) B_{n - k} (x) .

(26)

Then we have

Continuing this process, we get

\begin{aligned} \frac{d^{k} p (x)}{d x^{k}} & = p^{(k)} (x) = (n + 1) n \dots (n + 1 - k + 1) \sum_{l = k}^{n} B_{l - k} (x) B_{n - l} (x) \\ = \frac{(n + 1)!}{(n + 1 - k)!} \sum_{l = k}^{n} B_{l - k} (x) B_{n - l} (x) . \end{aligned}

(27)

From (27), we have

p^{(k - 1)} (1) = \frac{(n + 1)!}{(n + 2 - k)!} \sum_{l = k - 1}^{n} B_{l + 1 - k} (1) B_{n - l} (1) .

(28)

By (6), we get

\begin{aligned} B_{l + 1 - k} (1) B_{n - l} (1) & = (δ_{l + 1 - k, 1} + B_{l + 1 - k}) (δ_{n - l, 1} + B_{n - l}) \\ = {δ_{k, n - 1} + B_{n - k} + B_{n - k} + B_{l + 1 - k} B_{n - l}} . \end{aligned}

(29)

From (28) and (29), we have

p^{(k - 1)} (1) = \frac{(n + 1)!}{(n + 2 - k)!} {δ_{k, n - 1} + 2 B_{n - k} + \sum_{k - 1 \leq l \leq n} B_{l + 1 - k} B_{n - l}} .

(30)

By Theorem 1,

p (x) = \sum_{0 \leq k \leq n} B_{k} (x) B_{n - k} (x)

can be expressed by

p (x) = \sum_{1 \leq k \leq n + 1} b_{k} (x) G_{k} (x),

(31)

where

\begin{aligned} b_{k} & = \frac{1}{2 k!} {p^{(k - 1)} (1) + p^{(k - 1)} (0)} \\ = \frac{(n + 1)!}{2 k! (n + 2 - k)!} {δ_{k, n - 1} + 2 B_{n - k} + 2 \sum_{l = k - 1} B_{l + 1 - k} B_{n - l}} . \end{aligned}

(32)

Thus, by (31) and (32), we get

\begin{array}{rcl} p (x) & = & \frac{n (n + 1)}{12} G_{n - 1} (x) + \sum_{1 \leq k \leq n + 1} \frac{1}{k} (\binom{n + 1}{k - 1}) B_{n - k} G_{k} (x) \\ + \sum_{1 \leq k \leq n + 1} \frac{1}{k} (\binom{n + 1}{k - 1}) \sum_{l = k - 1}^{n} B_{l + 1 - k} B_{n - l} G_{k} (x) . \end{array}

(33)

Therefore, by (31) and (33), we obtain the following theorem.

Theorem 3 For

n \in N

, we have

\begin{array}{rcl} \sum_{k = 0}^{n} B_{k} (x) B_{n - k} (x) & = & \frac{n (n + 1)}{12} G_{n - 1} (x) + \sum_{1 \leq k \leq n + 1} \frac{1}{k} (\binom{n + 1}{k - 1}) B_{n - k} G_{k} (x) \\ + \sum_{1 \leq k \leq n + 1} (\sum_{k - 1 \leq l \leq n} \frac{1}{k} (\binom{n + 1}{k - 1}) B_{l + 1 - k} B_{n - l}) G_{k} (x) . \end{array}

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors would like to express their gratitude for the valuable comments and suggestions of referees. This research was supported by Kwangwoon University in 2013.

Authors’ Affiliations

(1)

Department of Mathematics, Kwangwoon University, Seoul, 139-701, South Korea

(2)

Department of Mathematics Education, Kyungpook National University, Daegu, 702-701, South Korea

(3)

Hanrimwon, Kwangwoon University, Seoul, 139-701, South Korea

(4)

Division of General Education, Kwangwoon University, Seoul, 139-701, South Korea

References

Araci S, Acikgöz M, Jolany H, Seo JJ: A unified generating function of the q -Genocchi polynomials with their interpolation functions. Proc. Jangjeon Math. Soc. 2012, 15(2):227–233.MATHMathSciNetGoogle Scholar
Araci S, Erdal D, Seo JJ: A study on the fermionic p -adic q -integral representation on $Z_{p}$ associated with weighted q -Bernstein and q -Genocchi polynomials. Abstr. Appl. Anal. 2011., 2011: Article ID 649248Google Scholar
Bayad A, Kim T: Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials. Adv. Stud. Contemp. Math. 2010, 20(2):247–253.MATHMathSciNetGoogle Scholar
Bayad A: Modular properties of elliptic Bernoulli and Euler functions. Adv. Stud. Contemp. Math. 2010, 20(3):389–401.MATHMathSciNetGoogle Scholar
Cangul IN, Kurt V, Ozden H, Simsek Y: On the higher-order w - q -Genocchi numbers. Adv. Stud. Contemp. Math. 2009, 19(1):39–57.MathSciNetGoogle Scholar
Ding D, Yang J: Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials. Adv. Stud. Contemp. Math. 2010, 20(1):7–21.MATHMathSciNetGoogle Scholar
Dolgy DV, Kim T, Lee B, Ryoo CS: On the q -analogue of Euler measure with weight. Adv. Stud. Contemp. Math. 2011, 21(4):429–435.MATHMathSciNetGoogle Scholar
Kim DS, Lee N, Na J, Park KH: Identities of symmetry for higher-order Euler polynomials in three variables (I). Adv. Stud. Contemp. Math. 2012, 22(1):51–74.MATHMathSciNetGoogle Scholar
Kim T: Some identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials. Adv. Stud. Contemp. Math. 2010, 20(1):23–28.MATHMathSciNetGoogle Scholar
Kim T: On the multiple q -Genocchi and Euler numbers. Russ. J. Math. Phys. 2008, 15(4):481–486. 10.1134/S1061920808040055MATHMathSciNetView ArticleGoogle Scholar
Kim T: Some identities on the q -Euler polynomials of higher order and q -Stirling numbers by the fermionic p -adic integral on $Z_{p}$ . Russ. J. Math. Phys. 2009, 16(4):484–491. 10.1134/S1061920809040037MATHMathSciNetView ArticleGoogle Scholar
Ozden H, Cangul IN, Simsek Y: Remarks on q -Bernoulli numbers associated with Daehee numbers. Adv. Stud. Contemp. Math. 2009, 18(1):41–48.MathSciNetGoogle Scholar
Rim S-H, Jeong J: On the modified q -Euler numbers of higher order with weight. Adv. Stud. Contemp. Math. 2012, 22(1):93–98.MATHMathSciNetGoogle Scholar
Ryoo CS: Calculating zeros of the twisted Genocchi polynomials. Adv. Stud. Contemp. Math. 2008, 17(2):147–159.MATHMathSciNetGoogle Scholar
Simsek Y: Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions. Adv. Stud. Contemp. Math. 2008, 16(2):251–278.MATHMathSciNetGoogle Scholar
Simsek Y: Theorems on twisted L-function and twisted Bernoulli numbers. Adv. Stud. Contemp. Math. 2005, 11(2):205–218.MATHMathSciNetGoogle Scholar
Kim DS, Kim T: Some identities of higher order Euler polynomials arising from Euler basis. Integral Transforms Spec. Funct. 2012. doi:10.1080/10652469.2012.754756Google Scholar

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