A Note on Poly-Genocchi Numbers and Polynomials - Hikari Ltd

Applied Mathematical Sciences, Vol. 8, 2014, no. 96, 4775 - 4781 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.46465

A Note on Poly-Genocchi Numbers and Polynomials Taekyun Kim Department of Mathematics Kwangwoon University Seoul 139-701, Republic of Korea Yu Seon Jang Department of Applied Mathematics Kangnam University Yongin 446-702, Republic of Korea Jong Jin Seo Department of Applied Mathematics Pukyong National University Pusan 698-737, Republic of Korea c 2014 Taekyun Kim, Yu Seon Jang and Jong Jin Seo. This is an open Copyright access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract In this paper, we introduce the poly-Genocchi numbers and polynomials and we give some identities of those polynomials related to the Stirling numbers of the second kind. Keywords: polylogarithmic function, poly-Genocchi numbers and polynomials, Stirling numbers of the second kind

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Taekyun Kim, Yu Seon Jang and Jong Jin Seo

1. Introduction The Genocchi polynomials are defined by the generating function to be ∞ 2t xt tn e , (see [1-14]). (1) = G (x) n et + 1 n! n=0 When x = 0, Gn = Gn (0) are called the Genocchi numbers. From (1), we note that Gn (x) =

n l=0

n n−l Gl x , and G0 = 0, (n ≥ 0). l

By (1), we easily get Gn = Bn − 2n+1 Bn , (n ≥ 0), (see [10-13]). where Bn are ordinary Bernoulli numbers. The classical polylogarithmic function Lik (z) is defined by Lik (z) =

∞ zn n=1

nk

, (k ∈ Z), (see [6,8,9]).

(2)

The poly-Bernoulli polynomials are defined by the generating function to be Lik (1 − e−t ) xt (k) tn e = Bn (x) , (see [6,8,9]). et − 1 n! n=0 ∞

(3)

(1)

When k = 1, Bn (x) = Bn (x), (n ≥ 0). The Stirling number of the second kind is given by (x)n = x(x − 1) · · · (x − n + 1) =

n

S2 (n, l)xl , (n ≥ 0).

l=0

In this paper, we consider poly-Genocchi numbers and polynomials and we give some formulae of those polynomials related to the Stirling numbers of the second kind. 2. poly-Genocchi numbers and polynomials Now, we define the poly-Genocchi polynomials as follows: 2Lik (1 − e−t ) xt (k) tn e = Gn (x) , (k ∈ Z). et + 1 n! n=0 ∞

(4)

A note on poly-Genocchi numbers and polynomials

4777

(k)

When x = 0, Gn = Gn (0) are called the poly-Genocchi numbers. By (4), we (k) easily get G0 = 0. For k = 1, from (4), we have 2t xt tn 2Li1 (1 − e−t ) xt e = t e = Gn (x) . et + 1 e +1 n! n=0 ∞

(5)

Thus, by (4) and (5), we get G(1) n (x) = Gn (x), (n ≥ 0). It is not difficult to show that ∞

G(k) n (x)

n=0

2Lik (1 − e−t ) xt tn = e n! et + 1 t t t 2 1 1 1 y = t ··· y dy · · · dy. y y y e +1 0 e −1 0 e −1 e −1 0 e −1 k−1−times

(6) In particular k = 2, we have ∞

∞ tm Bm y 2t xt dyext = e y t (m + 1)! e + 1 0 e −1 m=0

l Bm t tm Gl (x) (m + 1)! l! m=0 l=0

∞ n n Bl tn = Gn−l (x) . l l + 1 n! n=0 l=0

tn 2 G(2) (x) = t n n! e + 1 n=0

∞ =

t

Therefore, by (7), we obtain the following theorem. Theorem 2.1. For n ≥ 0, we have G(2) n (x) From (4), we have

n n Bl Gn−l (x). = l l+1 l=0

(7)

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∞

G(k) n (x)

n=0

tn 2Lik (1 − e−t ) xt = e n! et + 1

∞ ∞ (k) tl xm tm = Gl l! m! m=0 l=0

n ∞ n tn (k) n−l = . Gl x n! l n=0

(8)

l=0

Thus, by (8), we get

G(k) n (x)

=

n n l=0

l

(k)

Gl xn−l , (n ≥ 0).

(9)

From (9), we have (k) n−1 (k) Gn (x) n − 1 Gl+1 n−l−1 = x , (n ≥ 1). n l+1 l

(10)

l=0

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

Theorem 2.2. For n ≥ 1, the degree of Gn (x) is n − 1. That is, (k) n−1 (k) Gn (x) n − 1 Gl+1 n−l−1 . = x l n l + 1 l=0 From (4), we have

∞ n=0

Note that

tn G(k) (x) n

n!

=

Lik (1 − e−t ) t

2t xt e . et + 1

(11)


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1 (1 − e−t )l 1 (−1)l −t 1 = (e − 1)l Lik (1 − e−t ) = k k t t l=1 l t l=1 l ∞

∞

1 (−1)l tp p = l! (−1) S2 (p, l) t l=1 lk p! p=l ∞

∞

tp 1 (−1)l+p l!S (p, l) = 2 t p=1 l=1 lk p!

p+1 ∞ l+p+1 S2 (p + 1, l) tp (−1) l! = . lk p+1 p! p=0 l=1 ∞

p

(12)

Thus, by (1), (11) and (12), we get p+1

∞ ∞ n (−1)l+p+1 l!S2 (p + 1, l) tp Gm (x) m t (k) Gn (x) = × t k n! l (p + 1) p! m=0 m! n=0 p=0 l=1 (13)

n p+1 ∞ (−1)l+p+1 l!S2 (p + 1, l) n tn = Gn−p (x) . k (p + 1) l p n! n=0 p=0

∞

l=1

Therefore, by (13), we obtain the following theorem. Theorem 2.3. For n ≥ 0, we have G(k) n (x)

p+1 n (−1)l+p+1 l!S2 (p + 1, l) n = Gn−p (x). k (p + 1) l p p=0 l=1

We observe that 2Lik (1 − e−t ) (x+1)t 2Lik (1 − e−t ) xt e e + et + 1 et + 1 =2Lik (1 − e−t )ext

p ∞ (−1)l+p tp xt e 2 = l!S (p, l) 2 k l p! p=1 l=1

∞ p ∞ p m (−1)l+p x m t t = l!S2 (p, l) 2 × k l p! m! p=1 m=0 l=1

n p ∞ (−1)l+p n n−p tn . l!S2 (p, l) x 2 = p lk n! n=1 p=1 l=1

Therefore, by (14), we obtain the following theorem.

(14)

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Theorem 2.4. For n ≥ 1, we have p n n n−p (−1)l+p (k) (k) l!S2 (p, l) x . Gn (x + 1) + Gn (x) = 2 k p l p=1 l=1

For d ∈ N with d ≡ 1(mod 2), we have 2Lik (1 − e−t ) xt e et + 1

d−1 Lik (1 − e−t ) 2t a (a+x)t = (−1) e t edt + 1 a=0

∞ p+1 ∞ m d−1 (−1)l+p+1 S2 (p + 1, l) tp a + x t = dm−1 (−1)a Gm k l p+1 p! d m! p=0 m=0 a=0 l=0

n p+1 d−1 n ∞ (−1)l+p+1 S2 (p + 1, l) a+x n t n−p−1 a = . d (−1) Gn−p k l d p n! n=0 p=0 l=0 a=0 (15) By (4) and (15), we get G(k) n (x)

=

n n p=0

p

d

n−p−1

p+1 d−1 (−1)l+p+1 S2 (p + 1, l)

lk

l=0 a=0

a

(−1) Gn−p

a+x . d

3. Farther Remark Let us consider the modified poly-Genocchi polynomials as follows: tn Lik (1 − e−2t ) xt (k) e . = G (x) n,2 et + 1 n! n=0 ∞

(k)

(16) (k)

(k)

Thus, by (16), we easily get G0,2 (x) = 0. When x = 0, Gn,2 = Gn,2 (0) are called the modified Genocchi numbers. From (16), we can derive Lik (1 − e−2t ) t Lik (1 − e−2t ) xt e (e − 1)ext = et + 1 (et + 1)(et − 1) Lik (1 − e−2t ) ( x+1 )2t ( x2 )2t 2 = − e e e2t − 1 ∞ 2n tn x+1 (k) (k) x . Bn − Bn = 2 2 n! n=0 Thus, by (16) and (17), we get

(17)


(k) Gn,2 (x)

=2

n

Bn(k)

x+1 2

−

Bn(k)

x 2

.

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(18)

For k = 1, we note that 2t xt tn Li1 (1 − e−2t ) xt e e . = = G (x) n et + 1 et + 1 n! n=0 ∞

(1)

Hence, Gn,2 (x) = Gn (x), (n ≥ 0). References [1] S. Araci, M. Acikgoz, H. Jolany, and J. Seo, A unified generating function of the qGenocchi polynomials with their interpolation functions, Proc. Jangjeon Math. Soc. 15 (2012), 227-233. [2] A. Bayad and T. Kim, Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. 20 (2010), 247-253. [3] I. N. Cangul, V. Kurt, H. Ozden, and Y. Simsek, On the higher-order w-q-Genocchi numbers, Adv. Stud. Contemp. Math. 10 (2009), 39-57. [4] D. Ding and J. Yang, Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials, Adv. Stud. Contemp. Math. 20 (2010), 7-21. [5] S. Gaboury, R. Tremblay, and B.-J. Fugere, Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials, Proc. Jangjeon Math. Soc. 17 (2014), 115-123. [6] M. Kaneko, Poly-Bernoulli numbers, J. Theor. Nombres Bordeaux 9 (1997), 221-228. [7] D. Kang, J.-H, Jeong, B. J. Lee, S.-H. Rim, and S. H. Choi, Some identities of higher order Genocchi polynomials arising from higher order Genocchi basis, J. Comput. Anal. Appl. 17 (2014), 141-146. [8] D. S. Kim, T. Kim, and S. H. Lee, A note on poly-Bernoulli polynomials arising from umbral calculus, Adv. Stud. Theor. Phys. 7 (2013), no. 15, 731-744. [9] D. S. Kim and T. Kim, Higher-order Frobenius-Euler and poly-Bernoulli mixed-type polynomials, Adv. Difference Equ. 2013, 2013:251, 13 pp. [10] T. Kim, Some identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. 20 (2010), 23-28. [11] T. Kim, New approach to q-Euler, Genocchi numbers and their interpolation functions, Adv. Stud. Contemp. Math. 18 (2009), 105-112. [12] T. Kim, Note on q-Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. 17 (2008), 9-15. [13] T. Kim, On the multiple q-Genocchi and Euler numbers, Russ. J. Math. Phys. 15 (2008), 481-486. [14] B. Kurt and Y. Simsek, On the Hermite based Genocchi polynomials, Adv. Stud. Contemp. Math. 23 (2013), 13-17.

Received: June 5, 2014