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Volume 2007, Article ID 71452, 8 pagesdoi:10.1155/2007/71452 Research Article A Note on the q-Genocchi Numbers and Polynomials Taekyun Kim Received 15 March 2007; Revised 7 May 2007; Acc

Volume 2007, Article ID 71452, 8 pages

doi:10.1155/2007/71452

Research Article

A Note on the q-Genocchi Numbers and Polynomials

Taekyun Kim

Received 15 March 2007; Revised 7 May 2007; Accepted 24 May 2007

Recommended by Paolo Emilio Ricci

We discuss new concept of theq-extension of Genocchi numbers and give some relations

betweenq-Genocchi polynomials and q-Euler numbers.

Copyright © 2007 Taekyun Kim This is an open access article distributed under the Cre-ative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

The Genocchi numbersG n,n =0, 1, 2, , which can be defined by the generating

func-tion

2t

e t+ 1=

∞



n =0

G n t n n!, | t | < π, (1.1) have numerous important applications in number theory, combinatorics, and numerical analysis, among other areas, [1–13] It is easy to find the valuesG1=1,G3= G5= G7=

··· =0, and even coefficients are given by G2m =2(1−22n)B2n =2nE2n −1(0), whereB n

is a Bernoulli number andE n(x) is an Euler polynomial The first few Genocchi numbers

forn =2, 4, are −1,−3, 17,−155, 2073, The Euler polynomials are well known as

2

e t+ 1e

xt =

∞



n =0

E n(x) t n n!



see [1,3,7–9]

By (1.1) and (1.2) we easily see that

E n(x) =

n



k =0



n k



G k+1

k + 1 x

n − k, where



n k



= n(n −1)···(n − k + 1)

k!



cf [4–6]

.

(1.3)

Form, n ≥1 and,m odd, we have



n m − n

G m =

m−1

k =1



m k



n k G k Z m − k(n −1), (1.4) whereZ m(n) =1m −2m+ 3m − ···+ (−1)n −1n m, see [3,13] From (1.15) we derive

2t =∞

n =0



(G + 1) n+G nt n

where we use the technique method notation by replacingG m byG m(m ≥0), symboli-cally By comparing the coefficients on both sides in (1.5), we see that

G0=0, (G + 1) n+G n =

⎧

⎨

⎩

2 ifn =1,

0 ifn > 1. (1.6)

Let p be a fixed odd prime, and letCp denote the p-adic completion of the algebraic

closure ofQp(= p-adic number field ) For d is a fixed positive integer with (p, d) =1, let

X = X d =lim← −

N

Z

d p NZ,

X1= Z p,

X ∗ =

0<a<d p

(a,p) =1



a + d pZp

,

a + d p NZp =x ∈ X | x ≡ a (mod d)p N

,

(1.7)

wherea ∈ Zlies in 0≤ a < d p N

Ordinaryq-calculus is now very well understood from many different points of view Let us consider a complex numberq ∈ Cwith| q | < 1 (or q ∈ C pwith|1− q | p < p −1/(p −1))

as an indeterminate Theq-basic numbers are defined by

[x] q = q q x − −1

1 , [x] − q = −(− q + 1 q) x+ 1, forx ∈ R (1.8)

We say that f is a uniformly di fferentiable function at a point a ∈ Z pand denote this property by f ∈UD(Zp), if the difference quotients

F f(x, y) = f (x) − f (y)

have a limitl = f (a) as (x, y) →(a, a).

For f ∈UD(Zp), let us start with the expression

1

p N

q



0≤ j<p N

q j f ( j) = 

0≤ j<p N

f ( j)μ q

j + p NZp

(1.10)

representing aq-analogue of Riemann sums for f , (cf [5]) The integral of f onZpwill

be defined as limit (n → ∞) of those sums, when it exists The p-adic q-integral of the

function f ∈UD(Zp) is defined by

I q(f ) =



Zp

f (x)dμ q(x) = lim

N →∞

1

p N

q



0≤ x<p N

f (x)q x,

see [5,10–12]

. (1.11)

In the previous paper [4,9], the author constructed theq-extension of Euler polynomials

by usingp-adic q-fermionic integral onZpas follows:

E n,q(x) =



Zp

[t + x] n

q dμ − q(t), whereμ − q

x + p NZp



= (− q) x

p N

− q (1.12) From (1.12), we note that

E n,q(x) = [2]q

(1− q) n

n



l =0



n l



(−1)l

1 +q l+1 q lx, see [4]. (1.13) Theq-extension of Genocchi numbers is defined as

g q ∗(t) =[2]q t

∞



n =0 (−1)n q n e[n] q t =

∞



n =0

G ∗ n,q t n n!, see [4]. (1.14)

The following formula is well known in [4,7]:

E n,q(x) =

n



k =0



n k



[x] n − k

q q kx G ∗ k+1,q

The modifiedq-Euler numbers are defined as

ξ0,q =[2]q

2 , (qξ + 1)

k+ξ k,q =

⎧

⎨

⎩[2]q if

k =0,

with the usual convention of replacingξ ibyξ i,q, see [10] Thus, we derive the generating function ofξ n,qas follows:

F q(t) =[2]q

∞



k =0 (−1)k e[k] q t =

∞



n =0

ξ n,q t n

Now we also consider theq-Euler polynomials ξ n,q(x) as

F q(t, x) =[2]q

∞



k =0 (−1)k e[k+x] q t =

∞



n =0

ξ n,q(x) t n

From (1.18) we note that

ξ n,q(x) =

n



l =0



n l



ξ l,q q lx[x] n q − l, see [10]. (1.19)

In the recent, several authors studied theq-extension of Genocchi numbers and

polyno-mials (see [1,2,5–7,12]) In this paper we discuss the new concept of theq-extension of

Genocchi numbers and give the same relations betweenq-Genocchi numbers and q-Euler

numbers

2.q-extension of Genocchi numbers

In this section we assume thatq ∈ Cwith| q | < 1 Now we consider the q-extension of

Genocchi numbers as follows:

g q(t) =[2]q t

∞



k =0 (−1)k e[k] q t =

∞



n =0

G n,q t n

In (2.1), it is easy to show that limq →1g q(t) =2t/(e t+ 1)=∞ n =0G n(t n /n!) From (2.1) we derive

g q(t) =[2]q t

∞



k =0 (−1)k

∞



m =0

[k] m q

t m m! =[2]q

∞



k =0 (−1)k

∞



m =1

m[k] m −1

q

t m m!

=[2]q

∞



k =0 (−1)k

∞



m =0

m[k] m q −1t

m m! .

(2.2)

By (2.2), we easily see that

g q(t) =[2]q

∞



m =0



m

1− q

m − 1 m−1

l =0



m −1

l



(−1)l 1

1 +q l



t m m! . (2.3)

From (2.1) and (2.3) we note that

∞



m =0

G m,q t m m! =

∞



m =0



m[2] q

1− q

m − 1 m−1

l =0



m −1

l



(−1)l

1 +q l



t m m! . (2.4)

By comparing the coefficients on both sides in (2.4), we have the following theorem

Theorem 2.1 For m ≥ 0,

G m,q = m[2] q



1

1− q

m − 1 m−1

l =0



m −1

l



(−1)l

FromTheorem 2.1, we easily derive the following corollary

Corollary 2.2 For k ∈ N ,

G0,q =0, (qG + 1) k+G k,q =

⎧

⎪

⎪

[2]2

q

2 if k =1,

0 if k > 1,

(2.6)

with the usual convention of replacing G i by G i,q

Remark 2.3 We note thatCorollary 2.2is theq-extension of (1.6) By (1.15)–(1.19) and

Corollary 2.2, we obtain the following theorem

Theorem 2.4 For n ∈ N

ξ n,q = G n + 1 n+1,q (2.7) From (1.18) we derive

F q(x, t) =[2]q

∞



n =0

(−1)n e[n+x] q t = q x t[2]q

q x t e

[x] q t∞

n =0 (−1)n e q x[n] q t

= e[x] q t∞

n =0

q nx G n+1,q

n + 1

t n n! =∞

n =0

n

k =0



n k



[x] n − k

q q kx G k+1,q

k + 1



t n n! .

(2.8)

By (2.8), we easily see that

ξ n,q(x) =

n



k =0



n k



[x] n − k

q q kx G k+1,q

This formula can be considered as theq-extension of (1.3) Let us consider theq-analogue

of Genocchi polynomials as follows:

g q(x, t) =[2]q t

∞



k =0

(−1)k e[k+x] q t =∞

n =0

G n,q(x) t n

Thus, we note that limq →1g q(x, t) =(2t/(e t+ 1))e xt =∞ n =0G n(x)(t n /n!) From (2.10), we easily derive

G n,q(x) =[2]q n



1

1− q

n − 1 n−1

l =0

(−1)l

1 +q l q lx



n −1

l



By (2.10) we also see that

∞



n =0

G n,q(x) t n

n! =[2]q t

∞



k =0 (−1)k e[k+x] q t =[2]q t

m−1

a =0 (−1)a

∞



k =0 (−1)k e[k+(a+x)/m] qm[m] q t

= [2]q

[m] q[2]q m

m−1

a =0 (−1)a



[m] q t[2] q m

∞



k =0 (−1)k e[m] q t[k+(a+x)/m] qm



=∞

n =0



[2]q [m] q[2]q m

m−1

a =0

(−1)a[m] n

q G n,q m



x + a m



t n n!

=∞

n =0



[2]q [2]q m[m] n −1

q

m−1

a =0 (−1)a G n,q m



x + a m



t n n!, wherem ∈ Nodd.

(2.12) Therefore, we obtain the following theorem

Theorem 2.5 Let m( = odd) ∈ N Then the distribution of the q-Genocchi polynomials will

be as follows:

G n,q(x) = [2]q

[2]q m[m] n q −1

m−1

a =0 (−1)a G n,q m

x + a

m



where n is positive integer.

Theorem 2.5 will be used to construct the p-adic q-Genocchi measures which will

be treated in the next section Letχ be a primitive Dirichlet character with a conductor d( =odd)∈ N Then the generalizedq-Genocchi numbers attached to χ are defined as

g χ,q(t) =[2]q t

d−1

a =0

χ(n)( −1)n e[n] q t =∞

n =0

G n,χ,q t n

From (2.14), we derive

G n,χ,q = [2]q

[2]q d[d] n −1

q

d−1

a =0 (−1)a χ(a)G n,q d



a d



3.p-adic q-Genocchi measures

In this section we assume thatq ∈ C pwith|1− q | p < p −1/(p −1)so thatq x =exp(x log q).

Letχ be a primitive Dirichlet’s character with a conductor d( =odd)∈ N For any positive integersN, k, and d( =odd), letμ k = μ k,q;Gbe defined as

μ k



a + d p NZp



=(−1)a d p Nk −1

q

[2]q

[2]q d pN

G k,q d pN

d p N



By usingTheorem 2.5and (3.1), we show that

p−1

i =0

μ k

a + id p N+d p N+1Zp



= μ k

a + d p NZp



Therefore, we obtain the following theorem

Theorem 3.1 Let d be an odd positive integer For any positive integers N, k, and let μ k =

μ k,q;G be defined as

μ k

a + d p NZp



=(−1)a d p Nk −1

q

[2]q

[2]q d pN

G k,q d pN

d p N



Then μ k can be extended to a distribution on X.

From the definition ofμ kand (2.15) we note that



X χ(x)dμ k(x) = G k,χ,q (3.4)

By (2.1) and (2.3), it is not difficult to show that

G n,q(x) =

n



k =0



n k



[x] n q − k q kx G k,q (3.5) From (3.1) and (3.5) we derive

dμ k(a) = lim

N →∞ μ k

a + d p NZp



= k[a] k −1

q dμ − q(a). (3.6) Therefore, we obtain the following corollary

Corollary 3.2 Let k be a positive integer Then,

G k,χ,q =



X χ(x)dμ k(x) = k



X χ(x)[x] k q −1dμ − q(x). (3.7)

Moreover,

G k,q = k



X[x] k −1

Remark 3.3 In the recent paper (see [1]), Cenkci et al have studiedq-Genocchi

num-bers and polynomials andp-adic q-Genocchi measures Starting from T Kim, L.-C Jang,

and H K Pak’s construction ofq-Genocchi numbers [7], they employed the method de-veloped in a series of papers by Kim [see, e.g., [5,14–16]] and they considerd another

q-analogue of Genocchi numbers G k(q) as

G k(q) = q(1 + q)

(1− q) k −1

k



m =0



k m



m( −1)m+1

which is easily derived from the generating function

F(G)

q (t) =

∞



k =0

G k(q) t k k! = q(1 + q)t

∞



n =0 (−1)n q n e[n]t (3.10)

However, theseq-Genocchi numbers and generating function do not seem to be natural

ones; in particular, these numbers cannot be represented as a nice Witt’s type formula for thep-adic invariant integral onZpand the generating function does not seems to be sim-ple and useful for deriving many interesting identities related toq-Genocchi numbers By

this reason, we considerq-Genocchi numbers and polynomials which are different Our

q-Genocchi numbers and polynomials to treat in this paper can be represented by p-adic q-fermionic integral onZp [9,13] and this integral representation also can be consid-ered as Witt’s type formula forq-Genocchi numbers These formulae are useful to study

congruences and worthwhile identities forq-Genocchi numbers By using the generating

function of ourq-Genocchi numbers, we can derive many properties and identities as

same as ordinary Genocchi numbers which were well known

The author wishes to express his sincere gratitude to the referee for his/her valuable sug-gestions and comments and Professor Paolo E Ricci for his cooperations and helps This work was supported by Jangjeon Research Institute for Mathematical Science(JRIMS2005-005-C00001) and Jangjeon Mathematical Society

References

[1] M Cenkci, M Can, and V Kurt, “q-extensions of Genocchi numbers,” Journal of the Korean Mathematical Society, vol 43, no 1, pp 183–198, 2006.

[2] M Cenkci and M Can, “Some results onq-analogue of the Lerch zeta function,” Advanced Studies in Contemporary Mathematics, vol 12, no 2, pp 213–223, 2006.

[3] F T Howard, “Applications of a recurrence for the Bernoulli numbers,” Journal of Number

The-ory, vol 52, no 1, pp 157–172, 1995.

[4] T Kim, “A note onq-Volkenborn integration,” Proceedings of the Jangjeon Mathematical Society,

vol 8, no 1, pp 13–17, 2005.

[5] T Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol 9, no 3, pp.

288–299, 2002.

[6] T Kim, “A note onp-adic invariant integral in the rings of p-adic integers,” Advanced Studies in Contemporary Mathematics, vol 13, no 1, pp 95–99, 2006.

[7] T Kim, L.-C Jang, and H K Pak, “A note onq-Euler and Genocchi numbers,” Proceedings of the Japan Academy, Series A, vol 77, no 8, pp 139–141, 2001.

[8] T Kim, “A note on some formulas for theq-Euler numbers and polynomials,” Proceedings of the Jangjeon Mathematical Society, vol 9, pp 227–232, 2006.

[9] T Kim, J Y Choi, and J Y Sug, “Extendedq-Euler numbers and polynomials associated with

fermionicp-adic q-integrals onZp ,” Russian Journal of Mathematical Physics, vol 14, pp 160–

163, 2007.

[10] T Kim, “The modifiedq-Euler numbers and polynomials,” 2006,http://arxiv.org/abs/math/

0702523

[11] T Kim, “An invariantp-adic q-integral onZp ,” to appear in Applied Mathematics Letters.

[12] H M Srivastava, T Kim, and Y Simsek, “q-Bernoulli numbers and polynomials associated with

multipleq-zeta functions and basic L-series,” Russian Journal of Mathematical Physics, vol 12,

no 2, pp 241–268, 2005.

[13] M Schork, “Ward’s “calculus of sequences”,q-calculus and the limit q → − 1,” Advanced Studies

in Contemporary Mathematics, vol 13, no 2, pp 131–141, 2006.

[14] T Kim, “On aq-analogue of the p-adic log gamma functions and related integrals,” Journal of Number Theory, vol 76, no 2, pp 320–329, 1999.

[15] T Kim, “Non-Archimedeanq-integrals associated with multiple Changhee q-Bernoulli

polyno-mials,” Russian Journal of Mathematical Physics, vol 10, no 1, pp 91–98, 2003.

[16] T Kim, “On Euler-Barnes multiple zeta functions,” Russian Journal of Mathematical Physics,

vol 10, no 3, pp 261–267, 2003.

Taekyun Kim: Electrical Engineering Computer Science, Kyungpook National University,

Taegu 702-701, South Korea

Email addresses:tkim@knu.ac.kr ; tkim64@hanmail.net

In the recent, several authors studied the< i>q-extension of Genocchi numbers and

polyno-mials... class="text_page_counter">Trang 5

Remark 2.3 We note thatCorollary 2.2is the< i>q-extension of (1.6) By (1.15)–(1.19) and< /p>

Corollary 2.2, we obtain the. ..

(2.12) Therefore, we obtain the following theorem

Theorem 2.5 Let m( = odd)