Some Properties of Multiple Generalized q-Genocchi Polynomials with Weight [alpha] and Weak Weight [beta]

Abstract

The present paper deals with the various q-Genocchi numbers and polynomials. We define a new type of multiple generalized q-Genocchi numbers and polynomials with weight αand weak weight βby applying the method of p-adic q-integral. We will find a link between their numbers and polynomials with weight αand weak weight β. Also we will obtain the interesting properties of their numbers and polynomials with weight αand weak weight β. Moreover, we construct a Hurwitz-type zeta function which interpolates multiple generalized q-Genocchi polynomials with weight αand weak weight βand find some combinatorial relations.

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J. Y. Kang 1

Recommended by Cheon Ryoo

1, Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea

Received 28 May 2012; Revised 21 August 2012; Accepted 22 August 2012

This is an open 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.

1. Introduction

Let p be a fixed odd prime number. Throughout this paper _{... p} , _{... p} , ... , and _{... p} denote the ring of p -adic rational integers, the field of p -adic rational numbers, the complex number field, and the completion of the algebraic closure of _{... p} , respectively. Let ... be the set of natural numbers and _{... +} = ... ∪ {0 } . Let _{v p} be the normalized exponential valuation of _{... p} with |p _{| p} = ^{p - _{v p} (p )} =1 /p (see [ 1- 21]). When one talks of q -extension, q is variously considered as an indeterminate, a complex q ∈ ... , or a p -adic number q ∈ _{... p} . If q ∈ ... , then one normally assumes |q | <1 . If q ∈ _{... p} , then we assume that |q -1 _{| p} <1 .

Throughout this paper, we use the following notation: [figure omitted; refer to PDF] Hence _{lim q [arrow right]1} [x _{] q} =x for all x ∈ _{... p} (see [ 1- 14, 16, 18, 20, 21]).

We say that g : _{... p} [arrow right] _{... p} is uniformly differentiable function at a point a ∈ _{... p} and we write g ∈UD ( _{... p} ) if the difference quotients _{Φ g} : _{... p} × _{... p} [arrow right] _{... p} such that [figure omitted; refer to PDF] have a limit g ' (a ) as (x ,y ) [arrow right] (a ,a ) .

Let d be a fixed integer, and let p be a fixed prime number. For any positive integer N , we set [figure omitted; refer to PDF] where a ∈ ... lies in 0 ...4;a <d ^{p N} .

For any positive integer N , [figure omitted; refer to PDF] is known to be a distribution on X .

For g ∈UD ( _{... p} ) , Kim defined the q -deformed fermionic p -adic integral on _{... p} : [figure omitted; refer to PDF] (see [ 1- 13]), and note that [figure omitted; refer to PDF] We consider the case q ∈ ( -1,0 ) corresponding to q -deformed fermionic certain and annihilation operators and the literature given there in [ 9, 13, 14].

In [ 9, 12, 14, 19], we introduced multiple generalized Genocchi number and polynomials. Let χ be a primitive Dirichlet character of conductor f ∈ ... . We assume that f is odd. Then the multiple generalized Genocchi numbers, ^{G n , χ (r )} , and the multiple generalized Genocchi polynomials, ^{G n , χ (r )} (x ) , associated with χ , are defined by [figure omitted; refer to PDF] In the special case x =0 , ^{G n , χ (r )} = ^{G n , χ (r )} (0 ) are called the n th multiple generalized Genocchi numbers attached to χ .

Now, having discussed the multiple generalized Genocchi numbers and polynomials, we were ready to multiple-generalize them to their q -analogues. In generalizing the generating functions of the Genocchi numbers and polynomials to their respective q -analogues; it is more useful than defining the generating function for the Genocchi numbers and polynomials (see [ 12]).

Our aim in this paper is to define multiple generalized q -Genocchi numbers ^{G n , χ ,q ( α , β ,r )} and polynomials ^{G n , χ ,q ( α , β ,r )} (x ) with weight α and weak weight β . We investigate some properties which are related to multiple generalized q -Genocchi numbers ^{G n , χ ,q ( α , β ,r )} and polynomials ^{G n , χ ,q ( α , β ,r )} (x ) with weight α and weak weight β . We also derive the existence of a specific interpolation function which interpolate multiple generalized q -Genocchi numbers ^{G n , χ ,q ( α , β ,r )} and polynomials ^{G n , χ ,q ( α , β ,r )} (x ) with weight α and weak weight β at negative integers.

2. The Generating Functions of Multiple Generalized q -Genocchi Numbers and Polynomials with Weight α and Weak Weight β

Many mathematicians constructed various kinds of generating functions of the q -Gnocchi numbers and polynomials by using p -adic q -Vokenborn integral. First we introduce multiple generalized q -Genocchi numbers and polynomials with weight α and weak weight β .

Let us define the generalized q -Genocchi numbers ^{G n , χ ,q ( α , β )} and polynomials ^{G n , χ ,q ( α , β )} (x ) with weight α and weak weight β , respectively, [figure omitted; refer to PDF] By using the Taylor expansion of ^{e [x _{] q α} t} , we have [figure omitted; refer to PDF] By comparing the coefficient of both sides of ^{t n} /n ! in ( 2.2), we get [figure omitted; refer to PDF] From ( 2.2) and ( 2.3), we can easily obtain that [figure omitted; refer to PDF] Therefore, we obtain [figure omitted; refer to PDF]

Similarly, we find the generating function of generalized q -Genocchi polynomials with weight α and weak weight β : [figure omitted; refer to PDF] From ( 2.6), we have [figure omitted; refer to PDF] Observe that ^{F χ ,q ( α , β )} (t ) = ^{F χ ,q ( α , β )} (t ,0 ) . Hence we have ^{G n , χ ,q ( α , β )} = ^{G n , χ ,q ( α , β )} (0 ) . If q [arrow right]1 into ( 2.7), then we easily obtain _{F χ} (t ,x ) .

First, we define the multiple generalized q -Genocchi numbers ^{G n , χ ,q ( α , β ,r )} with weight α and weak weight β : [figure omitted; refer to PDF] Then we have [figure omitted; refer to PDF] where ( n +r r ) = (n +r ) ! / n !r ! .

By comparing the coefficients on the both sides of ( 2.9), we obtain the following theorem.

Theorem 2.1.

Let q ∈ _{... p} with |1 -q _{| p} <1 and n ∈ _{... +} . Then one has [figure omitted; refer to PDF]

From now on, we define the multiple generalized q -Genocchi polynomials ^{G n , χ ,q ( α , β ,r )} (x ) with weight α and weak weight β . [figure omitted; refer to PDF] Then we have [figure omitted; refer to PDF] where ( n +r r ) = (n +r ) ! / n !r ! .

By comparing the coefficients on the both sides of ( 2.12), we have the following theorem.

Theorem 2.2.

Let q ∈ _{... p} with |1 -q _{| p} <1 and n ∈ _{... +} . Then one has [figure omitted; refer to PDF]

In ( 2.11), we simply identify that [figure omitted; refer to PDF]

So far, we have studied the generating functions of the multiple generalized q -Genocchi numbers ^{G n , χ ,q ( α , β ,r )} and polynomials ^{G n , χ ,q ( α , β ,r )} (x ) with weight α and weak weight β .

3. Modified Multiple Generalized q -Genocchi Polynomials with Weight α and Weak Weight β

In this section, we will investigate about modified multiple generalized q -Genocchi numbers and polynomials with weight α and weak weight β . Also, we will find their relations in multiple generalized q -Genocchi numbers and polynomials with weight α and weak weight β .

Firstly, we modify generating functions of ^{G n , χ ,q ( α , β ,r )} and ^{G n , χ ,q ( α , β ,r )} (x ) . We access some relations connected to these numbers and polynomials with weight α and weak weight β . For this reason, we assign generating function of modified multiple generalized q -Genocchi numbers and polynomials with weight α and weak weight β which are implied by ^{G n , χ ,q ( α , β ,r )} and ^{G n , χ ,q ( α , β ,r )} (x ) . We give relations between these numbers and polynomials with weight α and weak weight β .

We modify ( 2.11) as follows: [figure omitted; refer to PDF] where ^{F χ ,q ( α , β ,r )} (t ,x ) is defined in ( 2.11).

From the above we know that [figure omitted; refer to PDF] After some elementary calculations, we attain [figure omitted; refer to PDF] where ^{F χ ,q ( α , β ,r )} (t ) is defined in ( 2.8).

From the above, we can assign the modified multiple generalized q -Genocchi polynomials ^{[straight epsilon] n , χ ,q ( α , β ,r )} (x ) with weight α and weak weight β as follows: [figure omitted; refer to PDF] Then we have [figure omitted; refer to PDF]

Theorem 3.1.

For r ∈ ... and n ∈ _{... +} , one has [figure omitted; refer to PDF]

Corollary 3.2.

For r ∈ ... and n ∈ _{... +} , by using ( 3.7), one easily obtains [figure omitted; refer to PDF]

Secandly, by using generating function of the multiple generalized q -Genocchi polynomials with weight α and weak weight β , which is defined by ( 2.11), we obtain the following identities.

By using ( 2.13), we find that [figure omitted; refer to PDF] Thus we have the following theorem.

Theorem 3.3.

Let q ∈ _{... p} with |1 -q _{| p} <1 and r ∈ ... . Then one has [figure omitted; refer to PDF]

By using ( 2.13), we have [figure omitted; refer to PDF] Thus we have [figure omitted; refer to PDF] By comparing the coefficients of both sides of (n +r ) ! / ^{t n +r} in the above, we arrive at the following theorem.

Theorem 3.4.

Let q ∈ _{... p} with |1 -q _{| p} <1 , r ∈ ... . Then one has [figure omitted; refer to PDF]

From ( 2.12), we easily know that [figure omitted; refer to PDF] From the above, we get the following theorem.

Theorem 3.5.

Let r ∈ ... , k ∈ _{... +} . Then one has [figure omitted; refer to PDF]

From ( 2.13), we have [figure omitted; refer to PDF] By using Cauchy product in ( 3.15), we obtain [figure omitted; refer to PDF] From ( 3.16), we have [figure omitted; refer to PDF] By comparing the coefficients of both sides of ^{t m +r +s} / (m +r +s ) ! in ( 3.17), we have the following theorem.

Theorem 3.6.

Let r ∈ ... and s ∈ _{... +} . Then one has [figure omitted; refer to PDF]

Corollary 3.7.

In ( 3.18) setting s =1 , one has [figure omitted; refer to PDF]

By using ( 2.13) we have the following theorem.

Theorem 3.8.

Distribution theorem is as follows: [figure omitted; refer to PDF]

4. Interpolation Function of Multiple Generalized q -Genocchi Polynomials with Weight α and Weak Weight β

In this section, we see interpolation function of multiple generalized q -Genocchi polynomials with weak weight α and find some relations.

Let us define interpolation function of the ^{G k +r ,q ( α , β ,r )} (x ) as follows.

Definition 4.1.

Let q ,s ∈ ... with |q | <1 and 0 <x ...4;1 . Then one defines [figure omitted; refer to PDF] We call ^{ζ q ( α , β ,r )} (s ,x ) the multiple generalized Hurwitz type q -zeta funtion.

In ( 4.1), setting r =1 , we have [figure omitted; refer to PDF]

Remark 4.2.

It holds that [figure omitted; refer to PDF] Substituting s = -n ,n ∈ _{... +} into ( 4.1), then we have, [figure omitted; refer to PDF]

Setting ( 3.14) into the above, we easily get the following theorem.

Theorem 4.3.

Let r ∈ ... , n ∈ _{... +} . Then one has [figure omitted; refer to PDF]

References

[1] T. Kim, "New approach to q -Euler, Genocchi numbers and their interpolation functions," Advanced Studies in Contemporary Mathematics , vol. 18, no. 2, pp. 105-112, 2009.

[2] T. Kim, "Note on the q -Euler numbers of higher order," Advanced Studies in Contemporary Mathematics , vol. 19, no. 1, pp. 25-29, 2009.

[3] T. Kim, "The modified q -Euler numbers and polynomials," Advanced Studies in Contemporary Mathematics , vol. 16, no. 2, pp. 161-170, 2008.

[4] T. Kim, "New approach to q -Euler polynomials of higher order," Russian Journal of Mathematical Physics , vol. 17, no. 2, pp. 218-225, 2010.

[5] T. Kim, "Barnes-type multiple q -zeta functions and q -Euler polynomials," Journal of Physics A , vol. 43, no. 25, 2010.

[6] T. Kim, "On the multiple q -Genocchi and Euler numbers," Russian Journal of Mathematical Physics , vol. 15, no. 4, pp. 481-486, 2008.

[7] T. Kim, J. Choi, Y. H. Kim, C. S. Ryoo, "A note on the weighted p -adic q -Euler measure on _...p ,", vol. 21, no. 1, pp. 35-40, 2011.

[8] C. S. Ryoo, "Some identities of the twisted q -Euler numbers and polynomials associated with q -Bernstein polynomials," Proceedings of the Jangjeon Mathematical Society , vol. 14, no. 2, pp. 239-248, 2011.

[9] C. S. Ryoo, "On the generalized Barnes type multiple q -Euler polynomials twisted by ramified roots of unity," Proceedings of the Jangjeon Mathematical Society , vol. 13, no. 2, pp. 255-263, 2010.

[10] C. S. Ryoo, "Some relations between twisted q -Euler numbers and Bernstein polynomials," Advanced Studies in Contemporary Mathematics , vol. 21, no. 2, pp. 217-223, 2011.

[11] C. S. Ryoo, "A numerical computation on the structure of the roots of q -extension of Genocchi polynomials," Applied Mathematics Letters , vol. 21, no. 4, pp. 348-354, 2008.

[12] C. S. Ryoo, T. Kim, J. Choi, B. Lee, "On the generalized q -Genocchi numbers and polynomials of higher-order," Advances in Difference Equations , vol. 2011, 2011.

[13] L.-C. Jang, "A study on the distribution of twisted q -Genocchi polynomials," Advanced Studies in Contemporary Mathematics , vol. 18, no. 2, pp. 181-189, 2009.

[14] L.-C. Jang, "On multiple generalized w -Genocchi polynomials and their applications," Mathematical Problems in Engineering , vol. 2010, 2010.

[15] A. Bayad, T. Kim, "Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials," Advanced Studies in Contemporary Mathematics , vol. 20, no. 2, pp. 247-253, 2010.

[16] A. Bayad, T. Kim, "Identities involving values of Bernstein, q -Bernoulli, and q -Euler polynomials," Russian Journal of Mathematical Physics , vol. 18, no. 2, pp. 133-143, 2011.

[17] B. Kurt, "The multiplication formulae for the Genocchi polynomials," Proceedings of the Jangjeon Mathematical Society , vol. 13, no. 1, pp. 89-96, 2010.

[18] M. Cenkci, M. Can, V. Kurt, " q -adic interpolation functions and Kummer-type congruences for q -twisted and q -generalized twisted Euler numbers," Advanced Studies in Contemporary Mathematics , vol. 9, no. 2, pp. 203-216, 2004.

[19] M. Domaratzki, "Combinatorial interpretations of a generalization of the Genocchi numbers," Journal of Integer Sequences , vol. 7, no. 3, article 04.3.6, 2004.

[20] Y. Simsek, I. N. Cangul, V. Kurt, D. Kim, " q -Genocchi numbers and polynomials associated with q -Genocchi-type l -functions," Advances in Difference Equations , vol. 2008, 2008.

[21] D. V. Dolgy, T. Kim, B. Lee, C. S. Ryoo, "On the q -analogue of Euler measure with weight α ," Advanced Studies in Contemporary Mathematics , vol. 21, no. 4, pp. 429-435, 2011.

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