International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 179385, 13 pages

http://dx.doi.org/10.1155/2012/179385

## Some Properties of Multiple Generalized *q*-Genocchi Polynomials with Weight and Weak Weight

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

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

Academic Editor: Cheon Ryoo

Copyright © 2012 J. Y. Kang. 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.

#### 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 *α* and weak weight *β* by applying the method of *p*-adic *q*-integral. We will find a link between their numbers and polynomials with weight *α* and weak weight *β*. Also we will obtain the interesting properties of their numbers and polynomials with weight *α* and weak weight *β*. Moreover, we construct a Hurwitz-type zeta function which interpolates multiple generalized *q*-Genocchi polynomials with weight *α* and weak weight *β* and find some combinatorial relations.

#### 1. Introduction

Let be a fixed odd prime number. Throughout this paper , and denote the ring of -adic rational integers, the field of -adic rational numbers, the complex number field, and the completion of the algebraic closure of , respectively. Let be the set of natural numbers and . Let be the normalized exponential valuation of with (see [1–21]). When one talks of -extension, is variously considered as an indeterminate, a complex , or a -adic number . If , then one normally assumes . If , then we assume that .

Throughout this paper, we use the following notation: Hence for all (see [1–14, 16, 18, 20, 21]).

We say that is uniformly differentiable function at a point and we write if the difference quotients such that have a limit as .

Let be a fixed integer, and let be a fixed prime number. For any positive integer , we set where lies in .

For any positive integer , is known to be a distribution on .

For , Kim defined the -deformed fermionic -adic integral on : (see [1–13]), and note that We consider the case corresponding to -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 be a primitive Dirichlet character of conductor . We assume that is odd. Then the multiple generalized Genocchi numbers, , and the multiple generalized Genocchi polynomials, , associated with , are defined by In the special case , are called the th multiple generalized Genocchi numbers attached to .

Now, having discussed the multiple generalized Genocchi numbers and polynomials, we were ready to multiple-generalize them to their -analogues. In generalizing the generating functions of the Genocchi numbers and polynomials to their respective -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 -Genocchi numbers and polynomials with weight and weak weight . We investigate some properties which are related to multiple generalized -Genocchi numbers and polynomials with weight and weak weight . We also derive the existence of a specific interpolation function which interpolate multiple generalized -Genocchi numbers and polynomials with weight and weak weight at negative integers.

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

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

Let us define the generalized -Genocchi numbers and polynomials with weight and weak weight , respectively, By using the Taylor expansion of , we have By comparing the coefficient of both sides of in (2.2), we get From (2.2) and (2.3), we can easily obtain that Therefore, we obtain

Similarly, we find the generating function of generalized -Genocchi polynomials with weight and weak weight : From (2.6), we have Observe that . Hence we have . If into (2.7), then we easily obtain .

First, we define the multiple generalized -Genocchi numbers with weight and weak weight : Then we have where .

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

Theorem 2.1. *Let with and . Then one has
*

*From now on, we define the multiple generalized -Genocchi polynomials with weight and weak weight .
Then we have
where .*

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

*Theorem 2.2. Let with and . Then one has
*

*In (2.11), we simply identify that
*

*So far, we have studied the generating functions of the multiple generalized -Genocchi numbers and polynomials with weight and weak weight .*

*3. Modified Multiple Generalized -Genocchi Polynomials with Weight and Weak Weight *

*3. Modified Multiple Generalized -Genocchi Polynomials with Weight and Weak Weight*

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

*
Firstly, we modify generating functions of and . We access some relations connected to these numbers and polynomials with weight and weak weight . For this reason, we assign generating function of modified multiple generalized -Genocchi numbers and polynomials with weight and weak weight which are implied by and . We give relations between these numbers and polynomials with weight and weak weight .*

*We modify (2.11) as follows:
where is defined in (2.11). *

*From the above we know that
After some elementary calculations, we attain
where is defined in (2.8). *

*From the above, we can assign the modified multiple generalized -Genocchi polynomials with weight and weak weight as follows:
Then we have
*

*Theorem 3.1. For and , one has
*

*Corollary 3.2. For and , by using (3.7), one easily obtains
*

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

*By using (2.13), we find that
Thus we have the following theorem.*

*Theorem 3.3. Let with and . Then one has
*

*
By using (2.13), we have
Thus we have
By comparing the coefficients of both sides of in the above, we arrive at the following theorem.*

*Theorem 3.4. Let with , . Then one has
*

*From (2.12), we easily know that
From the above, we get the following theorem.*

*Theorem 3.5. Let , . Then one has
*

*From (2.13), we have
By using Cauchy product in (3.15), we obtain
From (3.16), we have
By comparing the coefficients of both sides of in (3.17), we have the following theorem.*

*Theorem 3.6. Let and. Then one has
*

*Corollary 3.7. In (3.18) setting , one has
*

*By using (2.13) we have the following theorem.*

*Theorem 3.8. Distribution theorem is as follows:*

*4. Interpolation Function of Multiple Generalized -Genocchi Polynomials with Weight and Weak Weight *

*4. Interpolation Function of Multiple Generalized -Genocchi Polynomials with Weight and Weak Weight**In this section, we see interpolation function of multiple generalized -Genocchi polynomials with weak weight and find some relations.*

*Let us define interpolation function of the as follows.*

*Definition 4.1. * Let with and . Then one defines
We call the multiple generalized Hurwitz type -zeta funtion.

In (4.1), setting , we have

*Remark 4.2. *It holds that
Substituting into (4.1), then we have,

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

*Theorem 4.3. Let , . Then one has
*

*References*

*References*

- 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. View at Google Scholar - T. Kim, “Note on the $q$-Euler numbers of higher order,”
*Advanced Studies in Contemporary Mathematics*, vol. 19, no. 1, pp. 25–29, 2009. View at Google Scholar - T. Kim, “The modified $q$-Euler numbers and polynomials,”
*Advanced Studies in Contemporary Mathematics*, vol. 16, no. 2, pp. 161–170, 2008. View at Google Scholar - T. Kim, “New approach to $q$-Euler polynomials of higher order,”
*Russian Journal of Mathematical Physics*, vol. 17, no. 2, pp. 218–225, 2010. View at Publisher · View at Google Scholar - T. Kim, “Barnes-type multiple $q$-zeta functions and $q$-Euler polynomials,”
*Journal of Physics A*, vol. 43, no. 25, 11 pages, 2010. View at Publisher · View at Google Scholar - T. Kim, “On the multiple $q$-Genocchi and Euler numbers,”
*Russian Journal of Mathematical Physics*, vol. 15, no. 4, pp. 481–486, 2008. View at Publisher · View at Google Scholar - T. Kim, J. Choi, Y. H. Kim, and C. S. Ryoo, “A note on the weighted $p$-adic $q$-Euler measure on ${\mathbb{Z}}_{p}$,”
*Advanced Studies in Contemporary Mathematics*, vol. 21, no. 1, pp. 35–40, 2011. View at Google Scholar - 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. View at Google Scholar - 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. View at Google Scholar - 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. View at Google Scholar - 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. View at Publisher · View at Google Scholar - C. S. Ryoo, T. Kim, J. Choi, and B. Lee, “On the generalized $q$-Genocchi numbers and polynomials of higher-order,”
*Advances in Difference Equations*, vol. 2011, Article ID 424809, 8 pages, 2011. View at Google Scholar - 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. View at Google Scholar - L.-C. Jang, “On multiple generalized $w$-Genocchi polynomials and their applications,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 316870, 8 pages, 2010. View at Publisher · View at Google Scholar - A. Bayad and 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. View at Google Scholar · View at Zentralblatt MATH - A. Bayad and 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. View at Publisher · View at Google Scholar - B. Kurt, “The multiplication formulae for the Genocchi polynomials,”
*Proceedings of the Jangjeon Mathematical Society*, vol. 13, no. 1, pp. 89–96, 2010. View at Google Scholar · View at Zentralblatt MATH - M. Cenkci, M. Can, and 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. View at Google Scholar - M. Domaratzki, “Combinatorial interpretations of a generalization of the Genocchi numbers,”
*Journal of Integer Sequences*, vol. 7, no. 3, article 04.3.6, 2004. View at Google Scholar · View at Zentralblatt MATH - Y. Simsek, I. N. Cangul, V. Kurt, and D. Kim, “$q$-Genocchi numbers and polynomials associated with $q$-Genocchi-type $l$-functions,”
*Advances in Difference Equations*, vol. 2008, Article ID 815750, 12 pages, 2008. View at Google Scholar - D. V. Dolgy, T. Kim, B. Lee, and C. S. Ryoo, “On the $q$-analogue of Euler measure with weight $\alpha $,”
*Advanced Studies in Contemporary Mathematics*, vol. 21, no. 4, pp. 429–435, 2011. View at Google Scholar

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