## Abstract and Applied Analysis

### $q$-Genocchi Numbers and Polynomials Associated with Fermionic $p$-Adic Invariant Integrals on ${\mathbb{Z}}_{p}$

#### Abstract

The main purpose of this paper is to present a systemic study of some families of multiple Genocchi numbers and polynomials. In particular, by using the fermionic $p$-adic invariant integral on ${\mathbb{Z}}_{p}$, we construct $p$-adic Genocchi numbers and polynomials of higher order. Finally, we derive the following interesting formula: ${G}_{n+k,q}^{(k)}(x)={2}^{k}k!\big(\begin{smallmatrix}n+k\\ \vspace{0pt}k\end{smallmatrix}\big){\sum{}}_{l=0}^{\infty{}}{\sum{}}_{{d}_{0}+{d}_{1}+\cdots{}+{d}_{k}=k-1,{d}_{i}\in{}\mathbb{N}}{(-1)}^{l}{(l+x)}^{n}$, where ${G}_{n+k,q}^{(k)}(x)$ are the $q$-Genocchi polynomials of order $k$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 232187, 8 pages.

Dates
First available in Project Euclid: 9 September 2008

https://projecteuclid.org/euclid.aaa/1220969168

Digital Object Identifier
doi:10.1155/2008/232187

Mathematical Reviews number (MathSciNet)
MR2407281

Zentralblatt MATH identifier
1149.11010

#### Citation

Jang, Leechae; Kim, Taekyun. $q$ -Genocchi Numbers and Polynomials Associated with Fermionic $p$ -Adic Invariant Integrals on ${\mathbb{Z}}_{p}$. Abstr. Appl. Anal. 2008 (2008), Article ID 232187, 8 pages. doi:10.1155/2008/232187. https://projecteuclid.org/euclid.aaa/1220969168

#### References

• T. Kim, $q$-Euler numbers and polynomials associated with $p$-adic $q$-integrals,'' Journal of Nonlinear Mathematical Physics, vol. 14, no. 1, pp. 15--27, 2007.
• T. Kim, $q$-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients,'' Russian Journal Mathematical Physics, vol. 15, no. 1, pp. 51--57, 2008.
• T. Kim, $q$-Bernoulli numbers associated with $q$-stirling numbers,'' Advances in Difference Equations, vol. 2008, Article ID 743295, 10 pages, 2008.
• T. Kim, The modified $q$-Euler numbers and polynomials,'' Advanced Studies in Contemporary Mathematics, vol. 16, pp. 161--170, 2008.
• T. Kim, Euler numbers and polynomials associated with zeta functions,'' Abstract and Applied Analysis, vol. 2008, Article ID 581582, 13 pages, 2008.
• T. Ernst, Examples of a $q$-umbral calculus,'' Advanced Studies in Contemporary Mathematics, vol. 16, no. 1, pp. 1--22, 2008.
• H. Ozden, I. N. Cangul, and Y. Simsek, Multivariate interpolation functions of higher order $q$-Euler numbers and their applications,'' Abstract and Applied Analysis, vol. 2008, Article ID 390857, 16 pages, 2008.
• H. Ozden, Y. Simsek, S.-H. Rim, and I. N. Cangul, A note on $p$-adic $q$-Euler measure,'' Advanced Studies in Contemporary Mathematics, vol. 14, no. 2, pp. 233--239, 2007.
• H. Ozden, I. N. Cangul, and Y. Simsek, Remarks on sum of products of $(h,q)$-twisted Euler polynomials and numbers,'' Journal of Inequalities and Applications, vol. 2008, Article ID 816129, 8 pages, 2008.
• Y. Simsek, V. Kurt, and D. Kim, New approach to the complete sum of products of the twisted $(h,q)$-Bernoulli numbers and polynomials,'' Journal of Nonlinear Mathematical Physics, vol. 14, no. 1, pp. 44--56, 2007.
• 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.
• M. Cenkci, Y. Simsek, and V. Kurt, Further remarks on multiple $p$-adic $q$-$L$-function of two variables,'' Advanced Studies in Contemporary Mathematics, vol. 14, no. 1, pp. 49--68, 2007.
• T. Machide, Sums of products of Kronecker's double series,'' Journal of Number Theory, vol. 128, no. 4, pp. 820--834, 2008.