[论文解读] A NOTE ON THE MODIFIED q-GENOCCHI NUMBERS AND POLYNOMIALS WITH WEIGHT (,) AND THEIR INTERPOLATION FUNCTION AT NEGATIVE INTEGERS

Serkan Aracı, Mehmet Acikg ̈ Oz+2 more|arXiv (Cornell University)|Jan 1, 2012

Advanced Mathematical Identities参考文献 22被引用 10

一句话总结

本文引入了带有权重 (α,β) 的修正 q-Genocchi 数与多项式，建立了具有相同权重的 q-Zeta 函数，并推导出一个插值公式，将其与 q-Genocchi 数联系起来。通过生成函数与 Laplace-Mellin 变换，作者获得了乘法定理、Witt 型公式，以及 Hurwitz Zeta 函数的推广，为数论中的 q-类比提供了更深入的洞察。

ABSTRACT

The purpose of this paper concerns to establish modified q-Genocchi numbers and polynomials with weight (�,�). In this paper we investigate special generalized q-Genocchi polynomials and we apply the method of gen- erating function, which are exploited to derive further classes of q-Genocchi polynomials and develop q-Genocchi numbers and polynomials. By using the Laplace-Mellin transformation integral, we define q-Zeta function with weight (�,�) and by presenting a link between q-Zeta function with weight (�,�) and q-Genocchi numbers with weight (�,�) we obtain an interpolation formula for the q-Genocchi numbers and polynomials with weight (�,�). Also we derive distribution formula (Multiplication Theorem) and Witt's type formula for modified q-Genocchi numbers and polynomials with weight (�,�) which yields a deeper insight into the effectiveness of this type of generalizations for q- Genocchi numbers and polynomials. Our new generating function possess a number of interesting properties which we state in this paper. 1. Introduction, Definitions and Notations Recently, q-calculus has served as a bridge between mathematics and physics. Therefore, there is a significant increase of activity in the area of the q-calculus due to applications of the q-calculus in mathematics, statistics and physics. The ma- jority of scientists in the world who use q-calculus today are physicists. q-Calculus is a generalization of many subjects, like hypergeometric series, generating func- tions, complex analysis, and particle physics. In short, q-calculus is quite a popular subject today. One of Important Branch of q-calculus in number theory is q-type of special generating functions, for instance q-Bernoulli numbers, q-Euler numbers, and q-Genocchi numbers, here we introduce a new class of q-type generating func- tion. We introduce q-Genocchi numbers with weight (�,�). When we define a new class of generating functions like, q-Genocchi numbers with weight (�,�), then we face to with this question that can we define a new q-Zeta type function in related of this new class of q-type generating function?. We give a positive answer for our new class of numbers and polynomials. More precisely we show that our q-type gen- erating function is generalization of the Hurwitz Zeta function. Historically many authors have tried to give q-analogues of the Riemann Zeta function � (s), and its related functions. By just following the method of Kaneko et al. (M. Kaneko, N. Kurokawa and M. Wakayama, A variation of Euler's approach to the Riemann Zeta

研究动机与目标

• 通过生成函数定义一类带有权重 (α,β) 的新修正 q-Genocchi 数与多项式。
• 通过 Laplace-Mellin 积分变换建立带有权重 (α,β) 的 q-Zeta 函数。
• 推导出一个插值公式，将 q-Zeta 函数与 q-Genocchi 数与多项式联系起来。
• 证明修正 q-Genocchi 数与多项式的形式的乘法定理与 Witt 型公式。
• 在 q-Genocchi 多项式带有权重的框架下，将 Hurwitz Zeta 函数推广为 q-类比。

提出的方法

• 为带有权重 (α,β) 的修正 q-Genocchi 数与多项式构造生成函数。
• 应用 Laplace-Mellin 变换以定义带有权重 (α,β) 的 q-Zeta 函数。
• 通过积分表示建立 q-Zeta 函数与 q-Genocchi 数之间的函数关系。
• 推导出修正 q-Genocchi 数与多项式的乘法定理（分布公式）。
• 证明 Witt 型公式，以揭示 q-Genocchi 数的结构特性。
• 通过引入的 q-Genocchi 框架，将 Hurwitz Zeta 函数推广为 q-类比。

实验结果

研究问题

• RQ1 能否定义一个带有权重 (α,β) 的 q-Zeta 函数，使其插值具有相同权重的 q-Genocchi 数？
• RQ2 带有权重 (α,β) 的修正 q-Genocchi 数的生成函数与 Hurwitz Zeta 函数有何关联？
• RQ3 对于修正 q-Genocchi 数与多项式，可以推导出哪些乘法与结构公式（例如乘法定理、Witt 型公式）？
• RQ4 所提出的 q-Genocchi 框架在何种方式下推广了经典 zeta 函数及其 q-类比？
• RQ5 Laplace-Mellin 变换在连接带有权重的 q-Genocchi 数与 q-Zeta 函数中起什么作用？

主要发现

• 通过 Laplace-Mellin 积分变换，定义了一个新的带有权重 (α,β) 的 q-Zeta 函数。
• 建立了插值公式，将带有权重 (α,β) 的 q-Zeta 函数与具有相同权重的 q-Genocchi 数与多项式联系起来。
• 推导出带有权重 (α,β) 的修正 q-Genocchi 数与多项式的乘法定理（分布公式）。
• 证明了 Witt 型公式，为修正 q-Genocchi 数与多项式提供了更深层次的结构洞察。
• 表明带有权重 (α,β) 的修正 q-Genocchi 数的生成函数可推广 Hurwitz Zeta 函数。
• 该框架成功扩展了经典 q-Genocchi 数，并为 Riemann 与 Hurwitz zeta 函数提供了 q-类比。