特殊数与特殊函数的新型计算公式与关系：Debye函数的分析方法
Novel computational formulas and relations for special numbers and functions: approach to analysis of Debye functions
- Akdeniz University（安塔利亚大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文引入与Debye函数相关的新特殊函数族，利用生成函数法推导出涉及Debye函数、多重Hurwitz Lerch zeta函数等的新公式，并给出递推关系、级数展开和近似公式，应用于数学物理。
AI中文摘要：
本文的第一个目标是给出与Debye函数相关的一族新的特殊函数。通过使用这些函数和生成函数方法，我们推导出许多涉及Debye函数、多重Hurwitz Lerch zeta函数的新型公式和关系，后者在负整数处插值高阶Apostol-Bernoulli数。本文定理的证明方法不同于解析数论理论中的方法。本文的第二个目标是表明该函数的特殊值和积分表示与Debye函数的统一、负二项分布的矩生成函数、高阶Apostol型数的生成函数、Frobenius-Euler数、Bernoulli数以及Stirling数等密切相关。此外，我们给出了Debye函数统一的递推关系、Maclaurin级数展开和近似公式。最后，我们给出了关于本文结果在数学物理中的进一步评论、观察和应用。
英文摘要：
The first goal of this paper is to give a new family of special function related to the Debye functions. By using these function and generating function method, we derive many novel formulas and relations involving the Debye functions, multiple Hurwitz Lerch zeta function which interpolate the Apostol-Bernoulli numbers of higher order at negative integers. Proof method of theorems in this paper are different from those of theory of analytic numbers. The second goal of this paper is to show that special values and integral presentations of this function are closely related to unification of the Debye functions, the moment generating function for the negative binom distribution, the generating functions for the Apostol-type numbers of higher-order, and the Frobenius-Euler numbers, the Bernoulli numbers, and the Stirling numbers etc. Moreover, we give recurrence relations, Maclaurin's series expansion, and approximation formula for unification of the Debye functions. Finally, we give further remarks, observations and applications in mathematical physics on the results of this paper.
