Abstract

In this paper, we investigate degenerate versions of the generalized pth order Franel numbers which are certain finite sums involving powers of binomial coefficients. In more detail, we introduce degenerate generalized hypergeometric functions and study degenerate hypergeometric numbers of order p. These numbers involve powers of λ-binomial coefficients and λ-falling sequence, and can be represented by means of the degenerate generalized hypergeometric functions. We derive some explicit expressions and combinatorial identities for those numbers. We also consider several related special numbers like λ-hypergeometric numbers of order p and Apostol type λ-hypergeometric numbers of order p, of which the latter reduce in a limiting case to the generalized pth order Franel numbers.

中文翻译：

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退化二项式系数和退化超几何函数

摘要

在本文中，我们研究了广义p阶Franel数的简并形式，它们是涉及二项式系数幂的某些有限和。更详细地，我们介绍退化的广义超几何函数，并研究阶次p的退化的超几何数。这些数涉及λ-二项式系数和λ-下降序列的幂，并且可以通过简并的广义超几何函数表示。我们为这些数字得出一些明确的表达式和组合标识。我们还考虑了几个相关的特殊数字，例如λ - p阶的超几何数和Apostol类型λ-hypergeometric号码顺序的p，其中后者在限制的情况下，减少对广义p阶Franel号码。