Zeros of $p$-adic hypergeometric functions, $p$-adic analogues of Kummer's and Pfaff's identities

Values of $p$-adic hypergeometric functions, and $p$-adic analogue of Kummer's linear identity

Let $p$ be an odd prime and $\mathbb{F}_p$ be the finite field with $p$ elements. This paper focuses on the study of values of a generic family of hypergeometric functions in the $p$-adic setting which we denote by ${_{3n-1}G_{3n-1}}(p, t),$ where $n\geq1$ and $t\in\mathbb{F}_p$. These values are expressed in terms of numbers of zeros of certain polynomials over $\mathbb{F}_p$. These results lead to certain $p$-adic analogues of classical hypergeometric identities. Namely, we obtain $p$-adic analogues of particular cases of a Gauss' theorem and a Kummer's theorem. Moreover, we examine the zero…

重访普法夫方法

Pfaff's Method Revisited

本文重新审视1797年普法夫提出的超几何级数求和方法，并将其推广用于证明一系列终止型q-超几何级数-乘积恒等式，展示了该方法的广泛适用性。

Certain character sums and hypergeometric series

We prove two transformations for the $p$-adic hypergeometric series which can be described as $p$-adic analogues of a Kummer's linear transformation and a transformation of Clausen. We first evaluate two character sums, and then relate them to the $p$-adic hypergeometric series to deduce the transformations. We also find another transformation for the $p$-adic hypergeometric series from which many special values of the $p$-adic hypergeometric series as well as finite field hypergeometric functions are obtained.

Transformations of Well-Poised Hypergeometric Functions over Finite Fields

We define a hypergeometric function over finite fields which is an analogue of the classical generalized hypergeometric series. We prove that this function satisfies many transformation and summation formulas. Some of these results are analogous to those given by Dixon, Kummer and Whipple for the well-poised classical series. We also discuss this function's relationship to other finite field analogues of the classical series, most notably those defined by Greene and Katz.

Multiplicate inverse forms of terminating hypergeometric series

The multiplicate form of Gould--Hsu's inverse series relations enables to investigate the dual relations of the Chu-Vandermonde-Gauß's, the Pfaff-Saalschütz's summation theorems and the binomial convolution formula due to Hagen and Rothe. Several identitity and reciprocal relations are thus established for terminating hypergeometric series. By virtue of the duplicate inversions, we establish several dual formulae of Chu-Vandermonde-Gauß's and Pfaff-Saalschütz's summation theorems in Section (3)\cite{ChuVanGauss} and (4)\cite{PfaffSaalsch}, respectively. Finally, the last section is devoted to …

Recursion Rules for the Hypergeometric Zeta Functions

The hypergeometric zeta function is defined in terms of the zeros of the Kummer function M(a, a + b; z). It is established that this function is an entire function of order 1. The classical factorization theorem of Hadamard gives an expression as an infinite product. This provides linear and quadratic recurrences for the hypergeometric zeta function. A family of associated polynomials is characterized as Appell polynomials and the underlying distribution is given explicitly in terms of the zeros of the associated hypergeometric function. These properties are also given a probabilistic interpre…