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NEW SUPERCONGRUENCES INVOLVING PRODUCTS OF TWO BINOMIAL COEFFICIENTS

Part of: Elementary number theory Combinatorics Sequences and sets

Published online by Cambridge University Press:  23 August 2019

ZHI-HONG SUN
ZHI-HONG SUN*
Affiliation:
School of Mathematics and Statistics,Huaiyin Normal University, Huaian, Jiangsu 223300, PR China email zhsun@hytc.edu.cn
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Abstract

Let $p>3$ be a prime and let $a$ be a rational $p$-adic integer with $a\not \equiv 0\;(\text{mod}\;p)$. We evaluate

$$\begin{eqnarray}\mathop{\sum }_{k=1}^{(p-1)/2}\frac{1}{k}\binom{a}{k}\binom{-1-a}{k}\quad \text{and}\quad \mathop{\sum }_{k=0}^{(p-1)/2}\frac{1}{2k-1}\binom{a}{k}\binom{-1-a}{k}\end{eqnarray}$$
modulo $p^{2}$ in terms of Bernoulli and Euler polynomials.

Keywords

congruenceBernoulli polynomialEuler polynomial

MSC classification

Primary: 11A07: Congruences; primitive roots; residue systems
Secondary: 11B68: Bernoulli and Euler numbers and polynomials 05A10: Factorials, binomial coefficients, combinatorial functions 05A19: Combinatorial identities, bijective combinatorics
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 101 , Issue 3 , June 2020 , pp. 367 - 378
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

The author is supported by the National Natural Science Foundation of China (grant no. 11771173).

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