The Kazandzidis supercongruences. A simple proof and an application
type: Item
instance of: scholarly article
title: The Kazandzidis supercongruences. A simple proof and an application
zbMATH Open document ID: 0852.11001
author: Alain Robert
author: Maxime Zuber
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1996-12-05T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108373
review text: In 1969 \textit{G. S. Kazandzidis} [Bull. Soc. Math. Grèce, N. Ser. 10, 35-40 (1969; Zbl 0212.38801)] proved that \({{np} \choose {kp}}= {n\choose k}+ up^m nk (n-k) {n\choose k}\), where \(p\) is an odd prime, and \(m=3\) for \(p\geq 5\); \(m=2\) for \(p=3\). Here \(u\) is a \(p\)-adic integer (\(n\), \(k\) positive integers). The authors obtain a nice new proof, by using certain properties of the function \(x\mapsto \log \Gamma_p (x)\), with \(\Gamma_p (x)\) the \(p\)-adic gamma function [see e.g. \textit{W. H. Schikhof}, Ultrametric calculus. An introduction to \(p\)-adic analysis, Cambridge Univ. Press (1984; Zbl 0553.26006)]. With the aid of this theorem, a new supercongruence is obtained for the polynomial \(Q_n (t):= P_n (1+ 2t)+ P_{n-1} (1+ 2t)\) \((n\geq 1)\), where \(P_n (x)\) is the Legendre polynomial. Namely, for an odd prime \(p\), and \(n\geq 1\), one has \(Q_{np} (t) \equiv Q_n (t^p) \pmod {n^2 p^2 \mathbb{Z}_p [t]}\).
reviewed by: József Sándor
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 11S80
zbMATH DE Number: 889841
zbMATH Keywords: binomial coefficients
zbMATH Keywords: \(p\)-adic gamma function
zbMATH Keywords: supercongruence
zbMATH Keywords: Legendre polynomial
MaRDI profile type: Publication
cites work: Q5569569
cites work: Q5614048
cites work: Q3346515
cites work: Q3136256
cites work: Two congruence properties of Legendre polynomials
cites work: Q5522902
description: scientific article; zbMATH DE number 889841