The Kazandzidis supercongruences. A simple proof and an application
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 \(Q_{np} (t) \equiv Q_n (t^p) \pmod {n^2 p^2 \mathbb{Z}_p [t]}\).
scientific article; zbMATH DE number 889841