Robert, Alain; Zuber, Maxime The Kazandzidis supercongruences. A simple proof and an application. (English) Zbl 0852.11001 Rend. Semin. Mat. Univ. Padova 94, 235-243 (1995). In 1969 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. 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 \(Q_{np} (t) \equiv Q_n (t^p) \pmod {n^2 p^2 \mathbb{Z}_p [t]}\). Reviewer: József Sándor (Jud.Harghita) Cited in 2 Documents MSC: 11A07 Congruences; primitive roots; residue systems 11B65 Binomial coefficients; factorials; \(q\)-identities 05A10 Factorials, binomial coefficients, combinatorial functions 11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) Keywords:binomial coefficients; \(p\)-adic gamma function; supercongruence; Legendre polynomial Citations:Zbl 0212.38801; Zbl 0553.26006 PDFBibTeX XMLCite \textit{A. Robert} and \textit{M. Zuber}, Rend. Semin. Mat. Univ. Padova 94, 235--243 (1995; Zbl 0852.11001) Full Text: Numdam EuDML References: [1] G.S. Kazandzidis , A commentary on Lagrange’s congruence , D. Phil. Thesis, Oxford University 1948 , published version: Dept. of Mathematics, University of Ioannina , 1970 . MR 282914 [2] G.S. Kazandzidis , Congruences on the binomial coefficients , Bull. Soc. Math. Grèce , (N.S.) 9 , 1968 , pp. 1 - 12 . MR 265271 | Zbl 0179.06601 · Zbl 0179.06601 [3] G.S. Kazandzidis , On congruences in number theory , Bull. Soc. Math. Grèce , (N.S.) 10 , fasc. 1 ( 1969 ), pp. 35 - 40 . MR 279027 | Zbl 0212.38801 · Zbl 0212.38801 [4] W.H. Schikhof , Ultrametric Calculus-An Introduction to p-Adic Analysis , Cambridge Studies in Advanced Mathematics , 4 , Cambridge University Press ( 1984 ). MR 791759 | Zbl 0553.26006 · Zbl 0553.26006 [5] A. Robert , Polynômes de Legendre mod 4 , Comptes Rendus Acad. Sci. Paris , 316 , Série I ( 1993 ), pp. 1235 - 1240 . MR 1226106 | Zbl 0804.11064 · Zbl 0804.11064 [6] T. Honda , Two congruence properties of Legendre polynomials , Osaka J. Math. , 13 ( 1976 ), pp. 131 - 133 . MR 409485 | Zbl 0345.12101 · Zbl 0345.12101 [7] M. Zuber , Propriétés p-adiques de polynômes classiques , Thèse, Université de Neuchâtel ( 1992 ). [8] E. Artin , Collected Works , Addison-Wesley ( 1965 ). MR 176888 · Zbl 0146.00101 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.