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The Kazandzidis supercongruences. A simple proof and an application. (English) Zbl 0852.11001

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 \(Q_{np} (t) \equiv Q_n (t^p) \pmod {n^2 p^2 \mathbb{Z}_p [t]}\).

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.)
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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
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[8] E. Artin , Collected Works , Addison-Wesley ( 1965 ). MR 176888 · Zbl 0146.00101
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