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A binomial sum related to Wolstenholme’s theorem. (English) Zbl 1241.11016
M. Chamberland and K. Dilcher [J. Number Theory 120, No. 2, 349–371 (2006; Zbl 1139.11013)] had already found that a certain alternating sum \(u(n)\) of \(n+1\) products of two binomial coefficients has a property similar to Wolstenholme’s theorem, namely \(u(p)\equiv -\pmod {p^3}\) for all primes \(p\geq 5\).
In order to partly explain why such congruence also holds for certain composite integers \(p\) which appear to always have exactly two prime divisors, one of which is always 2 or 5, in the present paper the authors study the sums \[ u(n):=\sum_{k=0}^n (-1)^{k} \binom{n}{k} \binom{2n}{k} \] in greater details than they previously did. As a consequence of their accurate research, the composites in question are characterized and some new open problems are raised as well.
Beyond basic tools like Leibniz’s rule for higher derivatives of a product, Fermat’s Little Theorem, Kummer’s Theorem, Vandermonde’s Identity and the Chinese Remainder Theorem, in the proof the authors employ some remarkable results for binomial coefficients from A. Granville [Borwein, J. (ed.) et al., Organic mathematics. CMS Conf. Proc. 20, 253–276 (1997; Zbl 0903.11005)], the famous congruence discovered by H. Anton [Arch. Math. Phys. 49, 241–308 (1868; JFM 01.0047.04)], an important congruence obtained by V. Brun et al. [11. Skand. Mat.-Kongr., Trondheim 1949, 42–54 (1952; Zbl 0048.27204)] and generalized by K. Davis and W. Webb [J. Number Theory 43, No. 1, 20–23 (1993; Zbl 0769.11008)], a well-known theorem by E. Lucas [Bull. Soc. Math. Fr. 6, 49–54 (1878; JFM 10.0139.04)], a convergence property of linear recurrence sequences found by R. J. Kooman and R. Tijdeman [Nieuw Arch. Wiskd., IV. Ser. 8, No. 1, 13–25 (1990; Zbl 0713.11010)], an explicit formula for the Jacobi polynomials from [H. W. Gould, Combinatorial identities. Morgantown, W. Va.: Henry E. Gould (1972; Zbl 0241.05011)] and a second explicit expression for them from [M. Abramowitz (ed.) and I. A. Stegun (ed.), Handbook of mathematical functions with formulas, graphs and mathematical tables. Washington: U.S. Department of Commerce. (1964; Zbl 0171.38503)], the Davis-Webb congruence (mod 8) found by J. G. Huard, B. K. Spearman and K. S. Williams [Eur. J. Comb. 19, No. 1, 45–62 (1998; Zbl 0889.11007)], a type of congruences studied by R. J. McIntosh [Am. Math. Mon. 99, No. 3, 231–238 (1992; Zbl 0755.11001)] and the WZ algorithm by [M. Petkovšek, H. S. Wilf and D. Zeilberger, \(A=B\). Wellesley, MA: A. K. Peters (1996; Zbl 0848.05002)] implemented in Maple 9.5 (http://www.maplesoft.com).

MSC:
11B65 Binomial coefficients; factorials; \(q\)-identities
05A10 Factorials, binomial coefficients, combinatorial functions
Software:
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References:
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