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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
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##### References:
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