A congruential identity and the 2-adic order of lacunary sums of binomial coefficients. (English) Zbl 1089.11015

Tollisen, Gregory; Lengyel, Tamás

Integers 4, Paper A04, 8 p. (2004).

Summary: In this paper we obtain a universal lower bound on the 2-adic order of lacunary sums of binomial coefficients. By means of necessary and sufficient conditions, we determine the set of values for which the bound is achieved and show the periodicity of the set. We prove a congruential identity for the corresponding generating function. Our approach gives an alternative and transparent proof for some results derived recently by the second author and extends them. We also propose a conjecture that implies a recursion for calculating the \(2\)-adic order of the lacunary sums for almost all values. A congruence in the style of Lucas is proved for the lacunary sums considered.

MSC:

11B65

Binomial coefficients; factorials; \(q\)-identities