Proof of Two Conjectures on Supercongruences Involving Central Binomial coefficients

Abstract

In this note we use some $q$ -congruences proved by the method of ‘creative microscoping’ to prove two conjectures on supercongruences involving central binomial coefficients. In particular, we confirm the $m=5$ case of Conjecture 1.1 of Guo [‘Some generalizations of a supercongruence of Van Hamme’, Integral Transforms Spec. Funct.28 (2017), 888–899].

Several combinatorial identities derived from series expansions of powers of arcsine

“…The sequence of central binomial coefficients 2n n for n ≥ 0 is classical, simple, and elementary. This sequence has attracted many mathematicians who have published a number of papers such as [3,7,10,11,16,19,26,42,44] and closely related references therein. It is worth to mentioning that, the integral representation 2n n = 1 π ∞ 0 1 (1/4 + s 2 ) n+1 ds was derived in [33,Section 4.2].…”

Congruences concerning generalized central trinomial coefficients

“…It is known that sums involving central binomial coefficients sometimes have beautiful congruence properties (cf. e.g., [3,[5][6][7][8][15][16][17][18]). As mentioned above, the central binomial coefficients 2n n = T n (2, 1).…”