CONGRUENCES INVOLVINlG SUMS OF HARMONIC NUMBERS AND BINOMIAL COEFFICIENTS

Congruences involving sums of Harmonic numbers and binomial coefficients are considered in this paper. Recently, many great mathematicians have been interested to find congruences and relationships between these numbers such Sun & Tauraso, Koparal & Omur, Mao & Sun and Mestrovie & Andjie. In the present paper, some new combinatorial congruences are proved. These congruences are mainly determined modulo p(2) or p 3 (p in any prime) and they are motivated by a recent paper by Mestrovie and Andjie. The first main result (Theorem 1) presents the congruence modulo p(2) (p > 3 is any prime) involving sum of products of two binomial coefficients and Harmonic numbers. Two interesting congruences modulo a prime p > 3 (Corollary 2) involving Harmonic numbers H-k, Catalan numbers C-k and Fermat quotient q(2) := (2(p-1) - 1)/p are obtained as consequences of Theorem 1. The second main result (Theorem 2) presents the congruence modulo p(3) (p > 3 is any prime) involving sum of products of two binomial coefficients and Harmonic numbers.