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Let p be a prime greater than or equal to 5: In this paper, by using the harmonic numbers and Fermat quotient we establish congruences involving the sums... [ABSTRACT FROM AUTHOR]

In this paper, by the differential operator we give some congruences and new proof for some known congruence concerning the Bell polynomials, derangement polynomials and r-Lah polynomials. [ABSTRACT FROM AUTHOR]

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 & Ömür, Mao & Sun and Meštrović & Andjić. In the present paper, some new combinatorial congruences are proved. These congruences are mainly determined modulo p² or p³ (in any prime) and they are motivated by a recent paper by Meštrović and Andjić. The first main result (Theorem 1) presents the congruence modulo p² (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 Hk, Catalan numbers ck and Fermat quotient q2 := (2p-1 - 1)/p are obtained as consequences of Theorem 1. The second main result (Theorem 2) presents the congruence modulo p³(p > 3 is any prime) involving sum of products of two binomial coefficients and Harmonic numbers. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the umbral representation of the geometric polynomials wxn; = wn (x) to derive some properties involving these polynomials. Furthermore, for any prime number p and any polynomial f with integer coefficients, we show (f (wx))p≡ f (wx) (mod p) and we give other curious congruences. [ABSTRACT FROM AUTHOR]

In this paper we give some congruences on the r-derangement polynomials (defined below), Lah polynomials and some versions of Bell numbers and polynomials. [ABSTRACT FROM AUTHOR]