相关论文: Elementary proof of congruences involving sum of b…

We prove several congruences for trinomial coefficients.

In this paper, we pose many challenging conjectures on congruences involving binomial coefficients and Ap\'ery-like numbers.

We give an elementary probabilistic proof of a binomial identity. The proof is obtained by computing the probability of a certain event in two different ways, yielding two different expressions for the same quantity.

We prove congruences, modulo a power of a prime p, for certain finite sums involving central binomial coefficients $\binom{2k}{k}$.

We give some results and conjectures about recurrence relations for certain sequences of binomial sums.

We prove some 3-adic congruences for binomial sums, which were conjectured by Sun.

In this article, we derive a congruence property of particular sum rules involving prime numbers. The resulting expression involves Bernoulli numbers and polynomials, for which we obtain, as a consequence, a general congruence relation as…

In this short note, we establish some identities containing sums of binomials with coefficients satisfying third order linear recursive relations. As a result and in particular, we obtain general forms of earlier identities involving…

By using the Rodriguez-Villegas-Mortenson supercongruences, we prove four supercongruences on sums involving binomial coefficients, which were originally conjectured by Sun. We also confirm a related conjecture of Guo on integer-valued…

We consider a special class of binomial sums involving harmonic numbers and we prove three identities by using the elementary method of the partial fraction decomposition. Some applications to infinite series and congruences are given.

We give elementary proofs of some congruence criteria to compute binomial coefficients in modulo a prime. These criteria are analogues to the symmetry property of binomial coefficients. We give extended version of Lucas Theorem by using…

In this note, we give an alternate proof of the multinomial theorem using a probabilistic approach. Although the multinomial theorem is basically a combinatorial result, our proof may be simpler for a student familiar with only basic…

Binomial coefficients have been used for centuries in a variety of fields and have accumulated numerous definitions. In this paper, we introduce a new way of defining binomial coefficients as repeated sums of ones. A multitude of binomial…

We prove that the Eulerian polynomial satisfies certain polynomial congruences. Furthermore, these congruences characterize the Eulerian polynomial.

In the recent article arXiv:1606.03351, Apagodu and Zeilberger discuss some applications of an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequence. At the end they…

We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.

Modifying an idea of E. Brietzke we give simple proofs for the recurrence relations of some sequences of binomial sums which have previously been obtained by other more complicated methods.

In this note, we confirm two conjectural supercongruences on double sums of binomial coefficients due to El Bachraoui.

In this paper, we state and prove some congruence properties for the trinomial coeficients, one of which is similar to the Wolstenholme's theorem.

Using elementary methods, we establish old and new relations between binomial coefficients, Fibonacci numbers, Lucas numbers, and more.