相关论文: A Short Proof of Knuth's Old Sum

We extend the Reed Dawson identity for Knuth's old sum with a complex parameter, and we offer two separate hypergeometric series-based proofs of this generalization, and we apply this generalization to introduce binomial-harmonic sum…

We provide elementary proof of several congruences involving single sum and multisums of binomial coefficients.

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 paper we will give a proof of a certain summation formula for Gamma functions utilizing Gegenbauer polynomials.

We give combinatorial proofs for some identities involving binomial sums that have no closed form.

We give a combinatorial proof of a formula giving the partial sums of the $k$-bonacci sequence as alternating sums of powers of two multiplied by binomial coefficients. As a corollary we obtain a formula for the $k$-bonacci numbers.

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 present a different proof of the following identity due to Munarini, which generalizes a curious binomial identity of Simons. \begin{align*} \sum_{k=0}^{n}\binom{\alpha}{n-k}\binom{\beta+k}{k}x^k…

In this article we furnish a new simple proof of a hard identity from the theory of cubature formulas via the method of coefficients.

We develop a systematic and fully explicit approach to the evaluation of binomial sums involving reciprocals of binomial coefficients based on Beta integral techniques. Starting from a simple integral representation, we provide a derivation…

Using a probabilistic approach, we derive several interesting identities involving beta functions. Our results generalize certain well-known combinatorial identities involving binomial coefficients and gamma functions.

We perform certain alternating binomial summations with parameters that occur in the analysis of algorithms. A combination of integral and special function and special number representations is used. The results are sufficiently general to…

We present a unified approach which gives completely elementary proofs of three weighted sum formulae for double zeta values. This approach also leads to new evaluations of sums relating to the harmonic numbers, the alternating double zeta…

Using a probabilistic approach, we derive some interesting combinatorial identities involving gamma and beta functions. These results generalize certain well-known combinatorial identities involving binomial coefficients and special…

In this paper, we present a general framework for the derivation of interesting finite combinatorial sums starting with certain classes of polynomial identities. The sums that can be derived involve products of binomial coefficients and…

In this paper, we give a purely algebraic proof of an identity coming directly from Euler's reflection formula for the gamma function. Our proof uses Hoffman's harmonic algebra and some binomial identities.

We present an elementary proof of the generalization of the $k$-bonacci Binet formula, a closed form calculation of the $k$-bonacci numbers using the roots of the characteristic polynomial of the $k$-bonacci recursion.

We establish a simple identity and using it we find a new proof of a result of Kloosterman.

From an identity connecting a combinatorial sum and Legendre polynomials, we derive closed forms for a number of combinatorial sums. Some of them are obtained via results about the integrals of functions associated with Legendre…

In this note we present a method for obtaining a wide class of combinatorial identities. We give several examples, in particular, based on the Gamma and Beta functions. Some of them have already been considered by previously, and other are…