Related papers: A master identity for Horadam numbers
Classical binomial identities are established by giving probabilistic interpretations to the summands. The examples include Vandermonde identity and some generalizations.
We first establish the result that the Narayana polynomials can be represented as the integrals of the Legendre polynomials. Then we represent the Catalan numbers in terms of the Narayana polynomials by three different identities. We give…
In this note, we provide a conceptual explanation of a well-known polynomial identity used in algebraic number theory.
Based on a variant of Sury's polynomial identity we derive new expressions for various finite Fibonacci (Lucas) sums. We extend the results to Fibonacci and Chebyshev polynomials, and also to Horadam sequences. In addition to deriving sum…
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.
In this paper we give new identities involving q-Euler polynomials of higher order.
Some finite series of harmonic numbers involving certain reciprocals are evaluated. Products of such reciprocals are expanded in a sum of the individual reciprocals, leading to a computer program. A list of examples is provided.
Two new generalized Fibonacci number summation identities are stated and proved, and two other new generalized Fibonacci number summation identities are derived from these, of which two special cases are already known in literature.
In this note, we show how a combinatorial identity of Frisch can be applied to prove and generalize some well-known identities involving harmonic numbers. We also present some combinatorial identities involving odd harmonic numbers which…
A generalization of the Catalan numbers is considered. New results include binomial identities, recursive relations and a close formula for the multivariate generating function. A simple expression for the Catalan determinant is derived.
Two types of finite series of products of harmonic numbers involving nonnegative integer powers are evaluated, also yielding two other important harmonic number identities. The recursion formulas for these sums are derived, which are easily…
We prove a new universal identity for umbral operators. This motivates the definition of a subclass satisfying a simplified identity, which we fully characterize. The results are illustrated with common examples of the theory of umbral…
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…
The aim of this short note is to show how can be derived from the properties of fundamental interpolation polynomials some nice identities.
We give explicit formulae and study the combinatorics of an identity holding in all Rota-Baxter algebras. We describe the specialization of this identity for a couple of examples of Rota-Baxter algebras.
We discuss an algebraic identity, due to Sylvester, as well as related algebraic identities and applications.
We introduce several new identities combining basic hypergeometric sums and integrals. Such identities appear in the context of superconformal index computations for three-dimensional supersymmetric dual theories. We give both analytic…
A generalization of the Chu-Vandermonde convolution is presented and proved with the integral representation method. This identity can be transformed into another identity, which has as special cases two known identities. Another identity…
In this paper we derive some interesting identities arising from the orhtogonality of gegenbauer polynomials.
We derive an identity that relates a class of multiple integrals involving Vandermonde polynomials to divided differences. Alternatively the identity can be viewed as an integral formula for divided differences. As part of the derivation we…