Related papers: A Generalization of Tepper's Identity
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 give a direct combinatorial proof of a famous identity, $$ \sum_{i+j=n} m{2i}{i} \binom{2j}{j} = 4^n $$ by actually counting pairs of $k$-subsets of $2k$-sets. Then we discuss two different generalizations of the identity, and end the…
We use $q$-binomial theorem to prove three new polynomial identities involving $q$-trinomial coefficients. We then use summation formulas for the $q$-trinomial coefficients to convert our identities into another set of three polynomial…
We derive a combinatorial identity which is useful in studying the distribution of Fourier coefficients of L-functions by allowing us to pass from knowledge of moments of the coefficients to the distribution of the coefficients.
For $N \in \mathbb{N}$, let $T_{N}$ be the Chebyshev polynomial of the first kind. Expressions for the sequence of numbers $p_{\ell}^{(N)}$, defined as the coefficients in the expansion of $1/T_{N}(1/z)$, are provided. These coefficients…
Foster's network theorems and their extensions to higher orders involve resistance values and conductances. We establish identities concerning voltage values and conductances. Our identities are analogous to the extended Foster's…
This paper presents both a proof method and a result. The proof method presented is particularly suitable for uniformly proving families of identities satisfied by a family of recursive sequences. To illustrate the method, we study the…
Previously, we proved an identity for theta functions of degree eight, and several applications of it were also discussed. This identity is a natural extension of the addition formula for the Weierstrass sigma-function. In this paper we…
In a recent paper, Griffin, Ono and Warnaar present a framework for Rogers-Ramanujan type identities using Hall-Littlewood polynomials to arrive at expressions of the form \[\sum_{\lambda : \lambda_1 \leq m}…
The main aim of the present paper is to represent an exact and simple proof for FLT by using properties of the algebra identities and linear algebra.
We introduce poly-Bernoulli polynomials in two variables by using a generalization of Stirling numbers of the second kind that we studied in a previous work. We prove the bi-variate poly-Bernoulli polynomial version of some known results on…
We derive several symmetric identities for Bernoulli and Euler polynomials which imply some known identities. Our proofs depend on the new technique developed in part I and some identities obtained in [European J. Combin. 24(2003),…
We show that various identities from [1] and [3] involving Gould-Hopper polynomials can be deduced from the real but also complex orthogonal invariance of multivariate Gaussian distributions. We also deduce from this principle a useful…
By considering a limiting case of a Kronecker-type identity, we obtain an identity found by both Andrews and Crandall. We then use the Andrews-Crandall identity to give a new proof of a formula of Gauss for the representations of a number…
By polynomial (or extended binomial) coefficients, we mean the coefficients in the expansion of integral powers, positive and negative, of the polynomial $1+t +\cdots +t^{m}$; $m\geq 1$ being a fixed integer. We will establish several…
We present an algebraic generalization of Euler's theorem for quadrilaterals. Starting from the parallelogram identity in an inner product space, we derive Apollonius' identity and obtain Euler's quadrilateral identity in a unified vector…
We discuss some geometric invariants of polynomial identities of algebras deduced from Kemer's theory and deduce some quantitative information on codimension and co--length
We provide bijective proofs of two classic identities that are very simple to prove using generating functions, but surprisingly difficult to prove combinatorially. The problem of finding a bijective proof for the first identity was first…
The author proves variants of Buchstab's identity on sieve functions, refining the previous work on new iteration rules of Brady. The main tool used in the proof is a special form of combinatorial identities related to the binomial…
In this paper, we show combinatorial identities that represent powers of positive integers using multinomial coefficients, which do not come from the multinomial theorem and the multinomial Vandermonde's convolution.