Related papers: An identity for the Kloosterman sum

We give a new short proof of the most simple relation between consecutive power sums of the first m positive integers.

We discuss an algebraic identity, due to Sylvester, as well as related algebraic identities and applications.

In this paper we present a new identity and some of its variants which can be used for finding solutions while solving fractional infinite and finite series. We introduce another simple identity which is capable of generating solutions for…

We explain how the identity $$\sum_{i+j=n}\binom{2i}{i}\binom{2j}{j}\;=\;\displaystyle4^n$$ is an easy consequence of the inclusion-exclusion principle.

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…

In this paper, we give a short proof of a relation generalizing many identities for Bernoulli numbers.

In this work we prove a new combinatorial identity and applying it we establish many finite harmonic sum identities. Among many others, we prove that \begin{equation*}…

We derive an identity connecting any two second-order linear recurrence sequences having the same recurrence relation but whose initial terms may be different. Binomial and ordinary summation identities arising from the identity are…

We present a new and useful congruence identity satisfied by m-permutable varieties.

In this note, we provide a conceptual explanation of a well-known polynomial identity used in algebraic number theory.

We give a purely combinatorial proof of the Glaisher-Crofton identity which derives from the analysis of discrete structures generated by iterated second derivative. The argument illustrates utility of symbolic and generating function…

We generalize Menon's identity by considering sums representing arithmetical functions of several variables. As an application, we give a formula for the number of cyclic subgroups of the direct product of several cyclic groups of arbitrary…

We give a new proof of the existence of designs, which is much shorter and gives better bounds.

We show that the identity is the sum of two commutators in the algebra of all operators affiliated with a von Neumann algebra of type II$_1$, settling a question, in the negative, that had puzzled a number of us.

We prove a double binomial sum identity which differs from most binomial sum identities in that the summands involve the absolute value function. The identity is of interest because it can be used in proofs of lower bounds for the Hadamard…

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…

A probability method is provided to prove three classes of combinatorial identities. The method is extremely simple, only one step after the proper probability setup.

We derive a new Fibonacci identity. This single identity subsumes important known identities such as those of Catalan, Ruggles, Halton and others, as well as standard general identities found in the books by Vajda, Koshy and others. We also…

The transformations of the sum identities for generalized harmonic and oscillatory numbers, obtained earlier in our recent report [1], enable us to derive the new identities expressed in terms of the corresponding square roots of x. At…

In this note, using the derangement polynomials and their umbral representation, we give another simple proof of an identity conjectured by Lacasse in the study of the PAC-Bayesian machine learning theory.