Related papers: New proofs of Melzak's identity
In this article we provide with combinatorial proofs of some recent identities due to Sury and McLaughlin. We show that, the solution of a general linear recurrence with constant coefficients can be interpreted as a determinant of a matrix.…
Motivated by a polynomial identity of certain iterated integrals, first observed in [CGM20] in the setting of lattice paths, we prove an intriguing combinatorial identity in the shuffle algebra. It has a close connection to de Bruijn's…
We give an identity which is conjectured and proved by using an implementation in Multi-WZ.
In this article we obtain a general polynomial identity in $k$ variables, where $k\geq 2$ is an arbitrary positive integer. We use this identity to give a closed-form expression for the entries of the powers of a $k \times k$ matrix.…
We present a new "integral=series" type identity of multiple zeta values, and show that this is equivalent in a suitable sense to the fundamental theorem of regularization. We conjecture that this identity is enough to describe all linear…
Our paper deals about identities involving Bell polynomials. Some identities on Bell polynomials derived using generating function and successive derivatives of binomial type sequences. We give some relations between Bell polynomials and…
We establish three identities involving Dyck paths and alternating Motzkin paths, whose proofs are based on variants of the same bijection. We interpret these identities in terms of closed random walks on the halfline. We explain how these…
We present a list of equivalent expressions and extensions of Bougerol's celebrated identity in law, obtained by several authors. We recall well-known results and the latest progress of the research associated with this celebrated identity…
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 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 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…
In this paper we formulate combinatorial identities that give representation of positive integers as linear combination of even powers of 2 with binomial coefficients. We present side by side combinatorial as well as computer generated…
We prove an inverse relation and a family of convolution formulas involving partial Bell polynomials. Known and some presumably new combinatorial identities of convolution type are discussed. Our approach relies on an interesting…
A conjecture connected with quantum physics led N. Katz to discover some amazing mixed character sum identities over a field of q elements, where q is a power of a prime p > 3. His proof required deep algebro-geometric techniques, and he…
We extend Fibonacci numbers with arbitrary weights and generalize a dozen Fibonacci identities. As a special case, we propose an elliptic extension which extends the $q$-Fibonacci polynomials appearing in Schur's work. The proofs of most of…
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*}…
This note provide bijective proofs of two combinatorial identities involving generalized Catalan number $C_{m,5}(n)={m\over 5n+m}{5n+m\choose n}$ recently proposed by Sun.
We obtain a bijection between certain lattice paths and partitions. This implies a proof of polynomial identities conjectured by Melzer. In a limit, these identities reduce to Rogers--Ramanujan-type identities for the…
We examine a method to conjecture two very famous identities that were conjectured by Ramanujan, and later found to be known to Rogers.
We consider an identity relating Fibonacci numbers to Pascal's triangle discovered by G. E. Andrews. Several authors provided proofs of this identity, all of them rather involved or else relying on sophisticated number theoretical…