The new combinatorial identities of symmetric functions
Meryem Bouzeraib, Ali Boussayoud, Salah Boulaaras
TL;DR
The paper addresses the construction of novel convolution identities for symmetric functions and their connections to classical number sequences. It develops binomial and non-binomial convolution formulas via exponential generating functions and applies them to bivariate polynomials such as $F_n(y,t)$, $L_n(y,t)$, $B^*_n(y,t)$, and $C_n(y,t)$, as well as to Bernoulli, Euler, and Genocchi numbers and polynomials. The main contributions include two new theorems on symmetric-function convolutions, extensive corollaries for bivariate Fibonacci/Lucas and balancing polynomials, and numerous special-case identities that link Genocchi/Bernoulli/Euler with these polynomial families. By revealing these deep connections and providing explicit closed-form identities, the work broadens the toolkit for combinatorial and number-theoretic explorations of symmetric-function structures, with particular emphasis on specializations that recover Fibonacci/Lucas-type identities.
Abstract
In this paper, we provide some novel binomial convolution related to symmetric functions, as well as convolution sums without the binomial symbol. Moreover we give some new convolution sums of Bernoulli, Euler, and Genocchi numbers and polynomials with symmetric functions , by making use of the elementary methods including exponential generating functions. From these convolutions we deduce several new combinatorial identities for bivariate polynomials, and we establish several new identities related Bernoulli, Euler and Genocchi numbers and polynomials with certain bivariate polynomials such as bivariate Fibonacci, bivariate Lucas, bivariate balancing and bivariate balancing-Lucas polynomials.