New Convolution Identities for Classical Number Sequences via Symmetric Function Methods.

In this work, we present a detailed study of the interplay between even- and odd-indexed symmetric functions and classical number sequences, approached from both algebraic and combinatorial perspectives. Symmetric functions, which occupy a central role in algebraic combinatorics, provide a natural framework for exploring underlying structures in sequences of numbers and polynomials. By focusing on the families S_{2n-1}​ and ϕ2n​, we systematically examine how the parity of indices affects the algebraic behavior and combinatorial interpretations of these functions. This investigation is facilitated by constructing exponential generating functions associated with these families, enabling a precise encoding of their structure and revealing the intricate relationships among their terms.

Through this approach, we derive new convolution identities connecting classical sequences, such as Bernoulli, Euler, and Genocchi numbers, with important polynomial families, including bivariate Fibonacci, Lucas, Mersenne, and Balancing polynomials. These formulas not only generalize existing results but also provide closed-form generating functions and unified symmetric representations that encompass multiple polynomial structures within a single coherent framework. The distinction between even and odd indices emerges as a crucial factor, uncovering fundamental algebraic properties that govern the structure of these identities and illuminating the subtle effects of parity on recurrence relations and summation patterns.

Moreover, our methodology extends naturally to Gaussian (p,q)-Fibonacci and (p,q)-Lucas numbers, where the introduction of complex initial conditions enriches the symmetric relations and leads to novel convolution formulas. This extension highlights the versatility of the approach and establishes a systematic bridge between symmetric function theory and generalized number sequences. The results offer a unifying methodology for studying polynomial families, generating functions, and their combinatorial properties, opening new avenues for research in combinatorics, number theory, and the theory of special functions.