Symmetric and generating functions for certain products of numbers and polynomials with application to bifurcation analysis

In this paper, we establish a novel theorem pertaining to symmetric and generating functions. Utilizing this theorem, we derive new generating functions for products of k -Fibonacci numbers and Fibonacci polynomials with certain ( p , q ) -numbers, such as ( p , q ) -Fibonacci and ( p , q ) -Jacobsthal-Lucas numbers. Furthermore, we examine the bifurcation and chaotic behavior of the generating functions associated with ( p , q ) -Jacobsthal numbers for specific parameter values of p and q .