Simplicial $d$-Polytopic Numbers Defined on Generalized Fibonacci Polynomials
Ronald Orozco López
TL;DR
This work develops a unified framework for simplicial $d$-polytopic numbers defined on the $(s,t)$-generalized Fibonacci polynomials, enabling a common treatment of diverse Fibonacci-type sequences. It introduces the $s,t$-Fibonomial calculus, derives recurrences, generating functions, and $q$-analogs for these polytopic numbers, and defines an $(s,t)$-Zeta function to study reciprocal sums. The paper establishes explicit generating functions, Warnaar-type identities, and several specializations (Fibonacci, Pell, Jacobsthal, Mersenne) with corresponding reciprocal-sum evaluations and OEIS connections. By transferring classical identities to the $(s,t)$-Fibonacci setting, it broadens the combinatorial toolbox for figurate numbers and reveals new connections between polynomials, generating functions, and zeta-type sums across multiple integer sequences. The results provide both theoretical insights and concrete formulas for generating functions and reciprocal sums across generalized Fibonacci families, with potential applications in combinatorics and number theory.
Abstract
In this article, we introduce the simplicial $d$-polytopic numbers defined on generalized Fibonacci polynomials. We establish basic identities and find $q$-identities known. Furthermore, we find generating functions for the simplicial $d$-polytopic numbers and for the squares of the generalized triangular numbers. Finally, we compute sums of reciprocals of generalized Fibonacci polynomials and generalized triangular numbers. Here we introduce the Zeta function defined on generalized Fibonacci polynomials.