Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II

TL;DR

This paper introduces a new recurrence formula for Ward-Horadam H(x)-binomials, reviews related combinatorial interpretations, and explores transferring interpretations between p,q-binomial and q-binomial coefficients, connecting historical and modern perspectives.

Contribution

It presents a novel recurrence for H(x)-binomials, reviews existing combinatorial interpretations, and links p,q-binomial coefficients to q-binomial coefficients, enhancing understanding of their relationships.

Findings

New H(x)-binomials recurrence formula

Reviewed combinatorial interpretations of generalized binomials

Connected p,q-binomial and q-binomial interpretations

Abstract

We deliver here second new $\textit{H(x)}-binomials'$ recurrence formula, were $H(x)-binomials' $ array is appointed by $Ward-Horadam$ sequence of functions which in predominantly considered cases where chosen to be polynomials . Secondly, we supply a review of selected related combinatorial interpretations of generalized binomial coefficients. We then propose also a kind of transfer of interpretation of $p,q-binomial $ coefficients onto $q-binomial$ coefficients interpretations thus bringing us back to $Gy{\"{o}}rgy P\'olya $ and Donald Ervin Knuth relevant investigation decades ago.