A q-analog of Schl\"afli and Gould identities on Stirling numbers

TL;DR

This paper introduces q-analogs of classical identities relating Stirling numbers, extending their combinatorial relationships through inversion generating functions, with proofs based on computation.

Contribution

It provides new q-analog identities for Stirling numbers of both kinds, expanding the algebraic framework of these combinatorial objects.

Findings

Established q-analogs of Schl"afli and Gould identities

Proofs are computational, not combinatorial

Open problem: finding a combinatorial proof

Abstract

Stirling numbers of both kinds are linked to each other via two combinatorial identities due to Schl\"afli and Gould. Using q-analogs of Stirling numbers defined as inversion generating functions, we provide q-analogs of the two identities. The proof is computational and we leave open the problem of finding a more combinatorial one.