Generalized r-Lah numbers

TL;DR

This paper introduces a two-parameter polynomial generalization of r-Lah numbers, providing new identities, combinatorial proofs, and orthogonality relations that extend and unify existing combinatorial number concepts.

Contribution

It presents a novel two-parameter generalization of r-Lah numbers, along with new identities and combinatorial proofs that deepen understanding of these combinatorial structures.

Findings

Derived several identities for G_{a,b}(n,k;r)

Provided combinatorial proofs for existing identities

Established orthogonality relations for the generalized numbers

Abstract

In this paper, we consider a two-parameter polynomial generalization, denoted by G_{a,b}(n,k;r), of the r-Lah numbers which reduces to these recently introduced numbers when a=b=1. We present several identities for G_{a,b}(n,k;r) that generalize earlier identities given for the r-Lah and r-Stirling numbers. We also provide combinatorial proofs of some identities involving the r-Lah numbers which were established previously using algebraic methods. Generalizing these arguments yields orthogonality-type relations that are satisfied by G_{a,b}(n,k;r).