2024/01/31 by Vincent Brugidou, Brugidou, Vincent
Mathematics · #Advanced Mathematical Identities #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Mathematical Inequalities and Applications #Mathematical functions and polynomials
paper · pdf · doi:10.48550/arxiv.2401.17687
openalex publication_date 2024/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let λ=( λ1,λ2,...,λr) be an integer partition, and [pλ] the q-analog of the symmetric power function %pλ. This q-analogue has been defined as a special case, in the author's previous article: "A q-analog of certain symmetric functions and one of its specializations". Here, we prove that a large part of the classical relations between pλ, on one hand, and the elementary and complete symmetric functions en and hn, on the other hand, have q-analogues with [ pλ] . In particular, the generating functions E( t) =∑\nolimitsn≥ 0entn and H( t) =∑\nolimitsn≥ 0hntn are expressed in terms of [ pn] , using Gessel's q-exponential formula and a variant of it. A factorization of these generating functions into infinite q-products, which has no classical counterpart, is established. By specializing these results, we show that the q-binomial theorem is a special case of these infinite q-products. We also obtain new formulas for the tree inversions enumerators and for certain q-orthogonal polynomials, detailing the case of dicrete q-Hermite polynomials.