Determinants involving q-Stirling numbers

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Abstract

Let S[i, j] denote the q-Stirling numbers of the second kind. We show that the determinant of the matrix (S[s + i + j, s + j])0≤i,j≤n is given by the product q3(s+n+1)-3(s)·[s]0 · [s + 1]1 ⋯ [s + n]n. We give two proofs of this result, one bijective and one based upon factoring the matrix. We also prove an identity due to Cigler that expresses the Hankel determinant of q-exponential polynomials as a product. Lastly, a two variable version of a theorem of Sylvester and an application are presented.

Original languageEnglish
Pages (from-to)630-642
Number of pages13
JournalAdvances in Applied Mathematics
Volume31
Issue number4
DOIs
StatePublished - Nov 2003

Bibliographical note

Funding Information:
The author thanks Margaret Readdy for comments on an earlier version of this paper. The author was partially supported by National Science Foundation grant 0200624.

ASJC Scopus subject areas

  • Applied Mathematics

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