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Determinants involving \(q\)-Stirling numbers. (English) Zbl 1071.05011
Summary: Let \(S[i,j]\) denote the \(q\)-Stirling numbers of the second kind. We show that the determinant of the matrix \((S[s+ i+ j,s+ j])_{0\leq i,j\leq n}\) is given by the product \(q^{{s+n+1\choose 3}-{s\choose 3}}\cdot[s]^0\cdot [s+1]^1\cdots[s+ n]^n\). We give two proofs of this result, one bijective and one based upon factoring the matrix. We also prove an identity due to J. Cigler [Sitzungsber., Abt. II, Österr. Akad. Wiss. Math.-Naturwiss. Kl. 208, 143–157 (1999; Zbl 1023.33011)] 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.

05A30 \(q\)-calculus and related topics
05A10 Factorials, binomial coefficients, combinatorial functions
11B73 Bell and Stirling numbers
Full Text: DOI
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