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- W1635346723 abstract "We give simple proofs for the Hankel determinants of q − exponential polynomials. Let ( , ) S n k be the Stirling numbers of the second kind. Christian Radoux ([6] ) has shown that the Hankel determinants of the exponential polynomials 0 ( ) ( , ) n k n k B x S n k x = = ∑ are given by ( ) 1 1 2 , 0 0 det ( ) !. n n n i j i j j B x x j ⎛ ⎞ − ⎜ ⎟ − ⎝ ⎠ + = = = ∏ (1) In [2] I have proved some q − analogues of this result. Then Richard Ehrenborg [4] has given a combinatorial proof of one of these q − analogues. In this paper I want to show that these q − analogues in some sense have simpler proofs than the original case. We use the usual notations: For n∈ let 1 [ ] 1 n q n q − = − . The q − factorial is the product [1] [2] [ ] n ⋅ and the q − binomial coefficient n k ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ is defined by [ ]! [ ]![ ]! n n k k n k ⎡ ⎤ = ⎢ ⎥ − ⎣ ⎦ for 0 k n ≤ ≤ and 0 n k ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ else. The q − Stirling numbers [ , ] S n k of the second kind are defined by [ , ] [ 1, 1] [ ] [ 1, ] S n k S n k k S n k = − − + − (2) with [ ] [0, ] 0 S k k = = and [ ,0] [ 0]. S n n = =" @default.
- W1635346723 created "2016-06-24" @default.
- W1635346723 creator A5000249606 @default.
- W1635346723 date "2009-01-29" @default.
- W1635346723 modified "2023-09-27" @default.
- W1635346723 title "Hankel determinants of q-exponential polynomials" @default.
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