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Evaluations of some determinants of matrices related to the Pascal triangle. (English) Zbl 1049.33006

The author proves six evaluations of determinants of matrices \((a_{i,j})_{0\leq i,j\leq n-1}\). The entries of the first three ones are given by an extension of the Pascal triangle recurrence: \(a_{i,j}= a_{i-1,j}+ a_{i,j-1}+ xa_{i- 1,j-1}\), \(i,j\geq 1\), with various initial conditions. While the remaining ones are given respectively by \[ a_{i,j}= \begin{pmatrix} 2i+ 2j+ a\\ i\end{pmatrix}- \begin{pmatrix} 2i+ 2j+ a\\ i-1\end{pmatrix},\quad a_{i,j}= \begin{pmatrix} 2i+ 2j+ a\\ i+1\end{pmatrix}- \begin{pmatrix} 2i+ 2j+a\\ i\end{pmatrix}, \] \(a_{i,j}= (i-j){(X+ i+ j)!\over (Y+ i+ j)!}\), \(X\) and \(Y\) being arbitrary nonnegative integers.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
05A19 Combinatorial identities, bijective combinatorics
05A10 Factorials, binomial coefficients, combinatorial functions
11C20 Matrices, determinants in number theory
15A15 Determinants, permanents, traces, other special matrix functions

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