Krattenthaler, C. Evaluations of some determinants of matrices related to the Pascal triangle. (English) Zbl 1049.33006 Sémin. Lothar. Comb. 47, 19 p. (2002). 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. Reviewer: Y. Ben Cheikh (Monastir, Tunesia) Cited in 3 ReviewsCited in 8 Documents 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 Keywords:determinant; bivariate recurrent sequence; binomial coefficient; hypergeometric series Software:DODGSON PDFBibTeX XMLCite \textit{C. Krattenthaler}, Sémin. Lothar. Comb. 47, 19 p. (2002; Zbl 1049.33006) Full Text: arXiv EuDML