A connection between a generalized Pascal matrix and the hypergeometric function
The authors consider the generalized Pascal matrix \(P\) and a lower triangular matrix \(L\) defined as follows, \[ [P(t)]_{ij} = {_2F_1} \left[1-i,1-j; 1;\frac{1}{t^2}\right]t^{i+j-2},\qquad [L(t)]_{ij}=\frac{(i-1)!t^{i-j}}{(j-1)!(i-j)!}\,. \] By elementary methods they establish the Cholesky decomposition \(P(t) = L(t)\) \(L{t}^T\). From one of Gauss' relations between contiguous functions, a recurrence relation for \(P(t)\) easily follows; this relation is then used to obtain an algorithm for computing the elements of \(P(t)\).
scientific article; zbMATH DE number 2075534