The determinants of matrices with recursive entries

We prove a result concerning the evaluation of determinant of a matrix, whose entries are given by certain linear inhomogeneous recurrence relations. This result generalizes previous determinants evaluations due to Krattenthaler, Neuwirth and the authors in [C. Krattenthaler, Evaluations of some determinants of matrices related to the Pascal triangle, Sém. Lothar. Combin. 47 (2002), 19, Article B47g; E. Neuwirth, Treeway Galton arrays and generalized Pascal-like determinants, Technical Report, University of Vienna, 2004, pp. 1–16; A.R. Moghaddamfar, S.M.H. Pooya, S. Navid Salehy, S. Nima Salehy, Evaluating and generalizing certain determinants, submitted for publication]. The main tool for proving our result is LU-factorization of the recursive arrays introduced in [C. Krattenthaler, Advanced determinant calculus: a complement, Linear Algebra Appl. 411 (2005) 68–166].