×

Generating functions of convolution matrices. (English) Zbl 1130.11311

Howard, Frederic T. (ed.), Applications of Fibonacci numbers. Volume 9. Proceedings of the 10th international research conference on Fibonacci numbers and their applications, Northern Arizona University, Flagstaff, AZ, USA, June 24–28, 2002. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1938-6 ). 289-295 (2004).
The convolution matrix of two sequences \(\{a_n\}\) and \(\{b_n\}\) is the matrix \(C\), whose first column is \(\{a_n\}\) and whose \(i\)th column \((i=2, 3,\ldots)\) is the Cauchy convolution sequence of the \((i-1)\)th column with \(\{b_n\}\). The author finds a decomposition for \(C\) as \(C=SP^{a_1}\), where \(S\) is a lower triangular matrix and \(P\) is the upper triangular Pascal matrix, and applies this decomposition to obtain an explicit formula for the entries and the row generating functions of \(C\). For more extensive lists of references, see the author and J. Leida [Fibonacci Q. 40, No. 2, 136–145 (2002; Zbl 1043.11024) and Fibonacci Q. 42, No. 3, 205–215 (2004; Zbl 1083.11022)].
For the entire collection see [Zbl 1054.11005].

MSC:

11C20 Matrices, determinants in number theory
11B65 Binomial coefficients; factorials; \(q\)-identities
15A23 Factorization of matrices
PDFBibTeX XMLCite