Journal of Integer Sequences, Vol. 27 (2024), Article 24.5.3

Some Identities Involving Stirling Numbers Arising from Matrix Decompositions


M. Bahrami-Taghanaki
Faculty of Mathematics
K. N. Toosi University of Technology
P. O. Box 16765–3381
Tehran
Iran
and
Department of Applied Mathematics II
Faculty of Aeronautical and Space Engineering
University of Vigo
Ourense
Spain

A. R. Moghaddamfar
Faculty of Mathematics
K. N. Toosi University of Technology
P. O. Box 16765–3381
Tehran
Iran

Nima Salehy
Department of Mathematics and Statistics
Louisiana Tech University
Ruston, LA 71272
USA

Navid Salehy
Department of Mathematics
University of New Orleans
New Orleans, LA 70148
USA

Abstract:

An infinite matrix $\mathsf{S}=[\mathsf{S}_{i, j}]_{i, j\geq 1}$ is said to be Stirling-like if its entries satisfy the recurrence $\mathsf{S}_{i, j}=\mathsf{S}_{i-1, j-1}+j\mathsf{S}_{i-1, j}$ for $i, j\geq 2$. The aim of the present study is twofold. Firstly, we find some matrix decompositions for certain Stirling-like matrices and specifically evaluate their determinants. Secondly, using the obtained matrix decompositions, we derive some new combinatorial identities involving Stirling numbers.

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(Concerned with sequences A000522 A294352.)

Received October 15 2023; revised version received January 26 2024; April 29 2024. Published in Journal of Integer Sequences, May 10 2024.

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