Journal of Integer Sequences, Vol. 25 (2022), Article 22.3.6

Pseudo-Involutions in the Riordan Group

Alexander Burstein and Louis W. Shapiro
Department of Mathematics
Howard University
Washington, DC 20059


We consider pseudo-involutions in the Riordan group where the generating function g for the first column of a Riordan array satisfies a palindromic or near-palindromic functional equation. For those types of equations, we find, for very little work, the pseudo-involutory companion of g and have a pseudo-involution in a k-Bell subgroup. There are only slight differences in the ordinary and exponential cases. In many cases, we also develop a general method for finding B-functions of Riordan pseudo-involutions in k-Bell subgroups, and show that these B-functions involve Chebyshev polynomials. We apply our method for many families of Riordan arrays, both new and already known.

We also have some duality and reciprocity results. Since many of the examples we discuss have combinatorial significance, we conclude with a few remarks on the general framework for a combinatorial interpretation of some of the generating function results we obtain.

Full version:  pdf,    dvi,    ps,    latex    

(Concerned with sequences A000085 A000108 A000110 A000245 A000248 A000272 A000629 A001003 A001006 A002212 A004148 A006318 A007106 A007559 A025227 A027307 A038049 A038629 A049310 A068875 A069271 A086246 A089946 A106228 A109081 A111125 A127632 A153295 A153396 A156308 A166135 A182037 A200031 A212072 A215067 A216857 A238113 A344623 A347953 A348189 A348197 A349562)

Received December 29 2021; revised version received March 14 2022. Published in Journal of Integer Sequences, March 24 2022.

Return to Journal of Integer Sequences home page