Pseudo-Involutions in the Riordan Group
Alexander Burstein and Louis W. Shapiro
Department of Mathematics
Washington, DC 20059
We consider pseudo-involutions in the Riordan group where the generating
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
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
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,
(Concerned with sequences
Received December 29 2021; revised version received March 14 2022.
Published in Journal of Integer Sequences,
March 24 2022.
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