Abstract
In his paper, ‘Random walks and orthogonal polynomials: some challenges’, F. A. Grunbaum gave the polynomials orthogonal with respect to the weight
on
explicitly as
where
and
are, respectively, the Chebyshev polynomials of the first and second types. In this paper, similarly, we introduce the polynomials
defined by
where
and
are, respectively, the Chebyshev polynomials of the third and fourth types, then we give the three term recurrence relation of the polynomials
. Second, we give the kernel
where
is the Christoffel–Darboux formula for the polynomials
. Finally we give the integral of
function of
and we show how we deduce that
is orthogonal with respect to the weight
on
.
Subject classifications:
Acknowledgments
The authors are grateful to the editor and the two anonymous reviewers, whose insightful comments improved the paper considerably.
Disclosure statement
The authors reported no potential conflicts of interest relevant to this article.