On the restricted Chebyshev–Boubaker polynomials

Abstract

Using the language of Riordan arrays, we study a one-parameter family of orthogonal polynomials that we call the restricted Chebyshev-Boubaker polynomials. We characterize these polynomials in terms of the three term recurrences that they satisfy, and we study certain central sequences defined by their coefficient arrays. We give an integral representation for their moments, and we show that the Hankel transforms of these moments have a simple form. We show that the (sequence) Hankel transform of the row sums of the corresponding moment matrix is defined by a family of polynomials closely related to the Chebyshev polynomials of the second kind, and that these row sums are in fact the moments of another family of orthogonal polynomials.

Riordan arrays, Chebyshev polynomials, Fibonacci bases

“…Chebyshev polynomials and their modifications in connection with Riordan matrices are considered in [12] - [15].…”

On a transformation of Riordan moment sequences

“…In this note we study relationships between moment sequences that are defined by ordinary Riordan arrays [3,18,22] and by certain exponential Riordan arrays [3,19], which may be described as Eulerian. The theory of orthogonal polynomials [10,13,16,24] defined by Riordan arrays has been much studied [1,2,3,5,6]. For ordinary Riordan arrays, the associated orthogonal polynomials are generalized Chebyshev polynomials.…”

“…The Pascal-like (or centrally symmetric) variant A008292 that begins other two forms. To see this, we consider the bivariate generating function e x(1+y) (1 − y)2 (ye x − e yx ) 2 .…”