Title: Riordan arrays, orthogonal polynomials as moments, and Hankel transforms.
Abstract:In this note, we shall re-interpret some of the results of Ismail and Stanton [15, 16] in terms of Riordan arrays. These authors give functionals [15] whose moments are the Hermite, Laguerre, and vari...In this note, we shall re-interpret some of the results of Ismail and Stanton [15, 16] in terms of Riordan arrays. These authors give functionals [15] whose moments are the Hermite, Laguerre, and various Meixner families of polynomials. In this note, we shall confine ourselves to Legendre and Hermite polynomials. Indeed, the types of orthogonal polynomials representable with Riordan arrays is very limited (see below), but it is nevertheless instructive to show that a number of them can be exhibited as moments, again using (parameterized) Riordan arrays. The essence of the paper is to show that a Riordan array L (either ordinary or exponential) defines a family of orthogonal polynomials (via its inverse L−1) if and only if its production matrix [8, 9, 10] is tri-diagonal. The sequence of moments μn associated to the family of orthogonal polynomials then appears as the elements of the first column of L. In terms of generating functions, this means that if L = (g, f) (or L = [g, f ]), then g(x) is the generating function of the moment sequence. By defining suitable parameterized Riordan arrays, we can exhibit the Legendre and Hermite polynomials as such moment sequences.Read More
Publication Year: 2011
Publication Date: 2011-01-01
Language: en
Type: article
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Cited By Count: 39
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