Involve: A Journal of Mathematics

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  • Volume 8, Number 1 (2015), 39-61.

An elementary approach to characterizing Sheffer A-type 0 orthogonal polynomial sequences

Daniel Galiffa and Tanya Riston

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In 1939, Sheffer published “Some properties of polynomial sets of type zero”, which has been regarded as an indispensable paper in the theory of orthogonal polynomials. Therein, Sheffer basically proved that every polynomial sequence can be classified as belonging to exactly one type. In addition to various interesting and important relations, Sheffer’s most influential results pertained to completely characterizing all of the polynomial sequences of the most basic type, called A-type 0, and subsequently establishing which of these sets were also orthogonal. However, Sheffer’s elegant analysis relied heavily on several characterization theorems. In this work, we show all of the Sheffer A-type 0 orthogonal polynomial sequences can be characterized by using only the generating function that defines this class and a monic three-term recurrence relation.

Article information

Involve, Volume 8, Number 1 (2015), 39-61.

Received: 20 August 2012
Revised: 7 May 2013
Accepted: 27 December 2013
First available in Project Euclid: 22 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]

A-type 0 generating functions orthogonal polynomials recurrence relations Sheffer sequences


Galiffa, Daniel; Riston, Tanya. An elementary approach to characterizing Sheffer A-type 0 orthogonal polynomial sequences. Involve 8 (2015), no. 1, 39--61. doi:10.2140/involve.2015.8.39.

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