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https://www.nist.gov/itl/math/acmd-seminar-utility-series-and-integral-representations-askey-wilson-polynomials-and-their

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# ACMD Seminar: The utility of series and integral representations for the Askey-Wilson polynomials and their symmetric sub-families

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Howard Cohl*NIST Applied and Computational Mathematics Division*

Tuesday, June 29, 2021, 3:00 EDT (1:00 MDT)

**A video of this talk is available to NIST staff in the Math channel on NISTube, which is accessible from the NIST internal home page.**

**Abstract:**  The Askey-Wilson polynomials are a class of orthogonal polynomials which are symmetric in four free parameters which lie at the very top of the *q*-Askey scheme of basic hypergeometric orthogonal polynomials. These polynomials, and the polynomials in their subfamilies, are usually defined in terms of their finite series representations which are given in terms of terminating basic hypergeometric series. However, they also have nonterminating, *q*-integral, and integral representations. In this talk, we will explore some of what is known about the symmetry of these representations and how they have been used to compute their important properties such as  generating functions. This study led to an extension of interesting contour integral representations of sums of nonterminating basic hypergeometric functions initially studied by Bailey, Slater, Askey, Roy, Gasper and Rahman. We will also discuss how these contour integrals are deeply connected to the properties of the symmetric basic hypergeometric orthogonal polynomials.

**Bio:**  Howard S. Cohl is a Mathematician in the Applied and Computational Mathematics Division at the National Institute of Standards and Technology in Gaithersburg, Maryland. He is the Technical Editor for the NIST DLMF project and is the project leader for the NIST Digital Repository of Mathematical Formulae seeding and development projects. In this regard, he has been exploring mathematical knowledge management and the digital expression of mostly unambiguous context-free full semantic information for mathematical formulae. He is a research mathematician who is interested in the properties of the special functions of applied mathematics. His main focus is on generalized and basic hypergeometric functions, and as well on the Askey and *q*-Askey schemes of hypergeometric orthogonal polynomials. He also studies fundamental solutions of linear inhomogeneous partial differential equations on highly symmetric compact and noncompact manifolds of arbitrary dimensions---and as well, their eigenfunction expansions in curvilinear coordinate systems which separate these partial differential equations using orthogonal polynomials and related functions which provide a basis for functions on these manifolds.

**Note:**This talk will be recorded to provide access to NIST staff and associates who could not be present to the time of the seminar. The recording will be made available in the Math channel on NISTube, which is accessible only on the NIST internal network. This recording could be released to the public through a Freedom of Information Act (FOIA) request. Do not discuss or visually present any sensitive (CUI/PII/BII) material. Ensure that no inappropriate material or any minors are contained within the background of any recording. (To facilitate this, we request that cameras of attendees are muted except when asking questions.)

Mathematics and statistics

## Contacts

                -  Howard Cohl  howard.cohl@nist.gov      (202) 306-2705   
                -  Stephen Langer  stephen.langer@nist.gov      (301) 975-5423   

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