[Paper Review] Dual addition formulas associated with dual product formulas

Tom H. Koornwinder|arXiv (Cornell University)|Jul 20, 2016

Mathematical functions and polynomials4 citations

TL;DR

This paper introduces a dual addition formula for ultraspherical polynomials by recognizing their linearization coefficients as weights for Racah polynomials, enabling a Fourier–Racah expansion dual to the classical addition formula. The key result is a new orthogonal expansion that mirrors the product formula duality, with formal extensions to Gegenbauer and Hermite functions via Wilson polynomials and limit processes.

ABSTRACT

We observe that the linearization coefficients for ultraspherical polynomials are the orthogonality weights for Racah polynomials with special parameters. Then it turns out that the linearization sum with such a Racah polynomial as extra factor inserted, can also be evaluated. The corresponding Fourier--Racah expansion is an addition type formula which is dual to the well-known addition formula for ultraspherical polynomials. The limit to the case of Hermite polynomials of this dual addition formula is also considered. Similar results as for ultraspherical polynomials, although only formal, are given by taking the Ruijsenaars--Hallnäs dual product formula for Gegenbauer functions as a starting point and by working with Wilson polynomials.

Motivation & Objective

• To resolve Richard Askey's long-standing problem of finding an addition-type formula dual to the linearization formula for ultraspherical polynomials.
• To establish a duality between the classical addition formula (involving integration over angles) and a new expansion involving orthogonal polynomials in discrete variables.
• To extend the duality to the limit case of Hermite polynomials and to Gegenbauer functions using the Hallnäs–Ruijsenaars dual product formula.
• To identify the linearization coefficients of ultraspherical polynomials as orthogonality weights for Racah polynomials, enabling a new orthogonal expansion.
• To explore the possibility of group-theoretic interpretations, particularly in SU(2) tensor algebras, for the dual addition formula.

Proposed method

• Recognize that the linearization coefficients of ultraspherical polynomials correspond to the weight function of Racah polynomials with specific parameters.
• Insert a Racah polynomial as a factor in the linearization sum and evaluate the resulting sum using known orthogonality and transformation identities.
• Derive a Fourier–Racah expansion of the form $ P_n(x)P_n(y) = ext{sum over Racah polynomials} $, which constitutes the dual addition formula.
• Use the Hallnäs–Ruijsenaars dual product formula for Gegenbauer functions as a starting point and identify the weight function as that of Wilson polynomials.
• Apply formal contour integration and Fourier–Wilson inversion to derive a dual addition formula for Gegenbauer functions, expressed via Wilson polynomials.
• Take the limit $ u o ext{imaginary} $ and $ u o 0 $ to recover the Hermite case, observing that the orthogonality of Racah polynomials degenerates into a biorthogonality of shifted factorials.

Experimental results

Research questions

• RQ1 Can a dual addition formula be constructed for ultraspherical polynomials that mirrors the duality between the product formula and the linearization formula?
• RQ2 Are the linearization coefficients of ultraspherical polynomials identifiable as orthogonality weights for known orthogonal polynomials, such as Racah polynomials?
• RQ3 Does the Hallnäs–Ruijsenaars dual product formula for Gegenbauer functions suggest a dual addition formula involving Wilson polynomials?
• RQ4 What is the limiting behavior of the dual addition formula as the parameter $ u $ approaches imaginary infinity, leading to Hermite polynomials?
• RQ5 Can the dual addition formula be interpreted in a group-theoretic framework, such as in SU(2) representation theory?

Key findings

• The linearization coefficients of ultraspherical polynomials are identified as the orthogonality weights for Racah polynomials with parameters $ ho_1 = ho_2 = rac{1}{2} $, $ ho_3 = rac{1}{2} $, and $ ho_4 = rac{1}{2} $, enabling a new orthogonal expansion.
• A dual addition formula for ultraspherical polynomials is derived as $ P_n(x)P_n(y) = ext{sum over } R_k^{(n,n)}(j) imes ext{coefficients} $, where $ R_k^{(n,n)} $ are Racah polynomials.
• The dual addition formula for Gegenbauer functions is expressed as $ rac{1}{4 au} ext{integrand} = ext{sum over } W_k( u^2; ext{parameters}) imes ext{products of } ilde{ heta} $, with $ W_k $ being Wilson polynomials.
• The limit of the dual addition formula from ultraspherical to Hermite polynomials yields a known formula, with the orthogonality of special Racah polynomials degenerating into a biorthogonality of shifted factorials.
• The formal derivation of the dual addition formula for Gegenbauer functions relies on contour integration and Fourier–Wilson inversion, with convergence and analytic continuation issues left for future work.
• The paper suggests that the dual addition formula may have a group-theoretic interpretation in the context of SU(2) tensor algebras, particularly when $ u = 0 $.