[Paper Review] Addition formula for q-disk polynomials

Paul G. A. Floris|arXiv (Cornell University)|Nov 29, 1994

Algebraic structures and combinatorial models6 references4 citations

TL;DR

This paper presents a q-analogue of the addition formula for disk polynomials using quantum group theory, extending classical results by Sapiro and Koornwinder. It establishes an addition formula for q-disk polynomials—orthogonal polynomials on the unit disk—by leveraging quantum group representations, generalizing earlier work on Jacobi and disk polynomials to the q-deformed setting with applications in quantum algebra and special functions.

ABSTRACT

The purpose of this paper is to present an addition formula for so-called $q$-disk polynomials, using some quantum group theory. This result is a $q$-analogue of a result which was proved around 1970 by ${\breve{ ext S}}$apiro [S] and Koornwinder [Koo1,2] independently. They considered the homogeneous space $U(n)/U(n-1)$, were $U(n)$ denotes the group of unitary transformations of the vector space $\C^n$, and identified the corresponding zonal spherical functions as disk polynomials, in the terminology of [Koo2]. These are orthogonal polynomials in two variables whose orthogonality measure is supported by the closed unit disk in the complex plane, and which can be expressed in terms of Jacobi polynomials $P_n^{(\a,\b)} (x)$ for certain integer values of the parameters $\a$ and $\b$. The associated spher ical funtions were shown to be also expressible in terms of disk polynomials, and from this an addition formula was proved for (positive) integer values of $\a$ and $\b$. By an easy argument this identity was then extended to all complex values of $\a$ (and from this Koornwinder could even derive an addition formula for general Jacobi polynomials). And in fact the line of arguing and the results in this paper will be very similar to (part of) the ones in [Koo2].

Motivation & Objective

• To extend the classical addition formula for disk polynomials to the q-deformed setting using quantum group theory.
• To establish a q-analogue of the spherical function addition formula on the homogeneous space Uq(n)/Uq(n−1).
• To generalize results from classical Jacobi and disk polynomials to their q-analogue counterparts.
• To provide a quantum group-theoretic framework for orthogonal polynomials on the unit disk in the complex plane.

Proposed method

• Utilizes quantum group theory, specifically the quantum unitary group Uq(n), to derive the addition formula.
• Applies representation-theoretic techniques to the quantum homogeneous space Uq(n)/Uq(n−1).
• Identifies zonal spherical functions on the quantum group as q-disk polynomials.
• Employs the q-analogue of the classical method used by Sapiro and Koornwinder, adapting it to the quantum setting.
• Relies on the connection between q-disk polynomials and q-Jacobi polynomials for parameter values.
• Uses analytic continuation arguments to extend results from integer to complex parameters, as in classical theory.

Experimental results

Research questions

• RQ1 How can the classical addition formula for disk polynomials be generalized to a q-deformed setting?
• RQ2 What is the role of quantum group representations in constructing addition formulas for q-orthogonal polynomials?
• RQ3 How do q-disk polynomials arise as zonal spherical functions on quantum homogeneous spaces?
• RQ4 Can the method used for classical disk polynomials be adapted to the quantum group framework?
• RQ5 What is the relationship between q-disk polynomials and q-Jacobi polynomials in the context of addition formulas?

Key findings

• The paper derives a q-analogue of the addition formula for disk polynomials using quantum group theory.
• The zonal spherical functions on the quantum homogeneous space Uq(n)/Uq(n−1) are identified as q-disk polynomials.
• The addition formula for q-disk polynomials is established via quantum group representation theory.
• The result generalizes the classical addition formula of Sapiro and Koornwinder to the q-deformed case.
• The method allows extension of the formula to complex parameters through analytic continuation, mirroring classical results.
• The construction confirms the role of q-disk polynomials as orthogonal polynomials on the unit disk with respect to a q-deformed measure.