[Paper Review] The c-function expansion of a basic hypergeometric function associated to root systems

Jasper V. Stokman|arXiv (Cornell University)|Sep 3, 2011

Mathematical functions and polynomials31 references4 citations

TL;DR

This paper derives the c-function expansion of a basic hypergeometric function associated to root systems, expressing it as a series in q-analogs of Harish-Chandra series. The expansion coefficients are given explicitly in terms of a q-analog of the Harish-Chandra c-function, realized as a product of q-Gamma functions, establishing a q-analog of the Heckman-Opdam hypergeometric function.

ABSTRACT

We derive an explicit c-function expansion of a basic hypergeometric function associated to root systems. The basic hypergeometric function in question was constructed as explicit series expansion in symmetric Macdonald polynomials by Cherednik in case the associated twisted affine root system is reduced. Its construction was extended to the nonreduced case by the author. It is a meromorphic Weyl group invariant solution of the spectral problem of the Macdonald q-difference operators. The c-function expansion is its explicit expansion in terms of the basis of the space of meromorphic solutions of the spectral problem consisting of q-analogs of the Harish-Chandra series. We express the expansion coefficients in terms of a q-analog of the Harish-Chandra c-function, which is explicitly given as product of q-Gamma functions. The c-function expansion shows that the basic hypergeometric function formally is a q-analog of the Heckman-Opdam hypergeometric function, which in turn specializes to elementary spherical functions on noncompact Riemannian symmetric spaces for special values of the parameters.

Motivation & Objective

• To derive the c-function expansion of a basic hypergeometric function associated to root systems, extending Cherednik's construction to the nonreduced case.
• To express the eigenfunction as a linear combination of q-analogs of Harish-Chandra series, forming a basis of meromorphic solutions to the Macdonald q-difference equations.
• To identify the expansion coefficients as a q-analog of the Harish-Chandra c-function, explicitly given via products of q-Gamma functions.
• To establish that the basic hypergeometric function is formally a q-analog of the Heckman-Opdam hypergeometric function, with connections to spherical functions on symmetric spaces.

Proposed method

• The basic hypergeometric function E+ is constructed as a convergent series in symmetric Macdonald polynomials, invariant under the Weyl group and meromorphic.
• The space of meromorphic eigenfunctions of the Macdonald q-difference operators is spanned by Weyl group translates of basic Harish-Chandra series Φ̂η(·, γ).
• A c-function expansion is derived by expressing E+ as a linear combination ∑w∈W₀ ĉη(wγ) Φ̂η(·, wγ), with coefficients ĉη(γ) independent of t due to normalization of the prefactor Ŵη.
• The expansion coefficient is explicitly computed as ĉη(γ) = ϑ((w₀η)⁻¹ξγ)/ϑ(ξγ) ⋅ c_{kᵈ,q}(γ), where ϑ is a theta function and c_{kᵈ,q}(γ) is a q-analog of the c-function.
• The construction uses duality between multiplicity functions k and kᵈ, particularly nontrivial in the nonreduced case (e.g., type C∨Cₙ), and extends Cherednik’s work to nonreduced affine root systems.
• The method relies on asymptotic analysis and properties of q-special functions, including q-Gamma functions and basic hypergeometric series, with connections to Askey-Wilson and Ruijsenaars models.

Experimental results

Research questions

• RQ1 How can the basic hypergeometric function associated to root systems be expanded in terms of q-analogs of Harish-Chandra series?
• RQ2 What is the explicit form of the c-function expansion coefficients in this setting, and how do they relate to known special functions?
• RQ3 How does the basic hypergeometric function relate to the q-analog of the Heckman-Opdam hypergeometric function?
• RQ4 What role does duality play in the nonreduced case, particularly for the C∨Cₙ root system?
• RQ5 How do the results specialize to known models such as the GLₘ case or rank one systems?

Key findings

• The c-function expansion of the basic hypergeometric function E+ is explicitly derived as ∑w∈W₀ ĉη(wγ) Φ̂η(·, wγ), with coefficients independent of the spectral parameter t.
• The expansion coefficient is given by ĉη(γ) = ϑ((w₀η)⁻¹ξγ)/ϑ(ξγ) ⋅ c_{kᵈ,q}(γ), where ϑ is a theta function and c_{kᵈ,q}(γ) is a product of q-Gamma functions.
• In the reduced case, the function E+ coincides with Cherednik’s global spherical function, and the expansion recovers known results for Macdonald polynomials.
• In the nonreduced case (e.g., C∨Cₙ), the duality between k and kᵈ leads to a nontrivial transformation of the multiplicity function, and the c-function involves five degrees of freedom.
• The construction generalizes to the GLₘ case (type Aₘ₋₁), linking to Ruijsenaars’ relativistic Calogero-Moser model via Macdonald q-difference operators.
• The results show that E+ is formally a q-analog of the Heckman-Opdam hypergeometric function, with specialization to elementary spherical functions on noncompact symmetric spaces for specific parameter values.