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A generalized hypergeometric function. II: Asymptotics and \(D_4\) symmetry. (English) Zbl 1092.33017

[For part I cf. Commun. Math. Phys. 206, No. 3, 639–690 (1999; Zbl 0944.33014).]
The quantum dynamics that lead to \(_2F_1\)-eigenfunctions may be viewed as the center-of-mass two-particle Hamiltonians of suitable Calogero-Moser-Sutherland \(N\)-particle systems. These systems admit a relativistic generalization in terms of an analytic difference equation which can be written as \(\psi (z-i)+V_a(z)\psi (z+i)+V_b(z)\psi (z)=E\psi (z)\). The case of hyperbolic coefficients is associated with a function \(R(a_+,a_-,{\mathbf c};v,\hat v)\) that generalizes the hypergeometric function, defined by the author via a contour integral involving a hyperbolic gamma function.
In this paper the author studies the symmetry properties of a similarity-transformed function \(\mathcal{E}(a_+,a_-,\gamma ;v,\hat v)\), with parameters \(\gamma \in \mathbb{C}^4\) related to \({\mathbf c}\in \mathbb{C}^4\) by a shift depending on \(a_+\) and \(a_-\). It is shown that the \(\mathcal{E}\)-function is invariant under all maps \(\gamma \mapsto w(\gamma )\), with \(w\) in the Weyl group of type \(D_4\). Choosing \(a_+\), \(a_-\) positive and \(\gamma \), \(\hat v\) real, the author obtains detailed information on the \(| Re\, v| \rightarrow \infty \) asymptotics of the \(\mathcal{E}\)-function.

MSC:

33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
33Cxx Hypergeometric functions
33D60 Basic hypergeometric integrals and functions defined by them
39A13 Difference equations, scaling (\(q\)-differences)

Citations:

Zbl 0944.33014
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