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)

Zbl 0944.33014
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