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Polynomial solutions of \(q\)-Heun equation and ultradiscrete limit. (English) Zbl 1420.39006

The Gauss hypergeometric function arises from the standard form of the second-order Fuchsian differential equation that has three regular singularities \([0,1,\infty]\). The standard form of the second-order Fuchsian differential equation that has four regular singularities \([0,1,t,\infty]\) is called Heun differential equation, see Equation (4) in the paper. The prescribed conditions in (4) imply that \(\alpha\) and \(\beta\) occurring in the differential equation are the local exponents about \(z=\infty\). It is known that the parameter \(B\), occurring in Equation (4) is independent on the local exponents, and it is called accessory parameter.
The \(q\)-Heun equation considered in this paper is given by Equation (6). It has the parameter \(E\) as accessory parameter and it gives the continuous Heun Equation (4) by considering the limit \(q\to 1\). This paper is devoted to the investigation of polynomial-type solutions of the \(q\)-Heun equation. The ultradiscrete limit is also considered.

MSC:

39A13 Difference equations, scaling (\(q\)-differences)
33C05 Classical hypergeometric functions, \({}_2F_1\)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
33D60 Basic hypergeometric integrals and functions defined by them
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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References:

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