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§31.5 Solutions Analytic at Three Singularities: Heun Polynomials… ►
31.5.2… ►These solutions are the Heun polynomials. …
§31.16(ii) Heun Polynomial Products►Expansions of Heun polynomial products in terms of Jacobi polynomial (§18.3) products are derived in Kalnins and Miller (1991a, b, 1993) from the viewpoint of interrelation between two bases in a Hilbert space: ►
… ►The main functions treated in this chapter are , , , and the polynomial . …Sometimes the parameters are suppressed.
… ►The right-hand side may be evaluated at any convenient value, or limiting value, of in since it is independent of . ►For corresponding orthogonality relations for Heun functions (§31.4) and Heun polynomials (§31.5), see Lambe and Ward (1934), Erdélyi (1944), Sleeman (1966a), and Ronveaux (1995, Part A, pp. 59–64). ►
§31.9(ii) Double Orthogonality►Heun polynomials , , satisfy …
… ►Series of Type II (§31.11(iv)) are expansions in orthogonal polynomials, which are useful in calculations of normalization integrals for Heun functions; see Erdélyi (1944) and §31.9(i). … ►The case for nonnegative integer corresponds to the Heun polynomial . … ►…
… ►For integral equations satisfied by the Heun polynomial we have , . ►For suitable choices of the branches of the -symbols in (31.10.9) and the contour , we can obtain both integral equations satisfied by Heun functions, as well as the integral representations of a distinct solution of Heun’s equation in terms of a Heun function (polynomial, path-multiplicative solution). …
7: Gerhard Wolf
… ►Wolf has published papers on Mathieu functions, orthogonal polynomials, and Heun functions. …
… ►Here is a polynomial of degree in and of degree in , that is a solution of the third-order differential equation satisfied by a product of any two solutions of Heun’s equation. … ►When approaches the ends of the gaps, the solution (31.8.2) becomes the corresponding Heun polynomial. …
Hypergeometric expansions of Heun polynomials.
SIAM J. Math. Anal. 22 (5), pp. 1450–1459.
Addendum: “Hypergeometric expansions of Heun polynomials”.
SIAM J. Math. Anal. 22 (6), pp. 1803.
Orthogonal Polynomials on -spheres: Gegenbauer, Jacobi and Heun.
In Topics in Polynomials of One and Several Variables and their
10: Bibliography P
A new basis for the representation of the rotation group. Lamé and Heun polynomials.
J. Mathematical Phys. 14 (8), pp. 1130–1139.