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31.14.1 .… ►The three sets of parameters comprise the singularity parameters , the exponent parameters , and the free accessory parameters . … ►
31.2.1 .… ►All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, , can be transformed into (31.2.1). ►The parameters play different roles: is the singularity parameter; are exponent parameters; is the accessory parameter. …
31.15.6 ,… ►If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index , where each is a nonnegative integer, there is a unique Stieltjes polynomial with zeros in the open interval for each . … ►
§31.13 Asymptotic Approximations►For asymptotic approximations for the accessory parameter eigenvalues , see Fedoryuk (1991) and Slavyanov (1996). ►For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). ►For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).
… ►For a coalescing saddle point and endpoint see Olver (1997b, Chapter 9) and Wong (1989, Chapter 7); if the endpoint is an algebraic singularity then the uniform approximants are parabolic cylinder functions with fixed parameter, and if the endpoint is not a singularity then the uniform approximants are complementary error functions. …
… ►Sometimes the parameters are suppressed.
… ►This equation has regular singularities at with exponents and an irregular singularity of rank 1 at (if ). The equation contains three real parameters , , and . … … ►
30.2.2… ►With Equation (30.2.1) changes to …
31.11.3… ►and (31.11.1) converges outside the ellipse in the -plane with foci at 0, 1, and passing through the third finite singularity at . ►Every Heun function (§31.4) can be represented by a series of Type I convergent in the whole plane cut along a line joining the two singularities of the Heun function. … ►The expansion (31.11.1) with (31.11.12) is convergent in the plane cut along the line joining the two singularities and . In this case the accessory parameter is a root of the continued-fraction equation …
§31.4 Solutions Analytic at Two Singularities: Heun Functions►For an infinite set of discrete values , , of the accessory parameter , the function is analytic at , and hence also throughout the disk . … ►
31.4.1 .►The eigenvalues satisfy the continued-fraction equation … ►