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… ►The general second-order Fuchsian equation with regular singularities at , , and at , is given by ►
31.14.1 .►The exponents at the finite singularities are and those at are , where …With and the total number of free parameters is . … ►
… ►If is a zero of the Van Vleck polynomial , corresponding to an th degree Stieltjes polynomial , and are the zeros of (the derivative of ), then is either a zero of or a solution of the equation … ►The zeros , , of the Stieltjes polynomial are the critical points of the function , that is, points at which , , where … ►then there are exactly polynomials , each of which corresponds to each of the ways of distributing its zeros among intervals , . … ►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 . … ►Let and be Stieltjes polynomials corresponding to two distinct multi-indices and . …
… ► denotes the Stirling number of the first kind: times the number of permutations of with exactly cycles. … ►where is the Pochhammer symbol: . … ►For , … ►uniformly for . ►For asymptotic approximations for and that apply uniformly for as see Temme (1993) and Temme (2015, Chapter 34). …
… ►Miller (Bickley et al. (1952, pp. xvi–xvii)) that arbitrary “trial values” can be assigned to and , for example, and . … ►
Example 1. Bessel Functions… ►The Weber function satisfies … ►The values of for are the wanted values of . … ►For further information see Wimp (1984, Chapters 7–8), Cash and Zahar (1994), and Lozier (1980).
… ► is the number of paths from to that are composed of directed line segments of the form , , or . … ► is the number of lattice paths from to that stay on or above the line and are composed of directed line segments of the form , , or . … ► is the number of lattice paths from to that stay on or above the line , are composed of directed line segments of the form or , and for which there are exactly occurrences at which a segment of the form is followed by a segment of the form . … ► is the number of paths from to that stay on or above the diagonal and are composed of directed line segments of the form , , or . … ►