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1: 26.17 The Twelvefold Way
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2: 31.14 General Fuchsian Equation
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►The general second-order Fuchsian equation with regular singularities at , , and at , is given by
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31.14.1
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►The exponents at the finite singularities are and those at are , where
…With and the total number of free parameters is .
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31.14.3
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3: DLMF Project News
error generating summary4: 26.5 Lattice Paths: Catalan Numbers
5: 31.15 Stieltjes Polynomials
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►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
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►The zeros , , of the Stieltjes polynomial are the critical points of the function , that is, points at which , , where
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►then there are exactly
polynomials , each of which corresponds to each of the ways of distributing its zeros among intervals , .
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►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 .
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►Let and be Stieltjes polynomials corresponding to two distinct multi-indices and .
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6: 26.8 Set Partitions: Stirling Numbers
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denotes the Stirling number of the first kind: times the number of permutations of with exactly cycles.
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►where is the Pochhammer symbol: .
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►For ,
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►uniformly for .
►For asymptotic approximations for and that apply uniformly for as see Temme (1993) and Temme (2015, Chapter 34).
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7: 24.5 Recurrence Relations
8: Customize DLMF
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9: 3.6 Linear Difference Equations
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►Miller (Bickley et al. (1952, pp. xvi–xvii)) that arbitrary “trial values” can be assigned to and , for example, and .
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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).10: 26.6 Other Lattice Path Numbers
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is the number of paths from to that are composed of directed line segments of the form , , or .
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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 .
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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 .
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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 .
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26.6.10
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