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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: Preface
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►Lozier directed the NIST research, technical, and support staff associated with the project, administered grants and contracts, together with Boisvert compiled the Software sections for the Web version of the chapters, conducted editorial and staff meetings, represented the project within NIST and at professional meetings in the United States and abroad, and together with Olver carried out the day-to-day development of the project.
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►Among the research, technical, and support staff at NIST these are B.
…Zelen.
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4: 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: 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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