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31: 26.10 Integer Partitions: Other Restrictions
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denotes the number of partitions of into at most distinct parts.
…The set is denoted by .
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►It is known that for , , with strict inequality for sufficiently large, provided that , or ; see Yee (2004).
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►where is the modified Bessel function (§10.25(ii)), and
…The quantity is real-valued.
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32: 28.8 Asymptotic Expansions for Large
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►Also let and (§18.3).
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28.8.11
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►The approximations are expressed in terms of Whittaker functions and with ; compare §2.8(vi).
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►Subsequently the asymptotic solutions involving either elementary or Whittaker functions are identified in terms of the Floquet solutions (§28.12(ii)) and modified Mathieu functions (§28.20(iii)).
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33: 3.9 Acceleration of Convergence
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►A transformation of a convergent sequence with limit into a sequence is called limit-preserving if converges to the same limit .
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►This transformation is accelerating if is a linearly convergent
sequence, i.
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►Then the transformation of the sequence into a sequence is given by
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►Then .
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►We give a special form of Levin’s transformation in which the sequence of partial sums is transformed into:
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34: Bibliography S
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Recursive evaluation of - and - coefficients.
Comput. Phys. Comm. 11 (2), pp. 269–278.
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On integral representations for Lamé and other special functions.
SIAM J. Math. Anal. 11 (4), pp. 702–723.
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The Laplace transforms of products of Airy functions.
Dirāsāt Ser. B Pure Appl. Sci. 19 (2), pp. 7–11.
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A simple approach to asymptotic expansions for Fourier integrals of singular functions.
Appl. Math. Comput. 216 (11), pp. 3378–3385.
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Représentation asymptotique de la solution générale de l’équation de Mathieu-Hill.
Acad. Roy. Belg. Bull. Cl. Sci. (5) 51 (11), pp. 1415–1446.
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35: 21.1 Special Notation
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positive integers. | |
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th element of vector . | |
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Transpose of . | |
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set of all elements of the form “”. | |
set of all elements of , modulo elements of . Thus two elements of are equivalent if they are both in and their difference is in . (For an example see §20.12(ii).) | |
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36: Errata
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Chapters 1 Algebraic and Analytic Methods, 10 Bessel Functions, 14 Legendre and Related Functions, 18 Orthogonal Polynomials, 29 Lamé Functions
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Equation (34.7.4)
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Equation (5.17.5)
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Over the preceding two months, the subscript parameters of the Ferrers and Legendre functions, and the Laguerre polynomial, , were incorrectly displayed as superscripts. Reported by Roy Hughes on 2022-05-23
34.7.4
Originally the third symbol in the summation was written incorrectly as
Reported 2015-01-19 by Yan-Rui Liu.
Version 1.0.9 (August 29, 2014)
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5.17.5
Originally the term was incorrectly stated as .
Reported 2013-08-01 by Gergő Nemes and subsequently by Nick Jones on December 11, 2013.
Version 1.0.3 (Aug 29, 2011)
…37: Bibliography M
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The roots of
.
Quart. Appl. Math. 47 (2), pp. 375–378.
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Formulas and Theorems for the Special Functions of Mathematical Physics.
3rd edition, Springer-Verlag, New York-Berlin.
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Rational solutions of the Painlevé VI equation.
J. Phys. A 34 (11), pp. 2281–2294.
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Spheroidal eigenfunctions of the tidal equation.
Phys. Rev. Lett. 73 (11), pp. 1557–1560.
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Zeros of the function
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Differential Equations 11, pp. 797–811.
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38: 26.6 Other Lattice Path Numbers
39: 28.29 Definitions and Basic Properties
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►The basic solutions
, are defined in the same way as in §28.2(ii) (compare (28.2.5), (28.2.6)).
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►where the function is -periodic.
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►If
is a solution of (28.29.9), then , comprise a fundamental pair of solutions of Hill’s equation.
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►In the symmetric case
, is an even solution and is an odd solution; compare §28.2(ii).
…The -periodic or -antiperiodic solutions are multiples of , respectively.
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40: Bibliography C
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Note on Nörlund’s polynomial
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Proc. Amer. Math. Soc. 11 (3), pp. 452–455.
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The fourth Painlevé equation and associated special polynomials.
J. Math. Phys. 44 (11), pp. 5350–5374.
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Further formulas for calculating approximate values of the zeros of certain combinations of Bessel functions.
IEEE Trans. Microwave Theory Tech. 11 (6), pp. 546–547.
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Validated computation of certain hypergeometric functions.
ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
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Exact elliptic compactons in generalized Korteweg-de Vries equations.
Complexity 11 (6), pp. 30–34.
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