About the Project

Ferrers%20functions

AdvancedHelp

(0.002 seconds)

6 matching pages

1: 14.30 Spherical and Spheroidal Harmonics
§14.30 Spherical and Spheroidal Harmonics
Most mathematical properties of Y l , m ( θ , ϕ ) can be derived directly from (14.30.1) and the properties of the Ferrers function of the first kind given earlier in this chapter. …
Herglotz generating function
The following is the Herglotz generating function
2: 26.9 Integer Partitions: Restricted Number and Part Size
p k ( n ) denotes the number of partitions of n into at most k parts. … … A useful representation for a partition is the Ferrers graph in which the integers in the partition are each represented by a row of dots. … The conjugate partition is obtained by reflecting the Ferrers graph across the main diagonal or, equivalently, by representing each integer by a column of dots. …
§26.9(ii) Generating Functions
3: Bibliography V
  • C. G. van der Laan and N. M. Temme (1984) Calculation of Special Functions: The Gamma Function, the Exponential Integrals and Error-Like Functions. CWI Tract, Vol. 10, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam.
  • R. S. Varma (1941) An infinite series of Weber’s parabolic cylinder functions. Proc. Benares Math. Soc. (N.S.) 3, pp. 37.
  • R. Vidūnas (2005) Transformations of some Gauss hypergeometric functions. J. Comput. Appl. Math. 178 (1-2), pp. 473–487.
  • H. Volkmer (2004a) Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.
  • H. Volkmer (2021) Fourier series representation of Ferrers function 𝖯 .
  • 4: Software Index
    Open Source With Book Commercial
    20 Theta Functions
    ‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … In the list below we identify four main sources of software for computing special functions. …
  • Commercial Software.

    Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

  • The following are web-based software repositories with significant holdings in the area of special functions. …
    5: Bibliography C
  • R. Chelluri, L. B. Richmond, and N. M. Temme (2000) Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.
  • H. S. Cohl and R. S. Costas-Santos (2020) Multi-Integral Representations for Associated Legendre and Ferrers Functions. Symmetry 12 (10).
  • H. S. Cohl, J. Park, and H. Volkmer (2021) Gauss hypergeometric representations of the Ferrers function of the second kind. SIGMA Symmetry Integrability Geom. Methods Appl. 17, pp. Paper 053, 33.
  • M. Colman, A. Cuyt, and J. Van Deun (2011) Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
  • M. D. Cooper, R. H. Jeppesen, and M. B. Johnson (1979) Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.
  • 6: Errata
  • Chapters 1 Algebraic and Analytic Methods, 10 Bessel Functions, 14 Legendre and Related Functions, 18 Orthogonal Polynomials, 29 Lamé Functions

    Over the preceding two months, the subscript parameters of the Ferrers and Legendre functions, 𝖯 n , 𝖰 n , P n , Q n , 𝑸 n and the Laguerre polynomial, L n , were incorrectly displayed as superscripts. Reported by Roy Hughes on 2022-05-23

  • Equations (14.5.3), (14.5.4)

    The constraints in (14.5.3), (14.5.4) on ν + μ have been corrected to exclude all negative integers since the Ferrers function of the second kind is not defined for these values.

    Reported by Hans Volkmer on 2021-06-02

  • Equation (14.6.6)
    14.6.6 𝖯 ν m ( x ) = ( 1 x 2 ) m / 2 x 1 x 1 𝖯 ν ( x ) ( d x ) m

    The right-hand side has been corrected by replacing the Legendre function P ν ( x ) with the Ferrers function 𝖯 ν ( x ) .

  • Equation (14.2.7)

    The Wronskian was generalized to include both associated Legendre and Ferrers functions.

  • Chapters 8, 20, 36

    Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).