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relation to associated Legendre functions

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11: 14.32 Methods of Computation
§14.32 Methods of Computation
Essentially the same comments that are made in §15.19 concerning the computation of hypergeometric functions apply to the functions described in the present chapter. In particular, for small or moderate values of the parameters μ and ν the power-series expansions of the various hypergeometric function representations given in §§14.3(i)14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real. In other cases recurrence relations14.10) provide a powerful method when applied in a stable direction (§3.6); see Olver and Smith (1983) and Gautschi (1967). …
  • For the computation of conical functions see Gil et al. (2009, 2012), and Dunster (2014).

  • 12: 14.31 Other Applications
    §14.31(ii) Conical Functions
    §14.31(iii) Miscellaneous
    Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). … Legendre functions P ν ( x ) of complex degree ν appear in the application of complex angular momentum techniques to atomic and molecular scattering (Connor and Mackay (1979)).
    13: 10 Bessel Functions
    Chapter 10 Bessel Functions
    14: Errata
  • Chapters 10 Bessel Functions, 18 Orthogonal Polynomials, 34 3j, 6j, 9j Symbols

    The Legendre polynomial P n was mistakenly identified as the associated Legendre function P n in §§10.54, 10.59, 10.60, 18.18, 18.41, 34.3 (and was thus also affected by the bug reported below). These symbols now link correctly to their definitions. Reported by Roy Hughes on 2022-05-23

  • 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

  • Chapters 14 Legendre and Related Functions, 15 Hypergeometric Function

    The Gegenbauer function C α ( λ ) ( z ) , was labeled inadvertently as the ultraspherical (Gegenbauer) polynomial C n ( λ ) ( z ) . In order to resolve this inconsistency, this function now links correctly to its definition. This change affects Gegenbauer functions which appear in §§14.3(iv), 15.9(iii).

  • Chapter 25 Zeta and Related Functions

    A number of additions and changes have been made to the metadata to reflect new and changed references as well as to how some equations have been derived.

  • Equation (14.2.7)

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

  • 15: 14.17 Integrals
    §14.17(iii) Orthogonality Properties
    Orthogonality relations for the associated Legendre functions of imaginary order are given in Bielski (2013).
    §14.17(iv) Definite Integrals of Products
    §14.17(v) Laplace Transforms
    §14.17(vi) Mellin Transforms
    16: Bibliography B
  • S. L. Belousov (1962) Tables of Normalized Associated Legendre Polynomials. Pergamon Press, The Macmillan Co., Oxford-New York.
  • S. Bielski (2013) Orthogonality relations for the associated Legendre functions of imaginary order. Integral Transforms Spec. Funct. 24 (4), pp. 331–337.
  • G. Blanch and D. S. Clemm (1962) Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind. U.S. Government Printing Office, Washington, D.C..
  • G. Blanch and D. S. Clemm (1965) Tables Relating to the Radial Mathieu Functions. Vol. 2: Functions of the Second Kind. U.S. Government Printing Office, Washington, D.C..
  • T. H. Boyer (1969) Concerning the zeros of some functions related to Bessel functions. J. Mathematical Phys. 10 (9), pp. 1729–1744.
  • 17: 14.19 Toroidal (or Ring) Functions
    §14.19 Toroidal (or Ring) Functions
    §14.19(i) Introduction
    This form of the differential equation arises when Laplace’s equation is transformed into toroidal coordinates ( η , θ , ϕ ) , which are related to Cartesian coordinates ( x , y , z ) by …Most required properties of toroidal functions come directly from the results for P ν μ ( x ) and 𝑸 ν μ ( x ) . …
    §14.19(v) Whipple’s Formula for Toroidal Functions
    18: Bibliography S
  • B. I. Schneider, J. Segura, A. Gil, X. Guan, and K. Bartschat (2010) A new Fortran 90 program to compute regular and irregular associated Legendre functions. Comput. Phys. Comm. 181 (12), pp. 2091–2097.
  • J. Segura and A. Gil (1999) Evaluation of associated Legendre functions off the cut and parabolic cylinder functions. Electron. Trans. Numer. Anal. 9, pp. 137–146.
  • R. Szmytkowski (2009) On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 46 (1), pp. 231–260.
  • R. Szmytkowski (2011) On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 49 (7), pp. 1436–1477.
  • R. Szmytkowski (2012) On parameter derivatives of the associated Legendre function of the first kind (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Anal. Appl. 386 (1), pp. 332–342.
  • 19: Bibliography D
  • G. Delic (1979a) Chebyshev expansion of the associated Legendre polynomial P L M ( x ) . Comput. Phys. Comm. 18 (1), pp. 63–71.
  • C. F. Dunkl (1989) Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc. 311 (1), pp. 167–183.
  • T. M. Dunster (2003b) Uniform asymptotic expansions for associated Legendre functions of large order. Proc. Roy. Soc. Edinburgh Sect. A 133 (4), pp. 807–827.
  • T. M. Dunster (2004) Convergent expansions for solutions of linear ordinary differential equations having a simple pole, with an application to associated Legendre functions. Stud. Appl. Math. 113 (3), pp. 245–270.
  • P. L. Duren (1991) The Legendre Relation for Elliptic Integrals. In Paul Halmos: Celebrating 50 Years of Mathematics, J. H. Ewing and F. W. Gehring (Eds.), pp. 305–315.
  • 20: 14.10 Recurrence Relations and Derivatives
    §14.10 Recurrence Relations and Derivatives
    𝖰 ν μ ( x ) also satisfies (14.10.1)–(14.10.5).
    14.10.6 P ν μ + 2 ( x ) + 2 ( μ + 1 ) x ( x 2 1 ) 1 / 2 P ν μ + 1 ( x ) ( ν μ ) ( ν + μ + 1 ) P ν μ ( x ) = 0 ,
    Q ν μ ( x ) also satisfies (14.10.6) and (14.10.7). In addition, P ν μ ( x ) and Q ν μ ( x ) satisfy (14.10.3)–(14.10.5).