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expansions in series of Ferrers functions

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1: 30.8 Expansions in Series of Ferrers Functions
§30.8 Expansions in Series of Ferrers Functions
30.8.6 a n , k m ( γ 2 ) = ( n m ) ! ( n + m + 2 k ) ! ( n + m ) ! ( n m + 2 k ) ! a n , k m ( γ 2 ) .
2: 30.17 Tables
§30.17 Tables
3: 14.18 Sums
§14.18 Sums
For expansions of arbitrary functions in series of Legendre polynomials see §18.18(i), and for expansions of arbitrary functions in series of associated Legendre functions see Schäfke (1961b). … The formulas are also valid with the Ferrers functions as in §14.3(i) with μ = 0 . …
4: 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 relations (§14.10) provide a powerful method when applied in a stable direction (§3.6); see Olver and Smith (1983) and Gautschi (1967). …
  • Application of the uniform asymptotic expansions for large values of the parameters given in §§14.15 and 14.20(vii)14.20(ix).

  • 5: Bibliography V
  • H. Van de Vel (1969) On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23 (105), pp. 61–69.
  • R. S. Varma (1941) An infinite series of Weber’s parabolic cylinder functions. Proc. Benares Math. Soc. (N.S.) 3, pp. 37.
  • A. N. Vavreck and W. Thompson (1984) Some novel infinite series of spherical Bessel functions. Quart. Appl. Math. 42 (3), pp. 321–324.
  • H. Volkmer (1999) Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.
  • H. Volkmer (2021) Fourier series representation of Ferrers function 𝖯 .
  • 6: 14.15 Uniform Asymptotic Approximations
    §14.15 Uniform Asymptotic Approximations
    In other words, the convergent hypergeometric series expansions of 𝖯 ν μ ( ± x ) are also generalized (and uniform) asymptotic expansions as μ , with scale 1 / Γ ( j + 1 + μ ) , j = 0 , 1 , 2 , ; compare §2.1(v). … For asymptotic expansions and explicit error bounds, see Dunster (2003b). … For convergent series expansions see Dunster (2004). …
    7: 30.4 Functions of the First Kind
    §30.4(i) Definitions
    If γ = 0 , 𝖯𝗌 n m ( x , 0 ) reduces to the Ferrers function 𝖯 n m ( x ) : …
    §30.4(iii) Power-Series Expansion
    The expansion (30.4.7) converges in the norm of L 2 ( 1 , 1 ) , that is, …It is also equiconvergent with its expansion in Ferrers functions (as in (30.4.2)), that is, the difference of corresponding partial sums converges to 0 uniformly for 1 x 1 . …
    8: 14.13 Trigonometric Expansions
    §14.13 Trigonometric Expansions
    These Fourier series converge absolutely when μ < 0 . … In particular, …
    14.13.4 𝖰 n ( cos θ ) = 2 2 n + 1 ( n ! ) 2 ( 2 n + 1 ) ! k = 0 1 3 ( 2 k 1 ) k ! ( n + 1 ) ( n + 2 ) ( n + k ) ( 2 n + 3 ) ( 2 n + 5 ) ( 2 n + 2 k + 1 ) cos ( ( n + 2 k + 1 ) θ ) ,
    For other trigonometric expansions see Erdélyi et al. (1953a, pp. 146–147).
    9: Bibliography C
  • T. M. Cherry (1948) Expansions in terms of parabolic cylinder functions. Proc. Edinburgh Math. Soc. (2) 8, pp. 50–65.
  • 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.
  • J. P. Coleman and A. J. Monaghan (1983) Chebyshev expansions for the Bessel function J n ( z ) in the complex plane. Math. Comp. 40 (161), pp. 343–366.
  • R. M. Corless, D. J. Jeffrey, and D. E. Knuth (1997) A sequence of series for the Lambert W function. In Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (Kihei, HI), pp. 197–204.
  • 10: Errata
  • Expansion

    §4.13 has been enlarged. The Lambert W -function is multi-valued and we use the notation W k ( x ) , k , for the branches. The original two solutions are identified via Wp ( x ) = W 0 ( x ) and Wm ( x ) = W ± 1 ( x 0 i ) .

    Other changes are the introduction of the Wright ω -function and tree T -function in (4.13.1_2) and (4.13.1_3), simplification formulas (4.13.3_1) and (4.13.3_2), explicit representation (4.13.4_1) for d n W d z n , additional Maclaurin series (4.13.5_1) and (4.13.5_2), an explicit expansion about the branch point at z = e 1 in (4.13.9_1), extending the number of terms in asymptotic expansions (4.13.10) and (4.13.11), and including several integrals and integral representations for Lambert W -functions in the end of the section.

  • 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.