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relations to Lamé functions

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1: 31.7 Relations to Other Functions
§31.7(ii) Relations to Lamé Functions
2: 29.12 Definitions
§29.12(i) Elliptic-Function Form
There are eight types of Lamé polynomials, defined as follows: …
3: 31.8 Solutions via Quadratures
For m = ( m 0 , 0 , 0 , 0 ) , these solutions reduce to Hermite’s solutions (Whittaker and Watson (1927, §23.7)) of the Lamé equation in its algebraic form. …
4: 29.6 Fourier Series
In addition, if H satisfies (29.6.2), then (29.6.3) applies. … Consequently, Ec ν 2 m ( z , k 2 ) reduces to a Lamé polynomial; compare §§29.12(i) and 29.15(i). …
5: 29.1 Special Notation
The relation to the Lamé functions L c ν ( m ) , L s ν ( m ) of Jansen (1977) is given by …The relation to the Lamé functions Ec ν m , Es ν m of Ince (1940b) is given by …
6: 29.2 Differential Equations
For the Weierstrass function see §23.2(ii). …
7: 14 Legendre and Related Functions
Chapter 14 Legendre and Related Functions
8: Bibliography
  • V. S. Adamchik and H. M. Srivastava (1998) Some series of the zeta and related functions. Analysis (Munich) 18 (2), pp. 131–144.
  • T. M. Apostol and T. H. Vu (1984) Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.
  • F. M. Arscott and I. M. Khabaza (1962) Tables of Lamé Polynomials. Pergamon Press, The Macmillan Co., New York.
  • F. M. Arscott (1964a) Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.
  • F. M. Arscott (1964b) Periodic Differential Equations. An Introduction to Mathieu, Lamé, and Allied Functions. International Series of Monographs in Pure and Applied Mathematics, Vol. 66, Pergamon Press, The Macmillan Co., New York.
  • 9: 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

  • Source citations

    Specific source citations and proof metadata are now given for all equations in Chapter 25 Zeta and Related Functions.

  • 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).

  • Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9)

    The Kronecker delta symbols have been moved furthest to the right, as is common convention for orthogonality relations.

  • 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.

  • 10: 10 Bessel Functions
    Chapter 10 Bessel Functions