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Legendre relation for the hypergeometric function

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1: 15 Hypergeometric Function
Chapter 15 Hypergeometric Function
2: 14 Legendre and Related Functions
Chapter 14 Legendre and Related Functions
3: 14.3 Definitions and Hypergeometric Representations
§14.3 Definitions and Hypergeometric Representations
§14.3(ii) Interval 1 < x <
§14.3(iii) Alternative Hypergeometric Representations
14.3.14 w 2 ( ν , μ , x ) = 2 μ Γ ( 1 2 ν + 1 2 μ + 1 ) Γ ( 1 2 ν 1 2 μ + 1 2 ) x ( 1 x 2 ) μ / 2 𝐅 ( 1 2 1 2 ν 1 2 μ , 1 2 ν 1 2 μ + 1 ; 3 2 ; x 2 ) .
4: 14.21 Definitions and Basic Properties
§14.21(iii) Properties
5: 15.9 Relations to Other Functions
Legendre
§15.9(iv) Associated Legendre Functions; Ferrers Functions
6: 15.16 Products
Generalized Legendre’s Relation
7: Errata
  • 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).

  • 8: Software Index
    Open Source With Book Commercial
    14 Legendre and Related 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. …
    9: 14.5 Special Values
    §14.5 Special Values
    §14.5(v) μ = 0 , ν = ± 1 2
    10: 18.11 Relations to Other Functions
    §18.11 Relations to Other Functions
    See §§18.5(i) and 18.5(iii) for relations to trigonometric functions, the hypergeometric function, and generalized hypergeometric functions.
    Ultraspherical
    Laguerre
    Hermite