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Legendre relation for the hypergeometric function

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1: 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 ) .
2: 14.21 Definitions and Basic Properties
§14.21(iii) Properties
3: 15.9 Relations to Other Functions
Legendre
§15.9(iv) Associated Legendre Functions; Ferrers Functions
4: 15.16 Products
Generalized Legendre’s Relation
5: 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. …
    6: 14.5 Special Values
    §14.5 Special Values
    §14.5(v) μ = 0 , ν = ± 1 2
    7: 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
    8: 16.18 Special Cases
    §16.18 Special Cases
    The F 1 1 and F 1 2 functions introduced in Chapters 13 and 15, as well as the more general F q p functions introduced in the present chapter, are all special cases of the Meijer G -function. This is a consequence of the following relations: …As a corollary, special cases of the F 1 1 and F 1 2 functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer G -function. …
    9: 19.5 Maclaurin and Related Expansions
    §19.5 Maclaurin and Related Expansions
    where F 1 2 is the Gauss hypergeometric function (§§15.1 and 15.2(i)). …where F 1 ( α ; β , β ; γ ; x , y ) is an Appell function16.13). …
    10: 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).