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Legendre relation for the hypergeometric function

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11: 16.18 Special Cases
§16.18 Special Cases
β–ΊThe 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. …
12: 19.5 Maclaurin and Related Expansions
§19.5 Maclaurin and Related Expansions
β–Ίwhere F 1 2 is the Gauss hypergeometric function (§§15.1 and 15.2(i)). …where F 1 ⁑ ( Ξ± ; Ξ² , Ξ² ; Ξ³ ; x , y ) is an Appell function16.13). … β–Ί
13: 14.32 Methods of Computation
§14.32 Methods of Computation
β–ΊEssentially 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 ΞΌ and Ξ½ 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). … β–Ί
  • For the computation of conical functions see Gil et al. (2009, 2012), and Dunster (2014).

  • 14: Bille C. Carlson
    β–ΊIn his paper Lauricella’s hypergeometric function F D (1963), he defined the R -function, a multivariate hypergeometric function that is homogeneous in its variables, each variable being paired with a parameter. …This symmetry led to the development of symmetric elliptic integrals, which are free from the transformations of modulus and amplitude that complicate the Legendre theory. … β–ΊIn Symmetry in c, d, n of Jacobian elliptic functions (2004) he found a previously hidden symmetry in relations between Jacobian elliptic functions, which can now take a form that remains valid when the letters c, d, and n are permuted. This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions. In Permutation symmetry for theta functions (2011) he found an analogous hidden symmetry between theta functions. …
    15: Bibliography V
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  • G. Valent (1986) An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.
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  • R. VidΕ«nas (2005) Transformations of some Gauss hypergeometric functions. J. Comput. Appl. Math. 178 (1-2), pp. 473–487.
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  • N. Virchenko and I. Fedotova (2001) Generalized Associated Legendre Functions and their Applications. World Scientific Publishing Co. Inc., Singapore.
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  • H. Volkmer and J. J. Wood (2014) A note on the asymptotic expansion of generalized hypergeometric functions. Anal. Appl. (Singap.) 12 (1), pp. 107–115.
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  • H. Volkmer (1982) Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.
  • 16: Bibliography C
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  • B. C. Carlson (2006b) Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric R -functions. Math. Comp. 75 (255), pp. 1309–1318.
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  • W. J. Cody (1991) Performance evaluation of programs related to the real gamma function. ACM Trans. Math. Software 17 (1), pp. 46–54.
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  • M. W. Coffey (2009) An efficient algorithm for the Hurwitz zeta and related functions. J. Comput. Appl. Math. 225 (2), pp. 338–346.
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  • H. S. Cohl and R. S. Costas-Santos (2020) Multi-Integral Representations for Associated Legendre and Ferrers Functions. Symmetry 12 (10).
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  • S. Conde and S. L. Kalla (1981) On zeros of the hypergeometric function. Serdica 7 (3), pp. 243–249.
  • 17: Bibliography S
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  • 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.
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  • B. D. Sleeman (1966b) The expansion of Lamé functions into series of associated Legendre functions of the second kind. Proc. Cambridge Philos. Soc. 62, pp. 441–452.
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  • B. D. Sleeman (1968a) Integral equations and relations for Lamé functions and ellipsoidal wave functions. Proc. Cambridge Philos. Soc. 64, pp. 113–126.
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  • F. C. Smith (1939b) Relations among the fundamental solutions of the generalized hypergeometric equation when p = q + 1 . II. Logarithmic cases. Bull. Amer. Math. Soc. 45 (12), pp. 927–935.
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  • C. Snow (1952) Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory. National Bureau of Standards Applied Mathematics Series, No. 19, U. S. Government Printing Office, Washington, D.C..
  • 18: 14.19 Toroidal (or Ring) Functions
    β–Ί
    §14.19(i) Introduction
    β–ΊThis form of the differential equation arises when Laplace’s equation is transformed into toroidal coordinates ( Ξ· , ΞΈ , Ο• ) , which are related to Cartesian coordinates ( x , y , z ) by …Most required properties of toroidal functions come directly from the results for P Ξ½ ΞΌ ⁑ ( x ) and 𝑸 Ξ½ ΞΌ ⁑ ( x ) . … β–Ί
    §14.19(ii) Hypergeometric Representations
    β–ΊWith 𝐅 as in §14.3 and ΞΎ > 0 , …
    19: 18.5 Explicit Representations
    β–ΊRelated formula: … β–Ί
    §18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions
    β–Ί
    Laguerre
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    Hermite
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    Legendre
    20: Frank W. J. Olver
    β–Ί, Bessel functions, hypergeometric functions, Legendre functions). … β–ΊMost notably, he served as the Editor-in-Chief and Mathematics Editor of the online NIST Digital Library of Mathematical Functions and its 966-page print companion, the NIST Handbook of Mathematical Functions (Cambridge University Press, 2010). … β–Ί
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