Legendre relation for the hypergeometric function

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21—30 of 34 matching pages

… ►All derivatives are denoted by differentials, not by primes. ►The first set of main functions treated in this chapter are Legendre’s complete integrals …of the first, second, and third kinds, respectively, and Legendre’s incomplete integrals … ►In Abramowitz and Stegun (1964, Chapter 17) the functions (19.1.1) and (19.1.2) are denoted, in order, by

K(\alpha)

E(\alpha)

\Pi(n\backslash\alpha)

F(\phi\backslash\alpha)

E(\phi\backslash\alpha)

, and

\Pi(n;\phi\backslash\alpha)

, where

\alpha=\operatorname{arcsin}k

and

n

is the

\alpha^{2}

(not related to

k

) in (19.1.1) and (19.1.2). … ►

R_{-a}\left(b_{1},b_{2},\dots,b_{n};z_{1},z_{2},\dots,z_{n}\right)

is a multivariate hypergeometric function that includes all the functions in (19.1.3). …

23: 19.21 Connection Formulas

… ►Legendre’s relation (19.7.1) can be written … ►The complete cases of

R_{F}

and

R_{G}

have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). … ►If

0 <mrow> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>z</mi> </mrow> </math> and <math class=

, then as

p\to 0

(19.21.6) reduces to Legendre’s relation (19.21.1). … ►Change-of-parameter relations can be used to shift the parameter

p

R_{J}

from either circular region to the other, or from either hyperbolic region to the other (§19.20(iii)). … ►

19.21.12

(p-x)R_{J}\left(x,y,z,p\right)+(q-x)R_{J}\left(x,y,z,q\right)=3R_{F}\left(x,y,% z\right)-3R_{C}\left(\xi,\eta\right),

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\xi$ and $\eta$
Referenced by:: §19.15, §19.20(iii), §19.21(iii), §19.25(i), §19.26(i)
Permalink:: http://dlmf.nist.gov/19.21.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §19.21(iii), §19.21 and Ch.19

…

24: Bibliography H

… ►

P. I. Hadži (1969) Certain integrals that contain a probability function and degenerate hypergeometric functions. Bul. Akad. S̆tiince RSS Moldoven 1969 (2), pp. 40–47 (Russian).

ⓘ

External Links:: MathReview (Wazir Hasan Abdi)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

►

P. I. Hadži (1970) Some integrals that contain a probability function and hypergeometric functions. Bul. Akad. Štiince RSS Moldoven 1970 (1), pp. 49–62 (Russian).

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External Links:: MathReview (R. G. Langebartel)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

… ►

P. I. Hadži (1978) Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).

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External Links:: MathReview (M. Lawrence Glasser)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

… ►

N. Hale and A. Townsend (2016) A fast FFT-based discrete Legendre transform. IMA J. Numer. Anal. 36 (4), pp. 1670–1684.

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External Links:: ISSN 0272-4979, Document, Link, MathReview (Denis N. Sidorov)
Referenced by:: §18.16(iii).
Encodings:: BibTeX

… ►

Y. P. Hsu (1993) Development of a Gaussian hypergeometric function code in complex domains. Internat. J. Modern Phys. C 4 (4), pp. 805–840.

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External Links:: ISSN 0129-1831, Document, MathReview (Walter Gautschi), ZentralBlatt
Referenced by:: §15.20(iii).
Encodings:: BibTeX

…

25: Bibliography R

… ►

D. St. P. Richards (Ed.) (1992) Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications. Contemporary Mathematics, Vol. 138, American Mathematical Society, Providence, RI.

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External Links:: ISBN 0-8218-5159-4, MathReview, ZentralBlatt
Referenced by:: Ch.35, Profile Donald St. P. Richards.
Encodings:: BibTeX

►

D. St. P. Richards (2004) Total positivity properties of generalized hypergeometric functions of matrix argument. J. Statist. Phys. 116 (1-4), pp. 907–922.

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External Links:: ISSN 0022-4715, Document, MathReview (Stamatis Koumandos), ZentralBlatt
Referenced by:: §35.9.
Encodings:: BibTeX

… ►

L. Robin (1957) Fonctions sphériques de Legendre et fonctions sphéroïdales. Tome I. Gauthier-Villars, Paris.

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Notes:: Préface de H. Villat
External Links:: MathReview (R. Campbell), ZentralBlatt
Referenced by:: Ch.14.
Encodings:: BibTeX

… ►

Hans-J. Runckel (1971) On the zeros of the hypergeometric function. Math. Ann. 191 (1), pp. 53–58.

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External Links:: ISSN 0025-5831, Document, MathReview (R. C. Varma), ZentralBlatt
Referenced by:: §15.13.
Encodings:: BibTeX

►

J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.

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External Links:: ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt
Referenced by:: 5th item.
Encodings:: BibTeX

…

26: Bibliography G

… ►

F. Gao and V. J. W. Guo (2013) Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.

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External Links:: ISSN 1023-6198, Document, FullText, MathReview (Thomas Britz), ZentralBlatt
Referenced by:: §17.7(iii).
Encodings:: BibTeX

… ►

W. Gautschi (1959b) Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38 (1), pp. 77–81.

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External Links:: MathReview (H. O. Pollak), ZentralBlatt
Referenced by:: §5.6(i), §6.8, §7.8, §8.10.
Encodings:: BibTeX

… ►

W. Gautschi (2002a) Computation of Bessel and Airy functions and of related Gaussian quadrature formulae. BIT 42 (1), pp. 110–118.

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External Links:: ISSN 0006-3835, Document, MathReview (Khélifa Trimèche), ZentralBlatt
Referenced by:: §10.74(iii), §9.17(iii).
Encodings:: BibTeX

… ►

A. Gervois and H. Navelet (1985a) Integrals of three Bessel functions and Legendre functions. I. J. Math. Phys. 26 (4), pp. 633–644.

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External Links:: ISSN 0022-2488, Document, MathReview (B. D. Agrawal), ZentralBlatt
Referenced by:: §10.22(iv), §10.43(iv).
Encodings:: BibTeX

►

A. Gervois and H. Navelet (1985b) Integrals of three Bessel functions and Legendre functions. II. J. Math. Phys. 26 (4), pp. 645–655.

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External Links:: ISSN 0022-2488, Document, MathReview (B. D. Agrawal), ZentralBlatt
Referenced by:: §10.22(iv), §10.43(iv).
Encodings:: BibTeX

…

27: 19.25 Relations to Other Functions

§19.25 Relations to Other Functions

►

§19.25(i) Legendre’s Integrals as Symmetric Integrals

… ►

§19.25(iii) Symmetric Integrals as Legendre’s Integrals

… ►

§19.25(vii) Hypergeometric Function

… ►

28: Bibliography

… ►

J. Abad and J. Sesma (1995) Computation of the regular confluent hypergeometric function. The Mathematica Journal 5 (4), pp. 74–76.

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External Links:: ISSN 1047-5974, FullText
Referenced by:: §13.32(iii).
Encodings:: BibTeX

… ►

V. S. Adamchik and H. M. Srivastava (1998) Some series of the zeta and related functions. Analysis (Munich) 18 (2), pp. 131–144.

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External Links:: ISSN 0174-4747, MathReview (Jack Quine), ZentralBlatt
Referenced by:: (25.11.37), (25.11.38), (25.11.39), (25.11.40), §25.11, §25.11(xi), (25.8.1), (25.8.4), §25.8.
Encodings:: BibTeX

… ►

G. E. Andrews (1974) Applications of basic hypergeometric functions. SIAM Rev. 16 (4), pp. 441–484.

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External Links:: Document, MathReview (L. J. Slater), ZentralBlatt
Referenced by:: §17.16, Ch.17.
Encodings:: BibTeX

… ►

T. M. Apostol and T. H. Vu (1984) Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.

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External Links:: ISSN 0022-314X, Document, MathReview (Kai Man Tsang), ZentralBlatt
Referenced by:: (25.16.10), (25.16.11), (25.16.12), (25.16.14), (25.16.15), (25.16.5), (25.16.6), (25.16.7), (25.16.8), (25.16.9), §25.16(ii), §25.16(ii), §25.16.
Encodings:: BibTeX

… ►

F. M. Arscott (1964a) Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.

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External Links:: Document, MathReview (I. Marx), ZentralBlatt
Referenced by:: §29.8.
Encodings:: BibTeX

…

29: Bibliography D

… ►

A. Deaño, J. Segura, and N. M. Temme (2010) Computational properties of three-term recurrence relations for Kummer functions. J. Comput. Appl. Math. 233 (6), pp. 1505–1510.

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External Links:: ISSN 0377-0427, Document, FullText, MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §13.29(iv), §13.29(iv).
Encodings:: BibTeX

… ►

K. Dilcher (2002) Bernoulli Numbers and Confluent Hypergeometric Functions. In Number Theory for the Millennium, I (Urbana, IL, 2000), pp. 343–363.

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External Links:: ISBN 1-56881-126-8, MathReview (Arnold M. Adelberg), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

… ►

T. M. Dunster (1999) Asymptotic approximations for the Jacobi and ultraspherical polynomials, and related functions. Methods Appl. Anal. 6 (3), pp. 21–56.

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External Links:: ISSN 1073-2772, Pdf, MathReview (Walter Van Assche), ZentralBlatt
Referenced by:: §15.12(iii), §18.15(i), §18.15(vi).
Encodings:: BibTeX

… ►

T. M. Dunster (2001c) Uniform asymptotic expansions for the reverse generalized Bessel polynomials, and related functions. SIAM J. Math. Anal. 32 (5), pp. 987–1013.

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External Links:: ISSN 0036-1410, Document, MathReview (Hasan Taşeli), ZentralBlatt
Referenced by:: §10.49(ii), §18.34(iii).
Encodings:: BibTeX

… ►

P. L. Duren (1991) The Legendre Relation for Elliptic Integrals. In Paul Halmos: Celebrating 50 Years of Mathematics, J. H. Ewing and F. W. Gehring (Eds.), pp. 305–315.

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External Links:: ISBN 0-387-97509-8, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §19.7(i).
Encodings:: BibTeX

…

30: 18.17 Integrals

… ►

Legendre

… ►

Legendre

… ►

Legendre

… ►

Legendre

… ►

Legendre

…

Legendre relation for the hypergeometric function

21—30 of 34 matching pages

21: 18.30 Associated OP’s

§18.30(i) Associated Jacobi Polynomials

§18.30(ii) Associated Legendre Polynomials

22: 19.1 Special Notation

23: 19.21 Connection Formulas

24: Bibliography H

25: Bibliography R

26: Bibliography G

27: 19.25 Relations to Other Functions

§19.25 Relations to Other Functions

§19.25(i) Legendre’s Integrals as Symmetric Integrals

§19.25(iii) Symmetric Integrals as Legendre’s Integrals

§19.25(vii) Hypergeometric Function

28: Bibliography

29: Bibliography D

30: 18.17 Integrals

Legendre

Legendre

Legendre

Legendre

Legendre