relations to confluent hypergeometric functions and generalized hypergeometric functions

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11: 35.6 Confluent Hypergeometric Functions of Matrix Argument

§35.6 Confluent Hypergeometric Functions of Matrix Argument

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§35.6(i) Definitions

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Laguerre Form

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§35.6(ii) Properties

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§35.6(iii) Relations to Bessel Functions of Matrix Argument

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12: 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. … ►

Laguerre

… ►For the confluent hypergeometric functions $M(a,b,x)$ and $U(a,b,x)$, see §13.2(i), and for the Whittaker functions $M_{\kappa ,\mu }\left(x\right)$ and $W_{\kappa ,\mu }\left(x\right)$ see §13.14(i). ►

Hermite

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13: 8.19 Generalized Exponential Integral

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§8.19(vi) Relation to Confluent Hypergeometric Function

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14: 33.14 Definitions and Basic Properties

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§33.14(ii) Regular Solution $f(ϵ,\mathrm{\ell };r)$

… ►where $M_{\kappa ,\mu }\left(z\right)$ and $M(a,b,z)$ are defined in §§13.14(i) and 13.2(i), and … ►

§33.14(iii) Irregular Solution $h(ϵ,\mathrm{\ell };r)$

►For nonzero values of $ϵ$ and $r$ the function $h(ϵ,\mathrm{\ell };r)$ is defined by … ►

§33.14(v) Wronskians

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15: 13.31 Approximations

§13.31 Approximations

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§13.31(i) Chebyshev-Series Expansions

►Luke (1969b, pp. 35 and 25) provides Chebyshev-series expansions of $M(a,b,x)$ and $U(a,b,x)$ that include the intervals $0\le x\le \alpha$ and , respectively, where $\alpha$ is an arbitrary positive constant. … ►For a discussion of the convergence of the Padé approximants that are related to the continued fraction (13.5.1) see Wimp (1985). … ►

13.31.3 $$z^{a}U(a,1+a-b,z)=\underset{n\to \mathrm{\infty }}{lim}\frac{A_n(z)}{B_n(z)}.$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $n$: nonnegative integer, $z$: complex variable, $A_n(z)$: expansion and $B_n(z)$: expansion

Permalink:

http://dlmf.nist.gov/13.31.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.31(iii), §13.31 and Ch.13

16: 12.14 The Function $W(a,x)$

§12.14 The Function $W(a,x)$

… ►In other cases the general theory of (12.2.2) is available. … ►

§12.14(vii) Relations to Other Functions

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Bessel Functions

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Confluent Hypergeometric Functions

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17: 18.30 Associated OP’s

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§18.30(i) Associated Jacobi Polynomials

… ►where the generalized hypergeometric function ${}_4{}^{}F_3^{}$ is defined by (16.2.1). … ►For the confluent hypergeometric function $U$ see §13.2(i). … ►For Gauss’ hypergeometric function $F$ see (15.2.1). … ►More generally, the $k$th corecursive monic polynomials (defined with the initialization of (18.30.28) followed by the $c=k$ recurrence of (18.30.27)) are related to the $(k+1)$st monic associated polynomials by …

18: Bibliography W

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Asymptotic expansions of some matrix argument hypergeometric functions, with applications to macromolecules. Ann. Inst. Statist. Math. 45 (3), pp. 467–475.

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External Links:

ISSN 0020-3157, Document, MathReview (N. K. Thakare), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

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Some transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 26 (2), pp. 257–272.

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External Links:

Document, ZentralBlatt

Referenced by:

§16.6.

Encodings:

BibTeX

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Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials. Ph.D. Thesis, University of Wisconsin, Madison, WI.

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Referenced by:

§16.4(iii), §16.4(iii), §16.4(iii).

Encodings:

BibTeX

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On the zeros of a confluent hypergeometric function. Proc. Amer. Math. Soc. 16 (2), pp. 281–283.

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External Links:

FullText, Document, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§13.9(ii).

Encodings:

BibTeX

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The asymptotic expansion of the generalized hypergeometric function. Proc. London Math. Soc. (2) 46, pp. 389–408.

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External Links:

ISSN 0024-6115, Document, MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§16.11(ii).

Encodings:

BibTeX

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19: Bibliography S

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Confluent hypergeometric functions on tube domains. Math. Ann. 260 (3), pp. 269–302.

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External Links:

ISSN 0025-5831, Document, MathReview (A. B. Venkov), ZentralBlatt

Referenced by:

§35.6(ii).

Encodings:

BibTeX

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Uniform Asymptotic Expansions of Confluent Hypergeometric Functions and Whittaker Functions. Doctoral dissertation, University of Copenhagen, Vol. 1965, Jul. Gjellerups Forlag, Copenhagen.

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External Links:

MathReview (A. K. Agarwal), ZentralBlatt

Referenced by:

§13.21(iv).

Encodings:

BibTeX

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Generalized Hypergeometric Functions. Cambridge University Press, Cambridge.

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External Links:

ISBN 9780521090612, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

Ch.15, §16.2(iii), §16.4(ii), §16.4(iii), §16.4(v), Ch.16, §17.1, §17.4(i), Ch.17.

Encodings:

BibTeX

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Tables of Airy Functions and Special Confluent Hypergeometric Functions. Pergamon Press, New York.

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Notes:

Translated from the Russian by D. G. Fry

External Links:

MathReview (W. F. Freiberger), ZentralBlatt

Referenced by:

(9.13.19), (9.13.20), (9.13.21), §9.13(i), 1st item.

Encodings:

BibTeX

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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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External Links:

Document, MathReview (C. S. Meijer), ZentralBlatt

Referenced by:

§16.8(ii).

Encodings:

BibTeX

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20: Bibliography M

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Quadratic relations for confluent hypergeometric functions. Tohoku Math. J. (2) 52 (4), pp. 489–513.

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External Links:

ISSN 0040-8735, Document, MathReview (Takeshi Sasaki), ZentralBlatt

Referenced by:

§13.12.

Encodings:

BibTeX

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Euler-type transformations for the generalized hypergeometric function ${}_{r+2}{}^{}F_{r+1}^{}(x)$ . Z. Angew. Math. Phys. 62 (1), pp. 31–45.

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External Links:

ISSN 0044-2275, Document, FullText, MathReview (Mridula Garg), ZentralBlatt

Referenced by:

§16.6, §16.6.

Encodings:

BibTeX

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On a Kummer-type transformation for the generalized hypergeometric function ${}_2{}^{}F_2^{}$ . J. Comput. Appl. Math. 157 (2), pp. 507–509.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§16.6.

Encodings:

BibTeX

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A new symmetry related to $\mathrm{𝑆𝑈}(n)$ for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.

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External Links:

ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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A connection formula for the $q$-confluent hypergeometric function. SIGMA Symmetry Integrability Geom. Methods Appl. 9, pp. Paper 050, 13.

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External Links:

ISSN 1815-0659, Document, MathReview (Jeremy Lovejoy), ZentralBlatt

Referenced by:

§17.6(ii), §17.6(ii).

Encodings:

BibTeX

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