relations to confluent hypergeometric functions

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21: 16.18 Special Cases

§16.18 Special Cases

►The ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ functions introduced in Chapters 13 and 15, as well as the more general ${}_p{}^{}F_q^{}$ 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 ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ 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. …

22: 13.27 Mathematical Applications

§13.27 Mathematical Applications

►Confluent hypergeometric functions are connected with representations of the group of third-order triangular matrices. …The other group elements correspond to integral operators whose kernels can be expressed in terms of Whittaker functions. This identification can be used to obtain various properties of the Whittaker functions, including recurrence relations and derivatives. …

23: 13.29 Methods of Computation

… ►The recurrence relations in §§13.3(i) and 13.15(i) can be used to compute the confluent hypergeometric functions in an efficient way. …

24: 33.16 Connection Formulas

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§33.16(iii) $f$ and $h$ in Terms of $W_{\kappa ,\mu }\left(z\right)$ when

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§33.16(v) $s$ and $c$ in Terms of $W_{\kappa ,\mu }\left(z\right)$ when

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25: 31.12 Confluent Forms of Heun’s Equation

… ►Confluent forms of Heun’s differential equation (31.2.1) arise when two or more of the regular singularities merge to form an irregular singularity. This is analogous to the derivation of the confluent hypergeometric equation from the hypergeometric equation in §13.2(i). … ►

Confluent Heun Equation

… ►This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\mathrm{\infty }$. ►Mathieu functions (Chapter 28), spheroidal wave functions (Chapter 30), and Coulomb spheroidal functions (§30.12) are special cases of solutions of the confluent Heun equation. …

26: 13.14 Definitions and Basic Properties

… ►Standard solutions are: … ►In general $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$ are many-valued functions of $z$ with branch points at $z=0$ and $z=\mathrm{\infty }$. … ►Although $M_{\kappa ,\mu }\left(z\right)$ does not exist when $2\mu =-1,-2,-3,\mathrm{\dots }$, many formulas containing $M_{\kappa ,\mu }\left(z\right)$ continue to apply in their limiting form. … ►Except when $z=0$, each branch of the functions $M_{\kappa ,\mu }\left(z\right)/\mathrm{\Gamma }\left(2\mu +1\right)$ and $W_{\kappa ,\mu }\left(z\right)$ is entire in $\kappa$ and $\mu$. Also, unless specified otherwise $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$ are assumed to have their principal values. …

27: 13.3 Recurrence Relations and Derivatives

§13.3 Recurrence Relations and Derivatives

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§13.3(i) Recurrence Relations

… ►Kummer’s differential equation (13.2.1) is equivalent to … ►

§13.3(ii) Differentiation Formulas

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13.3.22 $$\frac{d}{dz}U(a,b,z)=-aU(a+1,b+1,z),$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $z$: complex variable

A&S Ref:

13.4.21

Permalink:

http://dlmf.nist.gov/13.3.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §13.3(ii), §13.3 and Ch.13

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28: 13.9 Zeros

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§13.9(i) Zeros of $M(a,b,z)$

… ►For fixed $a$ and $z$ in $ℂ$ the function $M(a,b,z)$ has only a finite number of $b$-zeros. ►

§13.9(ii) Zeros of $U(a,b,z)$

… ►Inequalities for the zeros of $U(a,b,x)$ are given in Gatteschi (1990). … ►For fixed $a$ and $z$ in $ℂ$, $U(a,b,z)$ has two infinite strings of $b$-zeros that are asymptotic to the imaginary axis as $|b|\to \mathrm{\infty }$.

29: Bibliography G

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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

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New inequalities for the zeros of confluent hypergeometric functions. In Asymptotic and computational analysis (Winnipeg, MB, 1989), pp. 175–192.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§13.9(i), §13.9(ii).

Encodings:

BibTeX

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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

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Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.

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External Links:

MathReview Entry, ZentralBlatt

Referenced by:

§3.8(vi).

Encodings:

BibTeX

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Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions. J. Comput. Appl. Math. 139 (1), pp. 173–187.

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External Links:

ISSN 0377-0427, Document, MathReview (Hasan Taşeli), ZentralBlatt

Referenced by:

§13.29(iii), §15.19(iii).

Encodings:

BibTeX

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30: Bibliography N

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Algorithm 707: CONHYP: A numerical evaluator of the confluent hypergeometric function for complex arguments of large magnitudes. ACM Trans. Math. Software 18 (3), pp. 345–349.

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Notes:

Double-precision Fortran, minimum accuracy 9S, maximum accuracy 13S.

External Links:

ISSN 0098-3500, Document, GAMS (module 707; package TOMS), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Numerical evaluation of the confluent hypergeometric function for complex arguments of large magnitudes. J. Comput. Appl. Math. 39 (2), pp. 193–200.

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Notes:

Extended-precision Fortran evaluator, minimum accuracy 9S, maximum accuracy 13S.

External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§13.32(iii).

Encodings:

BibTeX

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Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview, ZentralBlatt

Referenced by:

§28.1, §28.26(i), 6th item, Notations S, Notations S.

Encodings:

BibTeX

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Confluent hypergeometric equations and related solvable potentials in quantum mechanics. J. Math. Phys. 41 (12), pp. 7964–7996.

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External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§13.28(i).

Encodings:

BibTeX

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Hypergeometric functions. Acta Math. 94, pp. 289–349.

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External Links:

Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§16.10, §16.8(ii), §16.8(ii).

Encodings:

BibTeX

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