hypergeometric equation

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11: Bibliography S

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On the logarithmic solutions of the generalized hypergeometric equation when $p=q+1$ . Bull. Amer. Math. Soc. 45 (8), pp. 629–636.

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External Links:

Document, MathReview (H. Bateman), ZentralBlatt

Referenced by:

§16.8(ii).

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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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..

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External Links:

MathReview, ZentralBlatt

Referenced by:

§13.23(iv), Ch.14, §31.3(iii).

Encodings:

BibTeX

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12: 31.3 Basic Solutions

… ►The full set of 192 local solutions of (31.2.1), equivalent in 8 sets of 24, resembles Kummer’s set of 24 local solutions of the hypergeometric equation, which are equivalent in 4 sets of 6 solutions (§15.10(ii)); see Maier (2007).

13: 19.4 Derivatives and Differential Equations

… ►If $\varphi =\pi /2$, then these two equations become hypergeometric differential equations (15.10.1) for $K\left(k\right)$ and $E\left(k\right)$. …

14: 15.19 Methods of Computation

… ►A comprehensive and powerful approach is to integrate the hypergeometric differential equation (15.10.1) by direct numerical methods. …

15: 15.5 Derivatives and Contiguous Functions

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15.5.16_5 $$F(a,b;c;z)-F(a-1,b;c;z)-(b/c)zF(a,b+1;c+1;z)=0,$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.5(ii), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/15.5.E16_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §15.5(ii), §15.5 and Ch.15

… ►An equivalent equation to the hypergeometric differential equation (15.10.1) is …

16: 32.10 Special Function Solutions

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§32.10(vi) Sixth Painlevé Equation

… ►where the fundamental periods $2{\varphi }_1$ and $2{\varphi }_2$ are linearly independent functions satisfying the hypergeometric equation …

17: 16.13 Appell Functions

… ►The following four functions of two real or complex variables $x$ and $y$ cannot be expressed as a product of two ${}_2{}^{}F_1^{}$ functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1): …

18: 17.17 Physical Applications

… ►See Kassel (1995). …

19: 13.2 Definitions and Basic Properties

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13.2.1 $$z\frac{d^{2}w}{{dz}^2}+(b-z)\frac{dw}{dz}-aw=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $z$: complex variable

A&S Ref:

13.1.1

Referenced by:

§13.14(i), §13.14(v), §13.2(i), §13.2(v), §13.29(ii), §13.3(i), §13.3(i)

Permalink:

http://dlmf.nist.gov/13.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

… ►It can be regarded as the limiting form of the hypergeometric differential equation (§15.10(i)) that is obtained on replacing $z$ by $z/b$, letting $b\to \mathrm{\infty }$, and subsequently replacing the symbol $c$ by $b$. In effect, the regular singularities of the hypergeometric differential equation at $b$ and $\mathrm{\infty }$ coalesce into an irregular singularity at $\mathrm{\infty }$. … ►

13.2.7 $$U(-m,b,z)={(-1)}^m{\left(b\right)}_{m}M(-m,b,z)={(-1)}^m\sum _{s=0}^m\left(\genfrac{}{}{0.0pt}{}{m}{s}\right){\left(b+s\right)}_{m-s}{(-z)}^s.$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $m$: integer, $n$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Referenced by:

§13.2(i), Erratum (V1.0.10) for Equation (13.2.7)

Permalink:

http://dlmf.nist.gov/13.2.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.0.10):

The equality $U(-m,b,z)={(-1)}^m{\left(b\right)}_{m}M(-m,b,z)$ has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation. Note also that the notation $a=-n$ has been changed to $a=-m$.

See also:

Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

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13.2.8 $$U(a,a+n+1,z)=\frac{{(-1)}^n{\left(1-a-n\right)}_n}{z^{a+n}}M(-n,1-a-n,z)=z^{-a}\sum _{s=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{s}\right){\left(a\right)}_s{z}^{-s}.$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $n$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Referenced by:

§13.2(i), Erratum (V1.0.10) for Equation (13.2.8)

Permalink:

http://dlmf.nist.gov/13.2.E8

Encodings:

pMML, png, TeX

Addition (effective with 1.0.10):

The equality $U(a,a+n+1,z)=\frac{{(-1)}^n{\left(1-a-n\right)}_n}{z^{a+n}}M(-n,1-a-n,z)$ has been added to the original equation to express an explicit connection between the two standard solutions of Kummer’s equation.

Suggested 2014-02-10 by Adri Olde Daalhuis

See also:

Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

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20: Bibliography M

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On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.

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External Links:

ISSN 0022-0396, Document, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§31.7(i).

Encodings:

BibTeX

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