hypergeometric differential equation

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1: 15.10 Hypergeometric Differential Equation

§15.10 Hypergeometric Differential Equation

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§15.10(i) Fundamental Solutions

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15.10.1 $$z(1-z)\frac{d^{2}w}{{dz}^2}+\left(c-(a+b+1)z\right)\frac{dw}{dz}-abw=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.11(i), §15.11(i), §15.17(ii), §15.19(ii), §15.2(ii), §15.5(ii), §16.13, §17.6(iv), §19.18(ii), §19.4(ii)

Permalink:

http://dlmf.nist.gov/15.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(i), §15.10 and Ch.15

►This is the hypergeometric differential equation. … ► …

2: 15.11 Riemann’s Differential Equation

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15.11.3

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Defines:

: Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation

Symbols:

$z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $\alpha$: point, $\beta$: point, $\gamma$: point and $w$: solution

Permalink:

http://dlmf.nist.gov/15.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.11(i), §15.11 and Ch.15

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15.11.4

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Symbols:

: Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $w$: solution

Permalink:

http://dlmf.nist.gov/15.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §15.11(i), §15.11 and Ch.15

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15.11.6

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Symbols:

: Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $\alpha$: point, $\beta$: point, $\gamma$: point and $t$: mapping

Permalink:

http://dlmf.nist.gov/15.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §15.11(ii), §15.11 and Ch.15

►The reduction of a general homogeneous linear differential equation of the second order with at most three regular singularities to the hypergeometric differential equation is given by … ►

15.11.8

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Symbols:

: Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $\lambda$: constant and $\mu$: constant

Permalink:

http://dlmf.nist.gov/15.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §15.11(ii), §15.11 and Ch.15

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3: 16.8 Differential Equations

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§16.8(ii) The Generalized Hypergeometric Differential Equation

… ►When no $b_j$ is an integer, and no two $b_j$ differ by an integer, a fundamental set of solutions of (16.8.3) is given by … ►We have the connection formula … ►

§16.8(iii) Confluence of Singularities

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4: 31.11 Expansions in Series of Hypergeometric Functions

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31.11.1 $$w(z)=\sum _{j=0}^{\mathrm{\infty }}{c}_j{P}_j,$$

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Symbols:

$z$: complex variable, $j$: nonnegative integer, $P_j$: General solution of generalized hypergeometric differential equation and $c_j$: coefficients

Referenced by:

§31.11(iii), §31.11(iii), §31.11(iii), §31.11(iii), Erratum (V1.1.7) for Subsection 31.11(iii)

Permalink:

http://dlmf.nist.gov/31.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

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31.11.2

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Defines:

$P_j$: General solution of generalized hypergeometric differential equation (locally)

Symbols:

: Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $j$: nonnegative integer, $\lambda$ and $\mu$

Referenced by:

§31.11(iii)

Permalink:

http://dlmf.nist.gov/31.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

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31.11.3_1 $$P_j^5=\frac{{\left(\lambda \right)}_j{\left(1-\gamma +\lambda \right)}_j}{{\left(1+\lambda -\mu \right)}_{2j}}{z}^{-\lambda -j}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\lambda +j,1-\gamma +\lambda +j}{1+\lambda -\mu +2j};\frac{1}{z}),$$

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Defines:

$P_j^5$: first Kummer solution at $\mathrm{\infty }$ of generalized hypergeometric differential equation (locally)

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $\lambda$ and $\mu$

Referenced by:

§31.11(ii), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/31.11.E3_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

Suggested 2022-09-14 by Hans Volkmer

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

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31.11.3_2 $$P_j^6=\frac{{\left(\lambda -\mu \right)}_{2j}}{{\left(1-\mu \right)}_j{\left(\gamma -\mu \right)}_j}{z}^{-\mu +j}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\mu -j,1-\gamma +\mu -j}{1-\lambda +\mu -2j};\frac{1}{z}).$$

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Defines:

$P_j^6$: second Kummer solution at $\mathrm{\infty }$ of generalized hypergeometric differential equation (locally)

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $\lambda$ and $\mu$

Referenced by:

§31.11(ii), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/31.11.E3_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

Suggested 2022-09-14 by Hans Volkmer

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

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31.11.9 $$M_{-1}{P}_{-1}=0.$$

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Symbols:

$P_j$: General solution of generalized hypergeometric differential equation and $M_j$: coefficients

Referenced by:

§31.11(iii)

Permalink:

http://dlmf.nist.gov/31.11.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

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5: 15.17 Mathematical Applications

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§15.17(i) Differential Equations

… ►The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … … ►The three singular points in Riemann’s differential equation (15.11.1) lead to an interesting Riemann sheet structure. …These monodromy groups are finite iff the solutions of Riemann’s differential equation are all algebraic. …

6: 15.19 Methods of Computation

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

7: 15.5 Derivatives and Contiguous Functions

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

8: 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): …

9: 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 $E\left(k\right)$. …

10: 16.23 Mathematical Applications

§16.23 Mathematical Applications

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§16.23(i) Differential Equations

… ►These equations are frequently solvable in terms of generalized hypergeometric functions, and the monodromy of generalized hypergeometric functions plays an important role in describing properties of the solutions. … … ►

§16.23(iv) Combinatorics and Number Theory

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