DLMF: Untitled Document

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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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31.11.1

w(z)=\sum_{j=0}^{\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

P_{j}=P\begin{Bmatrix}0&1&\infty&\\ 0&0&\lambda+j&z\\ 1-\gamma&1-\delta&\mu-j&\end{Bmatrix},

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Defines:: $P_{j}$ : General solution of generalized hypergeometric differential equation (locally)
Symbols:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : 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}}\left({\lambda% +j,1-\gamma+\lambda+j\atop 1+\lambda-\mu+2j};\frac{1}{z}\right),

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Defines:: $P_{j}^{5}$ : first Kummer solution at $\infty$ of generalized hypergeometric differential equation (locally)
Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : 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}}\left({\mu-j,1-\gamma+% \mu-j\atop 1-\lambda+\mu-2j};\frac{1}{z}\right).

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Defines:: $P_{j}^{6}$ : second Kummer solution at $\infty$ of generalized hypergeometric differential equation (locally)
Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : 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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15.11.3

w=P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.

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Defines:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : 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

w=P\begin{Bmatrix}0&1&\infty&\\ 0&0&a&z\\ 1-c&c-a-b&b&\end{Bmatrix}

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Symbols:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : 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

P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}=P\begin{Bmatrix}\widetilde{\alpha}&\widetilde{% \beta}&\widetilde{\gamma}&\\ a_{1}&b_{1}&c_{1}&t\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.

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Symbols:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : 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

z^{\lambda}(1-z)^{\mu}P\begin{Bmatrix}0&1&\infty&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}=P\begin{Bmatrix}0&1&\infty&\\ a_{1}+\lambda&b_{1}+\mu&c_{1}-\lambda-\mu&z\\ a_{2}+\lambda&b_{2}+\mu&c_{2}-\lambda-\mu&\end{Bmatrix},

ⓘ

Symbols:: $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : 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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§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

…

… ►

31.10.9

\mathcal{K}(\theta,\phi)=P\begin{Bmatrix}0&1&\infty&\\[1.0pt] 0&\frac{1}{2}-\delta-\sigma&\alpha&{\cos}^{2}\theta\\[1.0pt] 1-\gamma&\frac{1}{2}-\epsilon+\sigma&\beta&\end{Bmatrix}\*P\begin{Bmatrix}0&1&% \infty&\\[1.0pt] 0&0&-\frac{1}{2}+\delta+\sigma&{\cos}^{2}\phi\\[1.0pt] 1-\epsilon&1-\delta&-\frac{1}{2}+\epsilon-\sigma&\end{Bmatrix},

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31.10.11

\mathcal{K}(z,t)=(zt-a)^{\frac{1}{2}-\delta-\sigma}\left(\ifrac{zt}{a}\right)^% {-\frac{1}{2}+\delta+\sigma-\alpha}\*{{}_{2}F_{1}}\left({\frac{1}{2}-\delta-% \sigma+\alpha,\frac{3}{2}-\delta-\sigma+\alpha-\gamma\atop\alpha-\beta+1};% \frac{a}{zt}\right)\*P\begin{Bmatrix}0&1&\infty&\\ 0&0&-\frac{1}{2}+\delta+\sigma&\dfrac{(z-a)(t-a)}{(1-a)(zt-a)}\\ 1-\epsilon&1-\delta&-\frac{1}{2}+\epsilon-\sigma&\end{Bmatrix}.

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Symbols:: ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ : $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$ : 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, $\epsilon$ : real or complex parameter, $a$ : complex parameter, $\alpha$ : real or complex parameter, $\beta$ : real or complex parameter, $\mathcal{K}(z,t)$ : kernel and $\sigma$ : separation constant
Permalink:: http://dlmf.nist.gov/31.10.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §31.10(i), §31.10(i), §31.10 and Ch.31

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31.10.22

\mathcal{K}(r,\theta,\phi)=r^{m}{\sin}^{2p}\theta P\begin{Bmatrix}0&1&\infty&% \\ 0&0&a&{\cos}^{2}\theta\\ \tfrac{1}{2}(3-\gamma)&c&b&\end{Bmatrix}\*P\begin{Bmatrix}0&1&\infty&\\ 0&0&a^{\prime}&{\cos}^{2}\phi\\ 1-\epsilon&1-\delta&b^{\prime}&\end{Bmatrix},

…

… ►Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and

q

-series. …

§16.25 Methods of Computation

►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. They are similar to those described for confluent hypergeometric functions, and hypergeometric functions in §§13.29 and 15.19. There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations). …

… ►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): …

… ►

§16.2(i) Generalized Hypergeometric Series

… ► … ►

Polynomials

… ►Note also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via … ►

§16.2(v) Behavior with Respect to Parameters

…

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C. G. Lambe and D. R. Ward (1934) Some differential equations and associated integral equations. Quart. J. Math. (Oxford) 5, pp. 81–97.

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External Links:: Document, ZentralBlatt
Referenced by:: §31.10(i), §31.9(i).
Encodings:: BibTeX

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E. W. Leaver (1986) Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics. J. Math. Phys. 27 (5), pp. 1238–1265.

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External Links:: ISSN 0022-2488, Document, MathReview (Basilis C. Xanthopoulos), ZentralBlatt
Referenced by:: §30.12, §31.17(ii).
Encodings:: BibTeX

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J. Letessier, G. Valent, and J. Wimp (1994) Some Differential Equations Satisfied by Hypergeometric Functions. In Approximation and Computation (West Lafayette, IN, 1993), Internat. Ser. Numer. Math., Vol. 119, pp. 371–381.

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External Links:: MathReview (M. E. Muldoon), ZentralBlatt
Referenced by:: §16.8(ii).
Encodings:: BibTeX

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N. A. Lukaševič (1971) The second Painlevé equation. Differ. Uravn. 7 (6), pp. 1124–1125 (Russian).

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Notes:: In Russian. English translation: Differential Equations 7(6), pp. 853–854
External Links:: MathReview (Z. Rubinstein), ZentralBlatt
Referenced by:: §32.7(ii).
Encodings:: BibTeX

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Y. L. Luke and J. Wimp (1963) Jacobi polynomial expansions of a generalized hypergeometric function over a semi-infinite ray. Math. Comp. 17 (84), pp. 395–404.

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External Links:: FullText, MathReview (I. A. Stegun), ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

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generalized hypergeometric differential equation

1—10 of 55 matching pages

1: 16.8 Differential Equations

§16.8(ii) The Generalized Hypergeometric Differential Equation

§16.8(iii) Confluence of Singularities

2: 31.11 Expansions in Series of Hypergeometric Functions

3: 15.11 Riemann’s Differential Equation

4: 16.23 Mathematical Applications

§16.23 Mathematical Applications

§16.23(i) Differential Equations

§16.23(iv) Combinatorics and Number Theory

5: 31.10 Integral Equations and Representations

6: Howard S. Cohl

7: 16.25 Methods of Computation

§16.25 Methods of Computation

8: 16.13 Appell Functions

9: 16.2 Definition and Analytic Properties

§16.2(i) Generalized Hypergeometric Series

Polynomials

§16.2(v) Behavior with Respect to Parameters

10: Bibliography L