16 Generalized Hypergeometric Functions & Meijer G-Function Generalized Hypergeometric Functions 16.7 Relations to Other Functions 16.9 Zeros

§16.8 Differential Equations

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Keywords:: differential equations, of arbitrary order
Permalink:: http://dlmf.nist.gov/16.8
See also:: Annotations for Ch.16

§16.8(i) Classification of Singularities
§16.8(ii) The Generalized Hypergeometric Differential Equation
§16.8(iii) Confluence of Singularities

§16.8(i) Classification of Singularities

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Keywords:: classification of singularities, differential equations, generalized hypergeometric differential equation, of arbitrary order, ordinary point, singularities
Referenced by:: §16.21, §2.7(i)
Permalink:: http://dlmf.nist.gov/16.8.i
See also:: Annotations for §16.8 and Ch.16

An ordinary point of the differential equation

16.8.1 $\frac{{\mathrm{d}}^{n}w}{{\mathrm{d}z}^{n}}+f_{n-1}(z)\frac{{\mathrm{d}}^{n-1}% w}{{\mathrm{d}z}^{n-1}}+f_{n-2}(z)\frac{{\mathrm{d}}^{n-2}w}{{\mathrm{d}z}^{n-% 2}}+\dots+f_{1}(z)\frac{\mathrm{d}w}{\mathrm{d}z}+f_{0}(z)w=0$

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , : complex variable and $f_{j}(z)$ : coefficients
Referenced by:: §16.8(i)
Permalink:: http://dlmf.nist.gov/16.8.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §16.8(i), §16.8 and Ch.16

is a value $z_{0}$ of at which all the coefficients $f_{j}(z)$ , $j=0,1,\dots,n-1$ , are analytic. If $z_{0}$ is not an ordinary point but $(z-z_{0})^{n-j}f_{j}(z)$ , $j=0,1,\dots,n-1$ , are analytic at $z=z_{0}$ , then $z_{0}$ is a regular singularity. All other singularities are irregular. Compare §2.7(i) in the case . Similar definitions apply in the case $z_{0}=\infty$ : we transform $\infty$ into the origin by replacing in (16.8.1) by ; again compare §2.7(i).

For further information see Hille (1976, pp. 360–370).

§16.8(ii) The Generalized Hypergeometric Differential Equation

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Keywords:: connection formula, differential equation, fundamental solutions, generalized hypergeometric differential equation, generalized hypergeometric functions
Notes:: See Luke (1969a, §5.1). For (16.8.9) see Bühring (1988).
Referenced by:: §2.7(i)
Permalink:: http://dlmf.nist.gov/16.8.ii
See also:: Annotations for §16.8 and Ch.16

With the notation

16.8.2

$D=\frac{\mathrm{d}}{\mathrm{d}z},$

$\vartheta=z\frac{\mathrm{d}}{\mathrm{d}z},$

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Defines:: : differential operator (locally) and $\vartheta$ : differential operator (locally)
Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to and : complex variable
Permalink:: http://dlmf.nist.gov/16.8.E2
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §16.8(ii), §16.8 and Ch.16

the function $w={{}_{p}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)$ satisfies the differential equation

16.8.3 $\left(\vartheta(\vartheta+b_{1}-1)\cdots(\vartheta+b_{q}-1)-z(\vartheta+a_{1})% \cdots(\vartheta+a_{p})\right)w=0.$

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Symbols:: : nonnegative integer, : nonnegative integer, : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters, $\vartheta$ : differential operator and : solution
Referenced by:: §16.8(ii), §16.8(ii), §16.8(ii)
Permalink:: http://dlmf.nist.gov/16.8.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §16.8(ii), §16.8 and Ch.16

Equivalently,

16.8.4 $z^{q}D^{q+1}w+\sum_{j=1}^{q}z^{j-1}(\alpha_{j}z+\beta_{j})D^{j}w+\alpha_{0}w=0,$ $p\leq q$ ,

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Symbols:: : nonnegative integer, : nonnegative integer, : complex variable, : differential operator, : solution, $\alpha_{j}$ : constant and $\beta_{j}$ : constant
Referenced by:: §16.8(ii)
Permalink:: http://dlmf.nist.gov/16.8.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §16.8(ii), §16.8 and Ch.16

16.8.5 $z^{q}(1-z)D^{q+1}w+\sum_{j=1}^{q}z^{j-1}(\alpha_{j}z+\beta_{j})D^{j}w+\alpha_{% 0}w=0,$

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Symbols:: : nonnegative integer, : nonnegative integer, : complex variable, : differential operator, : solution, $\alpha_{j}$ : constant and $\beta_{j}$ : constant
Referenced by:: §16.8(ii)
Permalink:: http://dlmf.nist.gov/16.8.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §16.8(ii), §16.8 and Ch.16

where $\alpha_{j}$ and $\beta_{j}$ are constants. Equation (16.8.4) has a regular singularity at , and an irregular singularity at $z=\infty$ , whereas (16.8.5) has regular singularities at , 1, and $\infty$ . In each case there are no other singularities. Equation (16.8.3) is of order $\max(p,q+1)$ . In Letessier et al. (1994) examples are discussed in which the generalized hypergeometric function satisfies a differential equation that is of order 1 or even 2 less than might be expected.

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

16.8.6

$w_{0}(z)={{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right),$

${w_{j}(z)=z^{1-b_{j}}{{}_{p}F_{q}}\left({1+a_{1}-b_{j},\dots,1+a_{p}-b_{j}% \atop 2-b_{j},1+b_{1}-b_{j},\ldots*\dots,1+b_{q}-b_{j}};z\right),}$ $j=1,\dots,q$ ,

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, : nonnegative integer, : nonnegative integer, : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and : solution
Permalink:: http://dlmf.nist.gov/16.8.E6
Encodings:: TeX, TeX, pMML, pMML, png, png
See also:: Annotations for §16.8(ii), §16.8 and Ch.16

where indicates that the entry $1+b_{j}-b_{j}$ is omitted. For other values of the $b_{j}$ , series solutions in powers of (possibly involving also $\ln z$ ) can be constructed via a limiting process; compare §2.7(i) in the case of second-order differential equations. For details see Smith (1939a, b), and Nørlund (1955).

When , and no two $a_{j}$ differ by an integer, another fundamental set of solutions of (16.8.3) is given by

16.8.7 $\widetilde{w}_{j}(z)=(-z)^{-a_{j}}{{}_{q+1}F_{q}}\left({a_{j},1-b_{1}+a_{j},% \dots,1-b_{q}+a_{j}\atop 1-a_{1}+a_{j},\ldots*\dots,1-a_{q+1}+a_{j}};\frac{1}{% z}\right),$ $j=1,\dots,q+1$ ,

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, : nonnegative integer, : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and : solution
Permalink:: http://dlmf.nist.gov/16.8.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §16.8(ii), §16.8 and Ch.16

where indicates that the entry $1-a_{j}+a_{j}$ is omitted. We have the connection formula

16.8.8 ${{}_{q+1}F_{q}}\left({a_{1},\dots,a_{q+1}\atop b_{1},\dots,b_{q}};z\right)=% \sum_{j=1}^{q+1}\left({\textstyle\ifrac{\prod\limits_{\begin{subarray}{c}k=1\\ k\neq j\end{subarray}}^{q+1}\frac{\Gamma\left(a_{k}-a_{j}\right)}{\Gamma\left(% a_{k}\right)}}{\prod\limits_{k=1}^{q}\frac{\Gamma\left(b_{k}-a_{j}\right)}{% \Gamma\left(b_{k}\right)}}}\right)\widetilde{w}_{j}(z),$ $|\operatorname{ph}\left(-z\right)|\leq\pi$ .

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, : nonnegative integer, : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and : solution
Referenced by:: §16.11(ii)
Permalink:: http://dlmf.nist.gov/16.8.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §16.8(ii), §16.8 and Ch.16

More generally if $z_{0}$ ( $\in\mathbb{C}$ ) is an arbitrary constant, $|z-z_{0}|>\max{(|z_{0}|,|z_{0}-1|)}$ , and $|\operatorname{ph}\left(z_{0}-z\right)|<\pi$ , then

16.8.9 $\left({\textstyle\ifrac{\prod\limits_{k=1}^{q+1}\Gamma\left(a_{k}\right)}{% \prod\limits_{k=1}^{q}\Gamma\left(b_{k}\right)}}\right){{}_{q+1}F_{q}}\left({a% _{1},\dots,a_{q+1}\atop b_{1},\dots,b_{q}};z\right)=\sum_{j=1}^{q+1}\left(z_{0% }-z\right)^{-a_{j}}\sum_{n=0}^{\infty}\frac{\Gamma\left(a_{j}+n\right)}{n!}\*% \left({\textstyle\ifrac{\prod\limits_{\begin{subarray}{c}k=1\\ k\neq j\end{subarray}}^{q+1}\Gamma\left(a_{k}-a_{j}-n\right)}{\prod\limits_{k=% 1}^{q}\Gamma\left(b_{k}-a_{j}-n\right)}}\right)\*{{}_{q+1}F_{q}}\left({a_{1}-a% _{j}-n,\dots,a_{q+1}-a_{j}-n\atop b_{1}-a_{j}-n,\dots,b_{q}-a_{j}-n};z_{0}% \right)\left(z-z_{0}\right)^{-n}.$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, : factorial (as in ), : nonnegative integer, : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.8(ii)
Permalink:: http://dlmf.nist.gov/16.8.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §16.8(ii), §16.8 and Ch.16

(Note that the generalized hypergeometric functions on the right-hand side are polynomials in $z_{0}$ .)

When and some of the $a_{j}$ differ by an integer a limiting process can again be applied. For details see Nørlund (1955). In this reference it is also explained that in general when no simple representations in terms of generalized hypergeometric functions are available for the fundamental solutions near . Analytical continuation formulas for ${{}_{q+1}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)$ near are given in Bühring (1987b) for the case , and in Bühring (1992) for the general case.

§16.8(iii) Confluence of Singularities

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Keywords:: confluence of singularities, generalized hypergeometric differential equation
Notes:: See Luke (1969a, §3.5).
Permalink:: http://dlmf.nist.gov/16.8.iii
See also:: Annotations for §16.8 and Ch.16

If $p\leq q$ , then

16.8.10 $\lim_{|\alpha|\to\infty}{{}_{p+1}F_{q}}\left({a_{1},\dots,a_{p},\alpha\atop b_% {1},\dots,b_{q}};\frac{z}{\alpha}\right)={{}_{p}F_{q}}\left({a_{1},\dots,a_{p}% \atop b_{1},\dots,b_{q}};z\right).$

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, : nonnegative integer, : nonnegative integer, : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.8.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §16.8(iii), §16.8 and Ch.16

Thus in the case the regular singularities of the function on the left-hand side at $\alpha$ and $\infty$ coalesce into an irregular singularity at $\infty$ .

Next, if $p\leq q+1$ and $|\operatorname{ph}\beta|\leq\pi-\delta$ ( $<\pi$ ), then

16.8.11 $\lim_{|\beta|\to\infty}{{}_{p}F_{q+1}}\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q},\beta};\beta z\right)={{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b% _{1},\dots,b_{q}};z\right),$

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, : nonnegative integer, : nonnegative integer, : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.8.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §16.8(iii), §16.8 and Ch.16

provided that in the case we have when $|\operatorname{ph}\beta|\leq\frac{1}{2}\pi$ , and $|z|<|\sin\left(\operatorname{ph}\beta\right)|$ when $\frac{1}{2}\pi\leq|\operatorname{ph}\beta|\leq\pi-\delta$ ( $<\pi$ ).

16.7 Relations to Other Functions 16.9 Zeros

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§16.8 Differential Equations

Contents

§16.8(i) Classification of Singularities

§16.8(ii) The Generalized Hypergeometric Differential Equation

§16.8(iii) Confluence of Singularities