§16.8 Differential Equations

An ordinary point of the differential equation

16.8.1		$$\frac{d^{n}w}{{dz}^n}+f_{n-1}(z)\frac{d^{n-1}w}{{dz}^{n-1}}+f_{n-2}(z)\frac{d^{n-2}w}{{dz}^{n-2}}+\mathrm{\cdots }+f_1(z)\frac{dw}{dz}+f_0(z)w=0$$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $z$: 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 $z$ at which all the coefficients $f_j(z)$, $j=0,1,\mathrm{\dots },n-1$, are analytic. If $z_0$ is not an ordinary point but ${(z-z_0)}^{n-j}{f}_j(z)$, $j=0,1,\mathrm{\dots },n-1$, 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 $n=2$. Similar definitions apply in the case $z_0=\mathrm{\infty }$: we transform $\mathrm{\infty }$ into the origin by replacing $z$ in (16.8.1) by $1/z$; again compare §2.7(i).

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

With the notation

16.8.2		$\mathit{D}$	$={\displaystyle \frac{d}{dz}},$
		$\mathit{\vartheta }$	$=z{\displaystyle \frac{d}{dz}},$
ⓘ Defines: $\mathit{D}$: differential operator (locally) and $\mathit{\vartheta }$: differential operator (locally) Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $z$: 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 $w={}_{p}F_q(\mathbf{a};\mathbf{b};z)$ satisfies the differential equation

16.8.3		$$\left(\mathit{\vartheta }(\mathit{\vartheta }+b_1-1)\mathrm{\cdots }(\mathit{\vartheta }+b_q-1)-z(\mathit{\vartheta }+a_1)\mathrm{\cdots }(\mathit{\vartheta }+a_p)\right)w=0.$$
ⓘ Symbols: $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters, $\mathit{\vartheta }$: differential operator and $w$: 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		$$z^q{\mathit{D}}^{q+1}w+\sum _{j=1}^q{z}^{j-1}({\alpha }_{j}z+{\beta }_j){\mathit{D}}^{j}w+{\alpha }_{0}w=0,$$
$p\le q$,
ⓘ Symbols: $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $\mathit{D}$: differential operator, $w$: 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

or

16.8.5		$$z^q(1-z){\mathit{D}}^{q+1}w+\sum _{j=1}^q{z}^{j-1}({\alpha }_{j}z+{\beta }_j){\mathit{D}}^{j}w+{\alpha }_{0}w=0,$$
$p=q+1$,
ⓘ Symbols: $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $\mathit{D}$: differential operator, $w$: 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 $z=0$, and an irregular singularity at $z=\mathrm{\infty }$, whereas (16.8.5) has regular singularities at $z=0$, $1$, and $\mathrm{\infty }$. 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		$w_0(z)$	$={}_{p}F_q({\displaystyle \genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q}};z),$
		$w_j(z)$	$=z^{1-b_j}{}_{p}F_q({\displaystyle \genfrac{}{}{0pt}{}{1+a_1-b_j,\mathrm{\dots },1+a_p-b_j}{2-b_j,1+b_1-b_j,\mathrm{\dots }*\mathrm{\dots },1+b_q-b_j}};z),$
	$j=1,\mathrm{\dots },q$,
ⓘ Symbols: ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_{p}F_q(\mathbf{a};\mathbf{b};z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{\mathbf{a}}{\mathbf{b}};z)$ generalized hypergeometric function, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters and $w$: 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 $z$ (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 $p=q+1$, and no two $a_j$ differ by an integer, another fundamental set of solutions of (16.8.3) is given by

16.8.7		$${\tilde{w}}_j(z)={(-z)}^{-a_j}{}_{q+1}F_q(\genfrac{}{}{0pt}{}{a_j,1-b_1+a_j,\mathrm{\dots },1-b_q+a_j}{1-a_1+a_j,\mathrm{\dots }*\mathrm{\dots },1-a_{q+1}+a_j};\frac{1}{z}),$$
$j=1,\mathrm{\dots },q+1$,
ⓘ Symbols: ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_{p}F_q(\mathbf{a};\mathbf{b};z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{\mathbf{a}}{\mathbf{b}};z)$ generalized hypergeometric function, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters and $w$: 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		$${}_{q+1}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{q+1}}{b_1,\mathrm{\dots },b_q};z)=\sum _{j=1}^{q+1}\left(\prod _{\begin{array}{c}\hfill k=1\hfill \\ \hfill k\ne j\hfill \end{array}}^{q+1}\frac{\mathrm{\Gamma }\left(a_k-a_j\right)}{\mathrm{\Gamma }\left(a_k\right)}/\prod _{k=1}^q\frac{\mathrm{\Gamma }\left(b_k-a_j\right)}{\mathrm{\Gamma }\left(b_k\right)}\right){\tilde{w}}_j(z),$$
$|\mathrm{ph}\left(-z\right)|\le \pi$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_{p}F_q(\mathbf{a};\mathbf{b};z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{\mathbf{a}}{\mathbf{b}};z)$ generalized hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters and $w$: 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 $z_0$ ($\in ℂ$) is an arbitrary constant, $|z-z_0|>\max (|z_0|,|z_0-1|)$, and , then

16.8.9		$$\begin{array}{l}\left(\prod _{k=1}^{q+1}\mathrm{\Gamma }\left(a_k\right)/\prod _{k=1}^q\mathrm{\Gamma }\left(b_k\right)\right){}_{q+1}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{q+1}}{b_1,\mathrm{\dots },b_q};z)\\ =\sum _{j=1}^{q+1}{\left(z_0-z\right)}^{-a_j}\sum _{n=0}^{\mathrm{\infty }}\begin{array}{l}\frac{\mathrm{\Gamma }\left(a_j+n\right)}{n!}\left(\prod _{\begin{array}{c}\hfill k=1\hfill \\ \hfill k\ne j\hfill \end{array}}^{q+1}\mathrm{\Gamma }\left(a_k-a_j-n\right)/\prod _{k=1}^q\mathrm{\Gamma }\left(b_k-a_j-n\right)\right)\\ \times {}_{q+1}F_q(\genfrac{}{}{0pt}{}{a_1-a_j-n,\mathrm{\dots },a_{q+1}-a_j-n}{b_1-a_j-n,\mathrm{\dots },b_q-a_j-n};z_0){\left(z-z_0\right)}^{-n}.\end{array}\end{array}$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_{p}F_q(\mathbf{a};\mathbf{b};z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{\mathbf{a}}{\mathbf{b}};z)$ generalized hypergeometric function, $!$: factorial (as in $n!$), $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: 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 $p=q+1$ 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 $q>1$ no simple representations in terms of generalized hypergeometric functions are available for the fundamental solutions near $z=1$. Analytical continuation formulas for ${}_{q+1}F_q(\mathbf{a};\mathbf{b};z)$ near $z=1$ are given in Bühring (1987b) for the case $q=2$, and in Bühring (1992) for the general case.

If $p\le q$, then

16.8.10		$$\underset{|\alpha |\to \mathrm{\infty }}{lim}{}_{p+1}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p,\alpha }{b_1,\mathrm{\dots },b_q};\frac{z}{\alpha })={}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z).$$
ⓘ Symbols: ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_{p}F_q(\mathbf{a};\mathbf{b};z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{\mathbf{a}}{\mathbf{b}};z)$ generalized hypergeometric function, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: 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 $p=q$ the regular singularities of the function on the left-hand side at $\alpha$ and $\mathrm{\infty }$ coalesce into an irregular singularity at $\mathrm{\infty }$.

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

16.8.11		$$\underset{|\beta |\to \mathrm{\infty }}{lim}{}_{p}F_{q+1}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q,\beta };\beta z)={}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z),$$
ⓘ Symbols: ${}_{p}F_q(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_{p}F_q(\mathbf{a};\mathbf{b};z)$ or ${}_{p}F_q(\genfrac{}{}{0pt}{}{\mathbf{a}}{\mathbf{b}};z)$ generalized hypergeometric function, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: 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 $p=q+1$ we have when $|\mathrm{ph}\beta |\le \frac{1}{2}\pi$, and when $\frac{1}{2}\pi \le |\mathrm{ph}\beta |\le \pi -\delta$ ().