§16.8 Differential Equations

Keywords:: differential equations
Permalink:: http://dlmf.nist.gov/16.8

§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

Keywords:: differential equations, generalized hypergeometric differential equation, singularities
Referenced by:: §16.21, §2.7(i)
Permalink:: http://dlmf.nist.gov/16.8.i

An ordinary point of the differential equation

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

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
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is a value $z_{0}$ of $z$ 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 $n=2$ . Similar definitions apply in the case $z_{0}=\infty$ : we transform $\infty$ 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).

§16.8(ii) The Generalized Hypergeometric Differential Equation

Notes:: See Luke (1969a, §5.1). For (16.8.9) see Bühring (1988).
Keywords:: generalized hypergeometric differential equation, generalized hypergeometric functions
Referenced by:: §2.7(i)
Permalink:: http://dlmf.nist.gov/16.8.ii

With the notation

16.8.2

$D=\frac{d}{dz},$

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

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $D$ : differential operator and $\vartheta$ : differential operator
Permalink:: http://dlmf.nist.gov/16.8.E2
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the function $w=\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left(\mathbf{a};\mathbf{b};z\right)$ satisfies the differential equation

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

Symbols:: $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : 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 $w$ : solution
Referenced by:: §16.8(ii), §16.8(ii), §16.8(ii)
Permalink:: http://dlmf.nist.gov/16.8.E3
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Equivalently,

16.8.4 $z^{q}D^{{q+1}}w+\sum _{{j=1}}^{q}z^{{j-1}}(\alpha _{j}z+\beta _{j})D^{j}w+\alpha _{0}w=0,$ $p\leq q$ ,

Symbols:: $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $D$ : differential operator, $w$ : solution, $\alpha _{j}$ : constant and $\beta _{j}$ : constant
Referenced by:: §16.8(ii)
Permalink:: http://dlmf.nist.gov/16.8.E4
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16.8.5 $z^{q}(1-z)D^{{q+1}}w+\sum _{{j=1}}^{q}z^{{j-1}}(\alpha _{j}z+\beta _{j})D^{j}w+\alpha _{0}w=0,$ $p=q+1$ ,

Symbols:: $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $D$ : differential operator, $w$ : solution, $\alpha _{j}$ : constant and $\beta _{j}$ : constant
Referenced by:: §16.8(ii)
Permalink:: http://dlmf.nist.gov/16.8.E5
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where $\alpha _{j}$ and $\beta _{j}$ are constants. Equation (16.8.4) has a regular singularity at $z=0$ , and an irregular singularity at $z=\infty$ , whereas (16.8.5) has regular singularities at $z=0$ , 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

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

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

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and $w$ : solution
Permalink:: http://dlmf.nist.gov/16.8.E6
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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 $\mathop{\ln\/}\nolimits z$ ) 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 $\widetilde{w}_{j}(z)=(-z)^{{-a_{j}}}\mathop{{{}_{{q+1}}F_{{q}}}\/}\nolimits\!\left({a_{j},1-b_{1}+a_{j},\dots,1-b_{q}+a_{j}\atop 1-a_{1}+a_{j},\ldots*\dots,1-a_{{q+1}}+a_{j}};\frac{1}{z}\right),$ $j=1,\dots,q+1$ ,

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and $w$ : solution
Permalink:: http://dlmf.nist.gov/16.8.E7
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where $*$ indicates that the entry $1-a_{j}+a_{j}$ is omitted. We have the connection formula

16.8.8 $\mathop{{{}_{{q+1}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{{q+1}}\atop b_{1},\dots,b_{q}};z\right)=\sum _{{j=1}}^{{q+1}}\left({\textstyle\ifrac{\prod\limits _{{\substack{k=1\\ k\neq j}}}^{{q+1}}\frac{\mathop{\Gamma\/}\nolimits\!\left(a_{k}-a_{j}\right)}{\mathop{\Gamma\/}\nolimits\!\left(a_{k}\right)}}{\prod\limits _{{k=1}}^{q}\frac{\mathop{\Gamma\/}\nolimits\!\left(b_{k}-a_{j}\right)}{\mathop{\Gamma\/}\nolimits\!\left(b_{k}\right)}}}\right)\widetilde{w}_{j}(z),$ $|\mathop{\mathrm{ph}\/}\nolimits\!\left(-z\right)|\leq\pi$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters and $w$ : solution
Referenced by:: ¶ ‣ §16.11(ii)
Permalink:: http://dlmf.nist.gov/16.8.E8
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More generally if $z_{0}$ ( $\in\Complex$ ) is an arbitrary constant, $|z-z_{0}|>\max{(|z_{0}|,|z_{0}-1|)}$ , and $|\mathop{\mathrm{ph}\/}\nolimits\!\left(z_{0}-z\right)|<\pi$ , then

16.8.9 $\left({\textstyle\ifrac{\prod\limits _{{k=1}}^{{q+1}}\mathop{\Gamma\/}\nolimits\!\left(a_{k}\right)}{\prod\limits _{{k=1}}^{q}\mathop{\Gamma\/}\nolimits\!\left(b_{k}\right)}}\right)\mathop{{{}_{{q+1}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{{q+1}}\atop b_{1},\dots,b_{q}};z\right)=\sum _{{j=1}}^{{q+1}}\left(z_{0}-z\right)^{{-a_{j}}}\sum _{{n=0}}^{\infty}\frac{\mathop{\Gamma\/}\nolimits\!\left(a_{j}+n\right)}{n!}\*\left({\textstyle\ifrac{\prod\limits _{{\substack{k=1\\ k\neq j}}}^{{q+1}}\mathop{\Gamma\/}\nolimits\!\left(a_{k}-a_{j}-n\right)}{\prod\limits _{{k=1}}^{q}\mathop{\Gamma\/}\nolimits\!\left(b_{k}-a_{j}-n\right)}}\right)\*\mathop{{{}_{{q+1}}F_{{q}}}\/}\nolimits\!\left({a_{1}-a_{j}-n,\dots,a_{{q+1}}-a_{j}-n\atop b_{1}-a_{j}-n,\dots,b_{q}-a_{j}-n};z_{0}\right)\left(z-z_{0}\right)^{{-n}}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $!$ : $n!$ : factorial, $q$ : nonnegative integer, $z$ : 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
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(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 $\mathop{{{}_{{q+1}}F_{{q}}}\/}\nolimits\!\left(\mathbf{a};\mathbf{b};z\right)$ near $z=1$ are given in Bühring (1987b) for the case $q=2$ , and in Bühring (1992) for the general case.

§16.8(iii) Confluence of Singularities

Notes:: See Luke (1969a, §3.5).
Keywords:: generalized hypergeometric differential equation
Permalink:: http://dlmf.nist.gov/16.8.iii

If $p\leq q$ , then

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

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : 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
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Thus in the case $p=q$ 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 $|\mathop{\mathrm{ph}\/}\nolimits\beta|\leq\pi-\delta$ ( $<\pi$ ), then

16.8.11 $\lim _{{|\beta|\to\infty}}\mathop{{{}_{{p}}F_{{q+1}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q},\beta};\beta z\right)=\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right),$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : 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
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provided that in the case $p=q+1$ we have $|z|<1$ when $|\mathop{\mathrm{ph}\/}\nolimits\beta|\leq\frac{1}{2}\pi$ , and $|z|<|\mathop{\sin\/}\nolimits\!\left(\mathop{\mathrm{ph}\/}\nolimits\beta\right)|$ when $\frac{1}{2}\pi\leq|\mathop{\mathrm{ph}\/}\nolimits\beta|\leq\pi-\delta$ ( $<\pi$ ).

§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