# §16.8(i) Classification of Singularities

An ordinary point of the differential equation

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

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

With the notation

 16.8.2 $\displaystyle D$ $\displaystyle=\frac{d}{dz},$ $\displaystyle\vartheta$ $\displaystyle=z\frac{d}{dz},$ 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 Encodings: TeX, TeX, pMML, pMML, png, png

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

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

Equivalently,

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

or

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

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 $\displaystyle w_{0}(z)$ $\displaystyle=\mathop{{{}_{p}F_{q}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b% _{1},\dots,b_{q}};z\right),$ ${w_{j}(z)=z^{1-b_{j}}\mathop{{{}_{p}F_{q}}\/}\nolimits\!\left({1+a_{1}-b_{j},% \dots,1+a_{p}-b_{j}\atop 2-b_{j},1+b_{1}-b_{j},\ldots*\dots,1+b_{q}-b_{j}};z% \right),}$ $j=1,\dots,q$,

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

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

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

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

(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

If $p\leq q$, then

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

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

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 $\frac{1}{2}\pi\leq|\mathop{\mathrm{ph}\/}\nolimits\beta|\leq\pi-\delta$ ($<\pi$).