An ordinary point of the differential equation

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 origen 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

the function $w={}_p{}^{}F_q^{}(𝐚;𝐛;z)$ satisfies the differential equation

Equivalently,

or

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

where $\ast$ 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

where $\ast$ indicates that the entry $1-a_j+a_j$ is omitted. We have the connection formula

More generally if $z_0$ ($\in ℂ$) is an arbitrary constant, $|z-z_0|>\max (|z_0|,|z_0-1|)$, and , then

(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 ${}_{q+1}{}^{}F_q^{}(𝐚;𝐛;z)$ 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

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

provided that in the case $p=q+1$ we have when $|\mathrm{ph}\beta |\le \frac{1}{2}\pi$, and when $\frac{1}{2}\pi \le |\mathrm{ph}\beta |\le \pi -\delta$ ().