# §16.21 Differential Equation

$w={G^{m,n}_{p,q}}\left(z;\mathbf{a};\mathbf{b}\right)$ satisfies the differential equation

 16.21.1 $\left((-1)^{p-m-n}z(\vartheta-a_{1}+1)\cdots(\vartheta-a_{p}+1)-(\vartheta-b_{% 1})\cdots(\vartheta-b_{q})\right)w=0,$

where again $\vartheta=z\ifrac{\mathrm{d}}{\mathrm{d}z}$. This equation is of order $\max(p,q)$. In consequence of (16.19.1) we may assume, without loss of generality, that $p\leq q$. With the classification of §16.8(i), when $p the only singularities of (16.21.1) are a regular singularity at $z=0$ and an irregular singularity at $z=\infty$. When $p=q$ the only singularities of (16.21.1) are regular singularities at $z=0$, $(-1)^{p-m-n}$, and $\infty$.

A fundamental set of solutions of (16.21.1) is given by

 16.21.2 ${G^{1,p}_{p,q}}\left(z{\mathrm{e}}^{(p-m-n-1)\pi\mathrm{i}};{a_{1},\dots,a_{p}% \atop b_{j},b_{1},\dots,b_{j-1},b_{j+1},\dots,b_{q}}\right),$ $j=1,\dots,q$.

For other fundamental sets see Erdélyi et al. (1953a, §5.4) and Marichev (1984).