§16.21 Differential Equation

$w=G_{p,q}^{m,n}(z;𝐚;𝐛)$ satisfies the differential equation

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

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

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