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16 Generalized Hypergeometric Functions and Meijer G-FunctionMeijer G-Function

§16.21 Differential Equation

w=Gp,qm,n(z;a;b) satisfies the differential equation

16.21.1 ((-1)p-m-nz(ϑ-a1+1)(ϑ-ap+1)-(ϑ-b1)(ϑ-bq))w=0,

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

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

16.21.2 Gp,q1,p(ze(p-m-n-1)πi;a1,,apbj,b1,,bj-1,bj+1,,bq),
j=1,,q.

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