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16 Generalized Hypergeometric Functions & 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−n⁢z⁢(ϑ−a1+1)⁢⋯⁢(ϑ−ap+1)−(ϑ−b1)⁢⋯⁢(ϑ−bq))⁢w=0,

where again ϑ=z⁢d/dz. This equation is of order max⁡(p,q). In consequence of (16.19.1) we may assume, without loss of generality, that p≤q. 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⁡(z⁢e(p−m−n−1)⁢π⁢i;a1,…,apbj,b1,…,bj−1,bj+1,…,bq),

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