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

§16.19 Identities

16.19.1 Gp,qm,n⁡(1z;a1,…,apb1,…,bq) =Gq,pn,m⁡(z;1−b1,…,1−bq1−a1,…,1−ap),
16.19.2 zμ⁢Gp,qm,n⁡(z;a1,…,apb1,…,bq) =Gp,qm,n⁡(z;a1+μ,…,ap+μb1+μ,…,bq+μ),
16.19.3 Gp+1,q+1m,n+1⁡(z;a0,…,apb1,…,bq,a0) =Gp,qm,n⁡(z;a1,…,apb1,…,bq),
16.19.4 Gp,qm,n⁡(z;a1,…,apb1,…,bq)=2p+1+b1+⋯+bq−m−n−a1−⋯−apπm+n−12⁢(p+q)×G2⁢p,2⁢q2⁢m,2⁢n⁡(22⁢p−2⁢q⁢z2;12⁢a1,12⁢a1+12,…,12⁢ap,12⁢ap+1212⁢b1,12⁢b1+12,…,12⁢bq,12⁢bq+12),
16.19.5 ϑGp,qm,n⁡(z;a1,…,apb1,…,bq)=Gp,qm,n⁡(z;a1−1,a2,…,apb1,…,bq)+(a1−1)⁢Gp,qm,n⁡(z;a1,…,apb1,…,bq),
16.19.6 ∫01t−a0⁢(1−t)a0−bq+1−1⁢Gp,qm,n⁡(z⁢t;a1,…,apb1,…,bq)⁢dt=Γ⁡(a0−bq+1)⁢Gp+1,q+1m,n+1⁡(z;a0,…,apb1,…,bq+1),

where again ϑ=z⁢d/dz. For conditions for (16.19.6) see Luke (1969a, Chapter 5). This reference and Mathai (1993, §§2.2 and 2.4) also supply additional identities.