Digital Library of Mathematical Functions
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16 Generalized Hypergeometric Functions and 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 zμGp,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)×G2p,2q2m,2n(22p-2qz2;12a1,12a1+12,,12ap,12ap+1212b1,12b1+12,,12bq,12bq+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+1-1Gp,qm,n(zt;a1,,apb1,,bq)t=Γ(a0-bq+1)Gp+1,q+1m,n+1(z;a0,,apb1,,bq+1),

where again ϑ=z/z. 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.