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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;1b1,,1bq1a1,,1ap),
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++bqmna1apπm+n12(p+q)×G2p,2q2m,2n(22p2qz2;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;a11,a2,,apb1,,bq)+(a11)Gp,qm,n(z;a1,,apb1,,bq),
16.19.6 01ta0(1t)a0bq+11Gp,qm,n(zt;a1,,apb1,,bq)dt=Γ(a0bq+1)Gp+1,q+1m,n+1(z;a0,,apb1,,bq+1),

where again ϑ=zd/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.