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16 Generalized Hypergeometric Functions & Meijer G-FunctionNotation

§16.1 Special Notation

(For other notation see Notation for the Special Functions.)

p,q nonnegative integers.
k,n nonnegative integers, unless
stated otherwise.
z complex variable.
a1,a2,,apb1,b2,,bq} real or complex parameters.
δ arbitrary small positive constant.
a vector (a1,a2,,ap).
b vector (b1,b2,,bq).
(a)k (a1)k(a2)k(ap)k.
(b)k (b1)k(b2)k(bq)k.
D d/dz.
ϑ zd/dz.

The main functions treated in this chapter are the generalized hypergeometric function Fqp(a1,,apb1,,bq;z), the Appell (two-variable hypergeometric) functions F1(α;β,β;γ;x,y), F2(α;β,β;γ,γ;x,y), F3(α,α;β,β;γ;x,y), F4(α,β;γ,γ;x,y), and the Meijer G-function Gp,qm,n(z;a1,,apb1,,bq). Alternative notations are Fqp(ab;z), Fqp(a1,,ap;b1,,bq;z), and Fqp(a;b;z) for the generalized hypergeometric function, F1(α,β,β;γ;x,y), F2(α,β,β;γ,γ;x,y), F3(α,α,β,β;γ;x,y), F4(α,β;γ,γ;x,y), for the Appell functions, and Gp,qm,n(z;a;b) for the Meijer G-function.