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

§16.17 Definition

Again assume a1,a2,,ap and b1,b2,,bq are real or complex parameters. Assume also that m and n are integers such that 0mq and 0np, and none of ak-bj is a positive integer when 1kn and 1jm. Then the Meijer G-function is defined via the Mellin–Barnes integral representation:

16.17.1 Gp,qm,n(z;a;b)=Gp,qm,n(z;a1,,apb1,,bq)=12πL(=1mΓ(b-s)=1nΓ(1-a+s)/(=mq-1Γ(1-b+1+s)=np-1Γ(a+1-s)))zss,

where the integration path L separates the poles of the factors Γ(b-s) from those of the factors Γ(1-a+s). There are three possible choices for L, illustrated in Figure 16.17.1 in the case m=1, n=2:

  1. (i)

    L goes from - to . The integral converges if p+q<2(m+n) and |phz|<(m+n-12(p+q))π.

  2. (ii)

    L is a loop that starts at infinity on a line parallel to the positive real axis, encircles the poles of the Γ(b-s) once in the negative sense and returns to infinity on another line parallel to the positive real axis. The integral converges for all z (0) if p<q, and for 0<|z|<1 if p=q1.

  3. (iii)

    L is a loop that starts at infinity on a line parallel to the negative real axis, encircles the poles of the Γ(1-a+s) once in the positive sense and returns to infinity on another line parallel to the negative real axis. The integral converges for all z if p>q, and for |z|>1 if p=q1.

See accompanying text See accompanying text See accompanying text
Case (i) Case (ii) Case (iii)
Figure 16.17.1: s-plane. Path L for the integral representation (16.17.1) of the Meijer G-function. Magnify

When more than one of Cases (i), (ii), and (iii) is applicable the same value is obtained for the Meijer G-function.

Assume pq, no two of the bottom parameters bj, j=1,,m, differ by an integer, and aj-bk is not a positive integer when j=1,2,,n and k=1,2,,m. Then

16.17.2 Gp,qm,n(z;a1,,apb1,,bq)=k=1mAp,q,km,n(z)Fq-1p(1+bk-a1,,1+bk-ap1+bk-b1,*,1+bk-bq;(-1)p-m-nz),

where * indicates that the entry 1+bk-bk is omitted. Also,

16.17.3 Ap,q,km,n(z)==1kmΓ(b-bk)=1nΓ(1+bk-a)zbk/(=mq-1Γ(1+bk-b+1)=np-1Γ(a+1-bk)).