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

Β§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 0≀m≀q and 0≀n≀p, and none of akβˆ’bj is a positive integer when 1≀k≀n and 1≀j≀m. 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)=12⁒π⁒i⁒∫L(βˆβ„“=1mΓ⁑(bβ„“βˆ’s)β’βˆβ„“=1nΓ⁑(1βˆ’aβ„“+s)/(βˆβ„“=mqβˆ’1Γ⁑(1βˆ’bβ„“+1+s)β’βˆβ„“=npβˆ’1Γ⁑(aβ„“+1βˆ’s)))⁒zs⁒ds,

where the integration path L separates the poles of the factors Γ⁑(bβ„“βˆ’s) from those of the factors Γ⁑(1βˆ’aβ„“+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 βˆ’i⁒∞ to i⁒∞. The integral converges if p+q<2⁒(m+n) and |ph⁑z|<(m+nβˆ’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⁑(bβ„“βˆ’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 (β‰ 0) if p<q, and for 0<|z|<1 if p=qβ‰₯1.

  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⁑(1βˆ’aβ„“+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=qβ‰₯1.

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 p≀q, no two of the bottom parameters bj, j=1,…,m, differ by an integer, and ajβˆ’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)=βˆ‘k=1mAp,q,km,n⁑(z)⁒Fqβˆ’1p⁑(1+bkβˆ’a1,…,1+bkβˆ’ap1+bkβˆ’b1,…*…,1+bkβˆ’bq;(βˆ’1)pβˆ’mβˆ’n⁒z),

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

16.17.3 Ap,q,km,n⁑(z)=βˆβ„“=1β„“β‰ kmΓ⁑(bβ„“βˆ’bk)β’βˆβ„“=1nΓ⁑(1+bkβˆ’aβ„“)⁒zbk/(βˆβ„“=mqβˆ’1Γ⁑(1+bkβˆ’bβ„“+1)β’βˆβ„“=npβˆ’1Γ⁑(aβ„“+1βˆ’bk)).