MeijerG1

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

The infinite contour of integration ℒ separates the poles of Γ(1-(Subscript[a, k])-s)at s=1-(Subscript[a, k])+j, j∈ℕ from the poles of Γ((Subscript[b, i])+ s) at s=-(Subscript[b, i])-l, l∈ℕ. Such a contour always exists in the cases  (Subscript[a, k])-(Subscript[b, i])-1∉ℕ.

There are three possibilities for the contour ℒ :

(i) ℒ runs from γ-i ∞  to γ+i ∞ (where Im(γ)=0) so that all poles of Γ((Subscript[b, i])+ s), i=1,...,m, are to the left, and all the poles of Γ(1-(Subscript[a, i])- s), i=1,...,n, to the right, of ℒ. 
This contour can be a straight line (γ-i ∞, γ+i ∞) if  Re((Subscript[b, i])-(Subscript[a, k]))>-1 (then  -Re((Subscript[b, i]))<γ<1-Re((Subscript[a, k]))). (In this case the integral converges if p+q<2(m+n) ,  |Arg(z)|<(m +n-((p+q)/2)) π. If m +n-((p+q)/2)=0, then z must be real and positive and additional condition  (q-p)γ+Re(μ)<0,μ==(∑_(l=1)^q)(Subscript[b, l])-(∑_(k=1)^p)(Subscript[a, k])+((p-q)/2)+1, should be added.)

(ii) ℒ is a left loop, starting and ending at -∞ and encircling all poles of Γ((Subscript[b, i])+ s), i=1,...,m, once in the positive direction, but none of the poles of  Γ(1-(Subscript[a, i])- s), i=1,...,n. 
(In this case the integral converges if q≥1 and either q>p or q=p and |z|<1 or q=p and |z|==1and  m +n-((p+q)/2)≥0 and  Re(μ)<0.)

(iii) ℒ is a right loop, starting and ending at +∞ and encircling all poles of Γ(1-(Subscript[a, i])-s), i=1,...,n, once in the negative direction, but none of the poles of Γ((Subscript[b, i])+s), i=1,...,m. 
(In this case the integral converges if p≥1 and either p>q or p=q and \[LeftBracketingBar]z\[RightBracketingBar]>1 or q=p and \[LeftBracketingBar]z\[RightBracketingBar]==1and  m +n-((p+q)/2)≥0 and  Re(μ)<0.)

Specific values

Specialized values

General case with restrictions on z and r

Case {m,n,p,q}={0,1,2,0}

Case {m,n,p,q}={0,1,3,1}

Case {m,n,p,q}={0,2,2,0}

Case {m,n,p,q}={0,2,2,1}

Case {m,n,p,q}={0,2,2,2}

Case {m,n,p,q}={0,2,3,1}

Case {m,n,p,q}={0,2,3,2}

Case {m,n,p,q}={0,2,4,2}

Case {m,n,p,q}={0,3,3,1}

Case {m,n,p,q}={0,4,4,2}

Case {m,n,p,q}={1,0,0,2}

Case {m,n,p,q}={1,0,0,4}

Case {m,n,p,q}={1,0,1,3}

Case {m,n,p,q}={1,1,1,2}

Case {m,n,p,q}={1,1,1,3}

Case {m,n,p,q}={1,1,2,1}

Case {m,n,p,q}={1,1,2,4}

Case {m,n,p,q}={1,1,3,1}

Case {m,n,p,q}={1,1,4,2}

Case {m,n,p,q}={1,2,2,1}

Case {m,n,p,q}={1,2,2,2}

Case {m,n,p,q}={1,2,3,1}

Case {m,n,p,q}={1,2,3,2}

Case {m,n,p,q}={1,2,3,5}

Case {m,n,p,q}={1,2,5,3}

Case {m,n,p,q}={1,3,4,2}

Case {m,n,p,q}={2,0,0,2}

Case {m,n,p,q}={2,0,1,2}

Case {m,n,p,q}={2,0,1,3}

Case {m,n,p,q}={2,0,2,2}

Case {m,n,p,q}={2,0,2,3}

Case {m,n,p,q}={2,0,2,4}

Case {m,n,p,q}={2,1,1,2}

Case {m,n,p,q}={2,1,1,3}

Case {m,n,p,q}={2,1,2,2}

Case {m,n,p,q}={2,1,2,3}

Case {m,n,p,q}={2,1,3,5}

Case {m,n,p,q}={2,2,2,4}

Case {m,n,p,q}={2,2,3,3}

Case {m,n,p,q}={2,2,4,2}

Case {m,n,p,q}={2,2,4,6}

Case {m,n,p,q}={2,2,6,4}

Case {m,n,p,q}={3,0,1,3}

Case {m,n,p,q}={3,0,2,6}

Case {m,n,p,q}={3,1,1,3}

Case {m,n,p,q}={3,1,2,4}

Case {m,n,p,q}={4,0,2,4}

General characteristics

Branch cuts

Branch cut locations: When p==q , then there is a branch cut along (-∞,0). If p≠q, however, then the existence of the branch cut depends on the parameters of MeijerG; if the branch cut exists, it will be along (-∞,0). In cases where p==q and m+n-p==0, there will be a jump discontinuity on the unit circle |z|==1.

Sets of discontinuity

(Subscript[G, p,q]^(m,n))(z,r|({{(Subscript[a, 1]),…,(Subscript[a, n]),(Subscript[a, n+1]),…,(Subscript[a, p])}, {(Subscript[b, 1]),…,(Subscript[b, m]),(Subscript[b, m+1]),…,(Subscript[b, q])}})) has discontinuity on the unit circle |z|==1in the case p==q==m+n.

Series representations

Generalized power series

Expansions at z==0

Expansions at z==∞

Residue representations

Integral representations

Contour integral representations

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Identities

Functional identities

Differentiation

Low-order differentiation

With respect to z

Symbolic differentiation

With respect to z

Fractional integro-differentiation

With respect to z

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Integral transforms

Mellin transforms

Representations through more general functions

Through hypergeometric functions

Hypergeometric functions are not more general than Meijer G function.

Involving HypergeometricPFQRegularized

Involving HypergeometricPFQ

Through Fox H

Through Meijer G

Classical cases

Classical Meijer G function is particular case of Meijer G function with parameter r.

History

J. Keiper and O.I. Marichev (1993) extended MeijerG function as part of the Mathematica system development effort.

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