MeijerG

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.)

Special notations for this file:

Specific values

Specialized values

Cases with arbitrary {m,n,p,q}

Cases with m==0

Cases with m==1

Cases with m==2

Cases with m==3

Cases with m==4

Cases with m==5

General characteristics

Domain and analyticity

(Subscript[G, p,q]^(m,n))(z|({{(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])}})) is an analytical function of (Subscript[a, 1]), …, (Subscript[a, p]), (Subscript[b, 1]), …, (Subscript[b, q]) and z which is defined in (ℂ^(p+q+1)). In the case p==q==m+nthis function with respect to  z is piecewise analytical function with discontinuity on unit circle |z|==1(it is continuos in the point z==1 if Re((∑_(k=1)^q)(Subscript[b, k])-(∑_(k=1)^p)(Subscript[a, k]))+1<0).

Symmetries and periodicities

Permutation symmetry

Periodicity

No periodicity

Branch points

The function (Subscript[G, p,q]^(m,n))(z|({{(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 two (z==0,∞ for p≠q) or three (z==0,(((-1))^(m+n-p)), ∞, for p==q) singular points. If  p<q (or p>q), then the point  z==0 (or z==∞) is a regular singular point, while  z==∞(or z==0) is a non-regular (essentual) singular point; if  p==q, then all three singular points are regular.

Branch cuts

The existence of the branch cut depends on the parameters of MeijerG. If the branch cut exists, it will be along (-∞,0) (or sometimes  (-∞,-1) or (-1,0)). For example, the function (Subscript[G, 1,1]^(1,1))(z|({{a}, {b}}))==Γ(1-a+b)(z^b) (((z+1))^(a-b-1)) has branch cut along (-∞,0) if b∉ℤ and b-a∉ℤ; in the case b∈ℤ it generically has branch cut along (-∞,-1).

If (Subscript[b, j])∉ℤ,1≤j≤m, and (Subscript[a, j])∉ℤ,1≤j≤n, the branch cut along (-∞,0) exists because it is produced by power functions (z^(Subscript[b, j]))and (z^((Subscript[a, j])-1)). In these cases the Meijer G function (Subscript[G, p,q]^(m,n))(z|({{(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])}})) is a single-valued function on the z-plane cut along the interval (-∞, 0), where it is continuous from above.

If p==q the branch cuts can appear not only from functions (z^(Subscript[b, j]))or (z^((Subscript[a, j])-1))but from other generalized hypergeometric factors in expansions of MeijerG  function (Subscript[G, p,q]^(m,n))(z|({{(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])}})). In these cases the behaviour is more complicated.

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. For example, (Subscript[G, 1,1]^(1,0))(z|({{1}, {0}}))==θ(1-|z|).

Series representations

Generalized power series

Expansions at generic point z==(Subscript[z, 0])

Expansions on branch cuts |||  Expansions on branch cuts

Expansions at z==0

Expansions at z==(((-1))^(m+n-q)) for p==q

The function (Subscript[G, q,q]^(m,n))(z|({{(Subscript[a, 1]),…,(Subscript[a, n]),(Subscript[a, n+1]),…,(Subscript[a, q])}, {(Subscript[b, 1]),…,(Subscript[b, m]),(Subscript[b, m+1]),…,(Subscript[b, q])}})) near the point z==(((-1))^(m+n-q)) has a rather complicated behabiour. The general formula (for
noninteger (Subscript[ψ, q])==-μ==(∑_(j=1)^q)((Subscript[a, j])-(Subscript[b, j]))-1/;p==q) includes two major terms that are analytical functions only in the case  m+n>q. In the opposite case m+n≤q they are the piecewise analytical functions with possible discontinuities on the unit circle |z|==1. Moreover, the first major term has representation of the form const (((1-(Subscript[z, 1])))^(Subscript[ψ, q])) (1+O((Subscript[z, 1])-1))/;(Subscript[z, 1])==(((-1))^(q-m-n))z and second one is bounded near the point (Subscript[z, 1])==1. A more detailed description of this behaviour is presented below.

If m+n-q>0, the function (Subscript[G, q,q]^(m,n))(z|({{(Subscript[a, 1]),…,(Subscript[a, n]),(Subscript[a, n+1]),…,(Subscript[a, q])}, {(Subscript[b, 1]),…,(Subscript[b, m]),(Subscript[b, m+1]),…,(Subscript[b, q])}})) is analytic in the sector |Arg(z)|<(m+n-q)π.

If m+n-q==1∧|Arg(z)|<π, at the singular point z==-1 the function (Subscript[G, q,q]^(m,n))(z|({{(Subscript[a, 1]),…,(Subscript[a, n]),(Subscript[a, n+1]),…,(Subscript[a, q])}, {(Subscript[b, 1]),…,(Subscript[b, m]),(Subscript[b, m+1]),…,(Subscript[b, q])}})) is continuous for Re((Subscript[ψ, q]))>0, bounded for Re((Subscript[ψ, q]))==0,(Subscript[ψ, q])≠0 and has, in general, a logarithmic singularity for (Subscript[ψ, q])==0, while for Re((Subscript[ψ, q]))<0, it has a power singularity of order -(Subscript[ψ, q]), to which, for integer (Subscript[ψ, q])a logarithmic singularity can also be added.

If m+n-q<1, the function (Subscript[G, q,q]^(m,n))(z|({{(Subscript[a, 1]),…,(Subscript[a, n]),(Subscript[a, n+1]),…,(Subscript[a, q])}, {(Subscript[b, 1]),…,(Subscript[b, m]),(Subscript[b, m+1]),…,(Subscript[b, q])}})) is a piecewise analytic function of z with a discontinuity on the circle |z|==1.
If m+n-q==0, at the singular point z==1it is continuous for Re((Subscript[ψ, q]))>0, bounded for Re((Subscript[ψ, q]))==0,(Subscript[ψ, q])≠0 and has, in general, a logarithmic singularity for (Subscript[ψ, q])==0,m n ≠0, while for Re((Subscript[ψ, q]))<0 it has a power singularity of order -(Subscript[ψ, q]), to which for integer (Subscript[ψ, q])a logarithmic singularity can also occur. Here the asymptotics of this function depends on the way z approaches 1 (from outside or inside the disc |z|≤1).

Expansions at z==∞

Asymptotic series expansions at z==0 for q<p

Expansions for p==q+1

Expansions for p==q+2

Expansions for p>q+2

Asymptotic series expansions at z==∞ for q>p

Expansions for q==p+1

Expansions for q==p+2

Expansions for q>p+2

General formulas of asymptotic series expansions

Main terms of asymptotic expansions

Expansions at z==0

Expansions at z==(((-1))^(m+n-q)) for p==q

Expansions at z==∞

Residue representations

Integral representations

Contour integral representations

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

The differential equation for Meijer G function  (Subscript[G, p,q]^(m,n))(z|({{(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 the order max(p,q). It has  two (z==0,z==(OverTilde[∞]) for p≠q) or three (z==0,z==(((-1))^(m+n-p)), z==(OverTilde[∞]), for p==q) singular points. If  p<q (or p>q), then the point  z==0 (or z==(OverTilde[∞])) is a regular singular point, while  z==(OverTilde[∞])(or z==0) is a nonregular (essential) singular point; if  p==q, then all three singular points are regular.

Below representation include functions of two kinds. The function (Subscript[G, q,q]^(q,0))((((-1))^(q-m-n)) z|({{(Subscript[a, 1]),…,(Subscript[a, q])}, {(Subscript[b, 1]),…,(Subscript[b, q])}})) is the piecewise analytical function with discontinuity on the unit circle |z|==1. It has a singularity near point z==(((-1))^(m+n-q))of the form const (((1-(Subscript[z, 1])))^(Subscript[ψ, q])) (1+O((Subscript[z, 1])-1))/;(Subscript[z, 1])==(((-1))^(q-m-n))z, when |z|<1. The functions (Subscript[G, q+2,q+2]^(2,q+2))((((-1))^(q-m-n))z|({{(Subscript[b, h]),(Subscript[b, j]),(Subscript[a, 1]),…,(Subscript[a, q])}, {(Subscript[b, h]),(Subscript[b, j]),(Subscript[b, 1]),…,(Subscript[b, q])}})) are the analytical functions and are bounded near point (Subscript[z, 1])==1.

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Products, sums, and powers of the direct function

Products of the direct function

Sums of the direct function

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Relations between contiguous functions

Relations of special kind

General cases

Relations with cut through poles

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

Definite integration

Classical and generalized Meijer's integrals from one G function

Classical Meijer's integral from two G functions

Generalization of classical Meijer's integral from two G functions

Integral from three G functions

Integral including G functions with shifted arguments

Integral by variable in parameters

Integrals for classical integral transforms

Integrals including G functions with rational arguments

Integral transforms

Fourier cos transforms

Fourier sin transforms

Laplace transforms

Mellin transforms

Hankel transforms

Hilbert transform

Operations

Limit operation

Representations through more general functions

Through hypergeometric functions

Hypergeometric functions are not more general than Meijer G function.

Involving HypergeometricPFQRegularized

Involving HypergeometricPFQ

Through G function of two variables

Through Fox H

Through Meijer G

Generalized cases

Theorems

The Wigner distribution function for the Liouville potential

The Wigner distribution function (Subscript[f, ε])(x,p) for the Liouville potential with Hamiltonian H==(p^2)+(e^(2x)) at the energy ε can be expressed as

(Subscript[f, ε])(x,p)==(1/(4(π^3)))sin(π(Sqrt[ε]))(2^(2i p))(e^(-x(1+2i p)))(Subscript[G, 0 4]^(4 0))(((e^(4x))/16)|((1+2i(Sqrt[ε]))/4),((1-2i(Sqrt[ε]))/4),((1+2i(Sqrt[ε])+4i p)/4),((1-2i(Sqrt[ε])+4i p)/4)).

History

S. Pincherle (1888); E.W. Barnes (1907, 1908); H. Mellin (1910); 
E.T. Whittaker and G.N. Watson (1927); A.L. Dixon and W.L.Ferrar (1936); C.S. 
Meijer (1936, 1941–1956); L.J. Slater (1966); Y.L. Luke (1969); A.M. 
Mathai and R.K. Saxena (1973); O.I. Marichev (1978); A.P. Prudnikov, Y.A. 
Brychkov and O.I. Marichev (1986).

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