Meijer-G-functions

Generalizations of the hypergeometric functions ${ {} _ {p} F _ {q} }$ of one variable (cf. also Hypergeometric function). They can be defined by an integral as

$$G _ {pq } ^ {mn } \left ( x \left | \begin{array}{c} a _ {1} \dots a _ {p} \\ b _ {1} \dots b _ {q} \\ \end{array} \right . \right ) =$$

$$= { \frac{1}{2 \pi i } } \int\limits _ { L } { { \frac{\prod _ {j = 1 } ^ { m } \Gamma ( b _ {j} - s ) \prod _ {j = 1 } ^ { n } \Gamma ( 1 - a _ {j} + s ) }{\prod _ {j = m + 1 } ^ { q } \Gamma ( 1 - b _ {j} + s ) \prod _ {j = n + 1 } ^ { p } \Gamma ( a _ {j} - s ) } } x ^ {s} } {ds } ,$$

where $0 \leq m \leq p$, $0 \leq n \leq q$ and the parameters $a _ {r}$, $b _ {r}$ are such that no pole of the functions $\Gamma ( b _ {j} - s )$ coincides with any pole of the functions $\Gamma ( 1 - a _ {j} + s )$. There are three possible choices for the contour $L$:

a) $L$ goes from $- i \infty$ to $+ i \infty$ remaining to the right of the poles of $\Gamma ( b _ {j} - s )$ and to the left of the poles of $\Gamma ( 1 - a _ {j} + s )$;

b) $L$ begins and ends at $+ \infty$, encircles counterclockwise all poles of $\Gamma ( b _ {j} - s )$ and does not encircle any pole of $\Gamma ( 1 - a _ {j} + s )$;

c) $L$ begins and ends at $- \infty$, encircles clockwise all poles of $\Gamma ( 1 - a _ {j} + s )$ and does not encircle any pole of $\Gamma ( b _ {j} - s )$.

The integral converges if $p + q < 2 ( m + n )$, $| { { \mathop{\rm arg} } x } | < ( m + n - {1 / 2 } ) ( p + q ) \pi$ in case a); if $q \geq 1$ and either $p < q$ or $p = q$ and $| x | < 1$ in case b); and if $p \geq 1$ and either $p > q$ or $p = q$ and $| x | > 1$ in case c).

The integral defining the Meijer $G$- functions can be calculated by means of the residue theorem and one obtains expressions for $G _ {pq } ^ {mn }$ in terms of the hypergeometric functions ${ {} _ {p} F _ {q - 1 } }$ or ${ {} _ {q} F _ {p - 1 } }$. The function $G _ {pq } ^ {mn }$ satisfies the linear differential equation

$${\mathcal L} y = 0,$$

where

$${\mathcal L} =$$

$$= \left [ ( - 1 ) ^ {p - m - n } \prod _ {j = 1 } ^ { p } \left ( x { \frac{d}{dx } } - a _ {j} + 1 \right ) \prod _ {j = 1 } ^ { q } \left ( x { \frac{d}{dx } } - b _ {j} \right ) \right ] .$$

Many functions of hypergeometric type and their products can be expressed in terms of Meijer $G$- functions, [a1]. For example,

$$J _ {a - b } ( 2 \sqrt x ) = x ^ {- ( a + b ) /2 } G _ {02 } ^ {10 } ( x \mid a,b ) ,$$

$$J _ {b - a } ( \sqrt x ) Y _ {b - a } ( \sqrt x ) = - \sqrt x x ^ {- a } G _ {13 } ^ {20 } \left ( x \left | \begin{array}{c} a + {1 / 2 } \\ b, a, 2a - b \\ \end{array} \right . \right ) .$$

Meijer $G$- functions appear in the theory of Lie group representations (cf. also Representation of a compact group) as transition coefficients for different bases of carrier spaces of representations [a2].

References

[a1]	A. Erdelyi, W. Magnus, F. Oberhettinger, F. Tricomi, "Higher transcendental functions" , 1 , McGraw-Hill (1953)
[a2]	N.J. Vilenkin, A.U. Klimyk, "Representation of Lie groups and special functions" , 2 , Kluwer Acad. Publ. (1993) (In Russian)