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)