Meijer-G-functions
Generalizations of the hypergeometric functions
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) |
Meijer-G-functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Meijer-G-functions&oldid=49297