Meijer G-function - Relationship Between The G-function and The Generalized Hypergeometric Function

Relationship Between The G-function and The Generalized Hypergeometric Function

If the integral converges when evaluated along the second path introduced above, and if no confluent poles appear among the Γ(b_j − s), j = 1, 2, ..., m, then the Meijer G-function can be expressed as a sum of residues in terms of generalized hypergeometric functions _pF_q−1 (Slater's theorem):

For the integral to converge along the second path one must have either p < q, or p = q and |z| < 1, and for the poles to be distinct no pair among the b_j, j = 1, 2, ..., m, may differ by an integer or zero. The asterisks in the relation remind us to ignore the contribution with index j = h as follows: In the product this amounts to replacing Γ(0) with 1, and in the argument of the hypergeometric function, if we recall the meaning of the vector notation,

this amounts to shortening the vector length from q to q−1.

Note that when m = 0, the second path does not contain any pole, and so the integral must vanish identically,

if either p < q, or p = q and |z| < 1.

Similarly, if the integral converges when evaluated along the third path above, and if no confluent poles appear among the Γ(1 − a_k + s), k = 1, 2, ..., n, then the G-function can be expressed as:

For this, either p > q, or p = q and |z| > 1 are required, and no pair among the a_k, k = 1, 2, ..., n, may differ by an integer or zero. For n = 0 one consequently has:

if either p > q, or p = q and |z| > 1.

On the other hand, any generalized hypergeometric function can readily be expressed in terms of the Meijer G-function:

where we have made use of the vector notation:

This holds unless a nonpositive integer value of at least one of its parameters a_p reduces the hypergeometric function to a finite polynomial, in which case the gamma prefactor of either G-function vanishes and the parameter sets of the G-functions violate the requirement a_k − b_j ≠ 1, 2, 3, ... for k = 1, 2, ..., n and j = 1, 2, ..., m from the definition above. Apart from this restriction, the relationship is valid whenever the generalized hypergeometric series _pF_q(z) converges, i. e. for any finite z when p ≤ q, and for |z| < 1 when p = q + 1. In the latter case, the relation with the G-function automatically provides the analytic continuation of _pF_q(z) to |z| ≥ 1 with a branch cut from 1 to ∞ along the real axis. Finally, the relation furnishes a natural extension of the definition of the hypergeometric function to orders p > q + 1. By means of the G-function we can thus solve the generalized hypergeometric differential equation for p > q + 1 as well.