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(bj − s), j = 1, 2, ..., m, then the Meijer G-function can be expressed as a sum of residues in terms of generalized hypergeometric functions pFq−1 (Slater's theorem):

G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) = \sum_{h=1}^m \frac{\prod_{j=1}^m \Gamma(b_j - b_h)^* \prod_{j=1}^n \Gamma(1+b_h - a_j) \; z^{b_h}} {\prod_{j=m+1}^q \Gamma(1+b_h - b_j) \prod_{j=n+1}^p \Gamma(a_j - b_h)} \times
\times \; _{p}F_{q-1} \!\left( \left. \begin{matrix} 1+b_h - \mathbf{a_p} \\ (1+b_h - \mathbf{b_q})^* \end{matrix} \; \right| \, (-1)^{p-m-n} \; z \right) .

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 bj, 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,

1 + b_h - \mathbf{b_q} = (1 + b_h - b_1), \,\dots, \,(1 + b_h - b_j), \,\dots, \,(1 + b_h - b_q),

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,

G_{p,q}^{\,0,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) = 0,

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 − ak + s), k = 1, 2, ..., n, then the G-function can be expressed as:

G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) = \sum_{h=1}^n \frac{\prod_{j=1}^n \Gamma(a_h - a_j)^* \prod_{j=1}^m \Gamma(1-a_h + b_j) \; z^{a_h-1}} {\prod_{j=n+1}^p \Gamma(1-a_h + a_j) \prod_{j=m+1}^q \Gamma(a_h - b_j)} \times
\times \; _{q}F_{p-1} \!\left( \left. \begin{matrix} 1-a_h + \mathbf{b_q} \\ (1-a_h + \mathbf{a_p})^* \end{matrix} \; \right| \, (-1)^{q-m-n} z^{-1} \right) .

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

G_{p,q}^{\,m,0} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) = 0,

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:

\; _{p}F_{q} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) = \frac {\Gamma(\mathbf{b_q})} {\Gamma(\mathbf{a_p})} \; G_{p,\,q+1}^{\,1,\,p} \!\left( \left. \begin{matrix} 1-\mathbf{a_p} \\ 0,1 - \mathbf{b_q} \end{matrix} \; \right| \, -z \right) = \frac {\Gamma(\mathbf{b_q})} {\Gamma(\mathbf{a_p})} \; G_{q+1,\,p}^{\,p,\,1} \!\left( \left. \begin{matrix} 1,\mathbf{b_q} \\ \mathbf{a_p} \end{matrix} \; \right| \, -z^{-1} \right),

where we have made use of the vector notation:

\Gamma(\mathbf{a_p}) = \prod_{j=1}^p \Gamma(a_j).

This holds unless a nonpositive integer value of at least one of its parameters ap 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 ak − bj ≠ 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 pFq(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 pFq(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.

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