Meijer G-function - Definite Integrals Involving The G-function

Definite Integrals Involving The G-function

Among definite integrals involving an arbitrary G-function one has:

\int_0^{\infty} x^{s - 1} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, \eta x \right) dx = \frac{\eta^{-s} \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)}.

Note that the restrictions under which this integral exists have been omitted here. It is, of course, no surprise that the Mellin transform of a G-function should lead back to the integrand appearing in the definition above.

Euler-type integrals for the G-function are given by:

\int_0^1 x^{-\alpha} \; (1-x)^{\alpha - \beta - 1} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z x \right) dx = \Gamma (\alpha - \beta) \; G_{p+1 ,\, q+1}^{\,m ,\, n+1} \!\left( \left. \begin{matrix} \alpha, \mathbf{a_p} \\ \mathbf{b_q}, \beta \end{matrix} \; \right| \, z \right),
\int_1^\infty x^{-\alpha} \; (x-1)^{\alpha - \beta - 1} \; G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z x \right) dx = \Gamma (\alpha - \beta) \; G_{p+1 ,\, q+1}^{\,m+1 ,\, n} \!\left( \left. \begin{matrix} \mathbf{a_p}, \alpha \\ \beta, \mathbf{b_q} \end{matrix} \; \right| \, z \right).

Here too, the restrictions under which the integrals exist have been omitted. Note that, in view of their effect on the G-function, these integrals can be used to define the operation of fractional integration for a fairly large class of functions (Erdélyi–Kober operators).

A result of fundamental importance is that the product of two arbitrary G-functions integrated over the positive real axis can be represented by just another G-function (convolution theorem):

\int_0^{\infty} G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, \eta x \right) G_{\sigma, \tau}^{\,\mu, \nu} \!\left( \left. \begin{matrix} \mathbf{c_{\sigma}} \\ \mathbf{d_\tau} \end{matrix} \; \right| \, \omega x \right) dx =
= \frac{1}{\eta} \; G_{q + \sigma ,\, p + \tau}^{\,n + \mu ,\, m + \nu} \!\left( \left. \begin{matrix} - b_1, \dots, - b_m, \mathbf{c_{\sigma}}, - b_{m+1}, \dots, - b_q \\ - a_1, \dots, -a_n, \mathbf{d_\tau}, - a_{n+1}, \dots, - a_p \end{matrix} \; \right| \, \frac{\omega}{\eta} \right) =
= \frac{1}{\omega} \; G_{p + \tau ,\, q + \sigma}^{\,m + \nu ,\, n + \mu} \!\left( \left. \begin{matrix} a_1, \dots, a_n, -\mathbf{d_\tau}, a_{n+1}, \dots, a_p \\ b_1, \dots, b_m, -\mathbf{c_{\sigma}}, b_{m+1}, \dots, b_q \end{matrix} \; \right| \, \frac{\eta}{\omega} \right) .

Again, the restrictions under which the integral exists have been omitted here. Note how the Mellin transform of the result merely assembles the gamma factors from the Mellin transforms of the two functions in the integrand. Many of the amazing definite integrals listed in tables or produced by computer algebra systems are nothing but special cases of this formula.

The convolution formula can be derived by substituting the defining Mellin–Barnes integral for one of the G-functions, reversing the order of integration, and evaluating the inner Mellin-transform integral. The preceding Euler-type integrals follow analogously.

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