Meijer G-function - Basic Properties of The G-function

Basic Properties of The G-function

As can be seen from the definition of the G-function, if equal parameters appear among the a_p and b_q determining the factors in the numerator and the denominator of the integrand, the fraction can be simplified, and the order of the function thereby be reduced. Whether the order m or n will decrease depends of the particular position of the parameters in question. Thus, if one of the a_k, k = 1, 2, ..., n, equals one of the b_j, j = m + 1, ..., q, the G-function lowers its orders p, q and n:

$G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} a_1, a_2, \dots, a_p \\ b_1, \dots, b_{q-1}, a_1 \end{matrix} \; \right| \, z \right) = G_{p-1,\,q-1}^{\,m,\,n-1} \!\left( \left. \begin{matrix} a_2, \dots, a_p \\ b_1, \dots, b_{q-1} \end{matrix} \; \right| \, z \right), \quad n,p,q \geq 1.$

For the same reason, if one of the a_k, k = n + 1, ..., p, equals one of the b_j, j = 1, 2, ..., m, then the G-function lowers its orders p, q and m:

$G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} a_1, \dots, a_{p-1}, b_1 \\ b_1, b_2, \dots, b_q \end{matrix} \; \right| \, z \right) = G_{p-1,\,q-1}^{\,m-1,\,n} \!\left( \left. \begin{matrix} a_1, \dots, a_{p-1} \\ b_2, \dots, b_q \end{matrix} \; \right| \, z \right), \quad m,p,q \geq 1.$

Starting from the definition, it is also possible to derive the following properties:

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

$G_{p+2,\,q}^{\,m,\,n+1} \!\left( \left. \begin{matrix} \alpha, \mathbf{a_p}, \alpha' \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) = (-1)^{\alpha'-\alpha} \; G_{p+2,\,q}^{\,m,\,n+1} \!\left( \left. \begin{matrix} \alpha', \mathbf{a_p}, \alpha \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right), \quad n \leq p, \; \alpha'-\alpha \in \mathbb{Z},$

$G_{p,\,q+2}^{\,m+1,\,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \beta, \mathbf{b_q}, \beta' \end{matrix} \; \right| \, z \right) = (-1)^{\beta'-\beta} \; G_{p,\,q+2}^{\,m+1,\,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \beta', \mathbf{b_q}, \beta \end{matrix} \; \right| \, z \right), \quad m \leq q, \; \beta'-\beta \in \mathbb{Z},$

$G_{p+1,\,q+1}^{\,m,\,n+1} \!\left( \left. \begin{matrix} \alpha, \mathbf{a_p} \\ \mathbf{b_q}, \beta \end{matrix} \; \right| \, z \right) = (-1)^{\beta-\alpha} \; G_{p+1,\,q+1}^{\,m+1,\,n} \!\left( \left. \begin{matrix} \mathbf{a_p}, \alpha \\ \beta, \mathbf{b_q} \end{matrix} \; \right| \, z \right), \quad m \leq q, \; \beta-\alpha = 0,1,2,\dots,$

$G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) = G_{q,p}^{\,n,m} \!\left( \left. \begin{matrix} 1-\mathbf{b_q} \\ 1-\mathbf{a_p} \end{matrix} \; \right| \, z^{-1} \right),$

$G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} \mathbf{a_p} \\ \mathbf{b_q} \end{matrix} \; \right| \, z \right) = \frac{h^{1+\nu+(p-q)/2}} {(2 \pi)^{(h-1) \delta}} \; G_{h p, \, h q}^{\, h m, \, h n} \!\left( \left. \begin{matrix} a_1/h, \dots, (a_1+h-1)/h, \dots, a_p/h, \dots, (a_p+h-1)/h \\ b_1/h, \dots, (b_1+h-1)/h, \dots, b_q/h, \dots, (b_q+h-1)/h \end{matrix} \; \right| \, \frac{z^h} {h^{h(q-p)}} \right), \quad h \in \mathbb{N}.$

The abbreviations ν and δ were introduced in the definition of the G-function above.

Read more about this topic: Meijer G-function

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