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 ap and bq 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 ak, k = 1, 2, ..., n, equals one of the bj, 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 ak, k = n + 1, ..., p, equals one of the bj, 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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