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normalized_fox-wright_function [2015/12/16 14:00] nikolaj |
normalized_fox-wright_function [2015/12/25 17:19] nikolaj |
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| ==== Function ==== | ==== Function ==== | ||
| | @#FF9944: definition | @#FF9944: $??$ | | | @#FF9944: definition | @#FF9944: $??$ | | ||
| - | | @#FF9944: definition | @#FF9944: ${}_p\Psi_q^*[\langle a_1, A_1\rangle,…,\langle a_p, A_p\rangle; \langle b_1, B_1\rangle,…,\langle a_q, A_q\rangle](z):= \sum_{n=0}^\infty c_n \dfrac{z^n}{n!}$ | | + | | @#FF9944: definition | @#FF9944: ${}_p\Psi_q^*[\langle a_1, A_1\rangle,…,\langle a_p, A_p\rangle; \langle b_1, B_1\rangle,…,\langle a_q, A_q\rangle](z):= \sum_{n=0}^\infty c_n z^n$ | |
| - | | @#BBDDEE: with | @#BBDDEE: $c_n = \dfrac{\prod_{m=1}^p \Gamma(a_m+A_m\cdot{n})\, /\, \Gamma(a_m)}{\prod_{j=1}^q \Gamma(b_j+B_m\cdot{n})\, /\, \Gamma(b_j)}$ | | + | | @#BBDDEE: with | @#BBDDEE: $c_n = \dfrac{1}{n!}\dfrac{\prod_{m=1}^p \Gamma(a_m+A_m\cdot{n})\, /\, \Gamma(a_m)}{\prod_{j=1}^q \Gamma(b_j+B_m\cdot{n})\, /\, \Gamma(b_j)}$ | |
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| So an expansion coefficient of ${}_p\Psi_q^*$ is a fraction of products with factors $\Gamma(a_m+A_m\cdot{n})\,/\,\Gamma(a_m)$, which are essentially also a product consisting of equidistant factors. | So an expansion coefficient of ${}_p\Psi_q^*$ is a fraction of products with factors $\Gamma(a_m+A_m\cdot{n})\,/\,\Gamma(a_m)$, which are essentially also a product consisting of equidistant factors. | ||
| - | E.g. at $n=5$, the context $\langle a_1,A_1\rangle=\langle 5,1\rangle$ gives a multiplicative of contribution | + | E.g. at $n=5$, the context $\langle a_1,A_1\rangle=\langle 5,1\rangle$ gives a multiplicative contribution |
| $\dfrac{\Gamma(4+5)}{\Gamma(4)} = 4 \cdot 5\cdot 6\cdot 7\cdot 8$. | $\dfrac{\Gamma(4+5)}{\Gamma(4)} = 4 \cdot 5\cdot 6\cdot 7\cdot 8$. | ||
