Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision Last revision Both sides next revision | ||
|
generalized_hypergeometric_function [2015/12/16 15:51] nikolaj |
generalized_hypergeometric_function [2016/03/20 00:14] nikolaj |
||
|---|---|---|---|
| Line 15: | Line 15: | ||
| Above we used the [[rising factorial]] | Above we used the [[rising factorial]] | ||
| - | |||
| $x^{\overline{n}} := x^{\overline{n},1}$, | $x^{\overline{n}} := x^{\overline{n},1}$, | ||
| - | |||
| where | where | ||
| Line 55: | Line 53: | ||
| $\dfrac{c_{n+1}}{c_n} = \dfrac{1}{n+1}$ | $\dfrac{c_{n+1}}{c_n} = \dfrac{1}{n+1}$ | ||
| - | With the polynomial $q(n) := n+1$. This means $c_n = \frac{1}{\prod_{k=1}^n q(k)} c_0 = \dfrac{1}{n!}$ and hence $f(x) = \sum_{n=0}^\infty \dfrac{1}{n!} x^n$. | + | With the polynomial $q(n) := n+1$, this means $c_n = \frac{1}{\prod_{k=1}^n q(k)} c_0 = \dfrac{1}{n!}$ and hence $f(x) = \sum_{n=0}^\infty \dfrac{1}{n!} x^n$. |
| Such an approach to solve a differential equation will often look like this. A whole lot of function have series coefficients $c_n$, such that | Such an approach to solve a differential equation will often look like this. A whole lot of function have series coefficients $c_n$, such that | ||
