# Differences

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generalized_hypergeometric_function [2016/03/20 00:14] nikolaj |
generalized_hypergeometric_function [2018/02/15 18:59] (current) nikolaj |
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$\dfrac{c_{n+1}}{c_n} = \dfrac{1}{n+1}$ | $\dfrac{c_{n+1}}{c_n} = \dfrac{1}{n+1}$ | ||

- | 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 $c_n = \frac{1}{\prod_{k=0}^{n-1} q(k)} c_0 = \dfrac{1}{n!}$ 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 | ||

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$\dfrac{c_{n+1}}{c_n} = \dfrac{p(n)}{q(n)}$ | $\dfrac{c_{n+1}}{c_n} = \dfrac{p(n)}{q(n)}$ | ||

- | where p and q are some polynomials. Any (arbitrary product of) polynomials in n can be written as a product of terms $(a_i-n)$. So define the generalized hypergeometric function | + | where p and q are some polynomials. Any (arbitrary product of) polynomials of an integer n can be written as a product of terms $(a_i-n)$. So define the generalized hypergeometric function |

${}_pF_q[a_1,…,a_p; b_1,…,b_q](z) :=1 + \dfrac{a_1\dots a_p}{b_1\dots b_q}\dfrac{z}{1!} + \dfrac{a_1(a_1+1)\dots a_p(a_p+1)}{b_1(b_1+1)\dots b_q(b_q+1)}\dfrac{z^2}{2!}+\dots$ | ${}_pF_q[a_1,…,a_p; b_1,…,b_q](z) :=1 + \dfrac{a_1\dots a_p}{b_1\dots b_q}\dfrac{z}{1!} + \dfrac{a_1(a_1+1)\dots a_p(a_p+1)}{b_1(b_1+1)\dots b_q(b_q+1)}\dfrac{z^2}{2!}+\dots$ |