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