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How to determinate the linearly independence between some special functions defined by ODE? For example:

  1. ${}_1F_1(a;b;x)$ , $x^{1-b}{}_1F_1(a-b+1;2-b;x)$ when $b$ is integer

  2. ${}_2F_1(a,b;c;x)$ , $x^{1-c}{}_2F_1(a-c+1,b-c+1;2-c;x)$ when $c$ is integer

  3. HeunC$(\alpha,\beta,\gamma,\delta,\eta;x)$ , $x^{-\beta}$HeunC$(\alpha,-\beta,\gamma,\delta,\eta;x)$ when $\beta$ is integer

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    ${}_mF_n$ is hypergeometric function?2012-05-17
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    You can write a hypergeometric function like this: $\displaystyle F\left( \left. { a_1,a_2,\dots,a_m \atop b_1,b_2,\dots,b_n } \right| z \right)$, or $F(a_1,a_2,\dots,a_m;b_1,b_2,\dots,b_n;z)$, so you might lose a semicolon in $F(a;b;x)$ and $F(a,b;c;x)$.2012-05-17
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    Please note that this question base on the information of http://eqworld.ipmnet.ru/en/solutions/ode/ode0210.pdf , http://eqworld.ipmnet.ru/en/solutions/ode/ode0222.pdf and http://www.maplesoft.com/support/help/Maple/view.aspx?path=HeunC , so this question should be correctly asked.2012-06-15

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