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Does anyone know a closed form for the following evaluations of the Hypergeometric function $$ {}_2F_1(h,-n, 2h; t^{-1}) $$ with $h>0,n\geq 0$ both integers and $0\leq t\leq 1$ a real. For the most part I'm interested in $t=1/2$ case.

Context: I just found these in my conformal field theory study of quantum Hall states. I don't know much about hypergeometric functions.

Alternative Formula: In case it matters, this is the original sum that I found: $$ {}_2F_1(h,-n, 2h; x) = \sum_{k=0}^n \frac{\binom{n}{k}\binom{h+k-1}{k}}{\binom{2h+n-1}{k}}\frac{(-1)^k}{x^k} $$ which Wolfram Mathematica idetified as the hypergeometric function.

Some Numerical Observations: From the numerical checks I have done, it seems when $n$ is odd, then ${}_2F_1(h,-n, 2h; 2)=0$. For the first few even values, it seems like for $n=2m$ the trend is $$ {}_2F_1(h,-2m, 2h; 2) = \prod_{j=0}^{m-1}\frac{2j+1}{2(h+j)+1} $$

Are these observations for $n=$odd and $n=$even true generally? Is there a proof somewhere?

Hamed
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1 Answers1

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The value of ${}_2F_1(h,-n; 2h;2) $ is given here: \begin{equation} {}_2F_1(-n,h; 2h;2)=2^{-n - 1}\frac{n! }{(n/2)!}\frac{(1 + (-1)^n)\Gamma(h + 1/2)}{\Gamma(h + (n + 1)/2)} \end{equation} It vanishes if $n$ is odd. If $n=2m$, \begin{equation} {}_2F_1(-2m,h; 2h;2)=2^{-2m}\frac{(2m)! }{m!}\frac{\Gamma(h + 1/2)}{\Gamma(h + m+1/2)} \end{equation} which can be written as \begin{align} {}_2F_1(-2m,h; 2h;2)&=2^{-2m}\frac{2^mm!1.3.5\cdots(2m-1)}{m!}\frac{\Gamma(h + 1/2)}{\Gamma(h + 1/2).\left( h+3/2).(h+5/2)\cdots\left( h+m+1/2) \right) \right)}\\ &= \prod_{j=0}^{m-1}\frac{2j+1}{2(h+j)+1} \end{align} as proposed.

For general values of the argument, we can use the representation in terms of the Gegenbauer polynomials given here \begin{equation} {}_2F_1(-n,h; 2h;z)=\frac{2^{-2 n}n! z^n}{ \left( h + 1/2 \right)_n} C_n^{1/2-h-n}\left( 1 - \frac{2}{z} \right) \end{equation}

Paul Enta
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