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Through some calculations in a graph counting problem, I have checked that for many values of N (a positive integer), the following is true:
$$ \,_2F_1 [\frac{1}{2},-N;-N+ \frac{1}{2} ; z=1] = \frac{4^N (N!)^2}{(2N)!} $$
I would like to prove that this correct for arbitrary N, but I cannot find anywhere an expression for this particular combination of parameters (note that ##c-a-b=0##, which is very unfortunate as it prevents the use of one well known identity). I have checked Abramowitz and Stegun and several other sources. No luck. Does anyone know if this is true or if they have some good source of identities to recommend?
$$ \,_2F_1 [\frac{1}{2},-N;-N+ \frac{1}{2} ; z=1] = \frac{4^N (N!)^2}{(2N)!} $$
I would like to prove that this correct for arbitrary N, but I cannot find anywhere an expression for this particular combination of parameters (note that ##c-a-b=0##, which is very unfortunate as it prevents the use of one well known identity). I have checked Abramowitz and Stegun and several other sources. No luck. Does anyone know if this is true or if they have some good source of identities to recommend?