Graduate Is this a known result? (hypergeometric function at special values)

[nrqed]

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?

 

[jasonRF]

It is a result that follows very easily using identities available at the DLMF:
https://dlmf.nist.gov/

The starting point is 15.4.24 (http://dlmf.nist.gov/15.4.E24)
$_2F_1\left(-N,b;c;1\right)=\frac{{\left(c-b\right)_{N}}}{{\left(c\right)_{N}}}$
where the Pochhammer symbols are $(a)_N = \Gamma(a+N)/\Gamma(a)$. You then use a few identities for those symbols to get your result. I used 5.2.7, 5.2.6, 5.2.5, 5.5.5 and 5.4.6 (see https://dlmf.nist.gov/5) .

jason

 

[nrqed]

It is a result that follows very easily using identities available at the DLMF:
https://dlmf.nist.gov/

The starting point is 15.4.24 (http://dlmf.nist.gov/15.4.E24)
$_2F_1\left(-N,b;c;1\right)=\frac{{\left(c-b\right)_{N}}}{{\left(c\right)_{N}}}$
where the Pochhammer symbols are $(a)_N = \Gamma(a+N)/\Gamma(a)$. You then use a few identities for those symbols to get your result. I used 5.2.7, 5.2.6, 5.2.5, 5.5.5 and 5.4.6 (see https://dlmf.nist.gov/5) .

jason

Wow,

Thank you so much! I had seen the equation given in 15.4.24 but I had not realized that one could define $(-n)_n$ as is given in 5.2.7.

Thanks!

 

[nrqed]

It is a result that follows very easily using identities available at the DLMF:
https://dlmf.nist.gov/

The starting point is 15.4.24 (http://dlmf.nist.gov/15.4.E24)
$_2F_1\left(-N,b;c;1\right)=\frac{{\left(c-b\right)_{N}}}{{\left(c\right)_{N}}}$
where the Pochhammer symbols are $(a)_N = \Gamma(a+N)/\Gamma(a)$. You then use a few identities for those symbols to get your result. I used 5.2.7, 5.2.6, 5.2.5, 5.5.5 and 5.4.6 (see https://dlmf.nist.gov/5) .

jason

Hello Jason,

I am now obtaining many conjectured identities that I cannot find in the literature. These involve generalized hypergeometric functions $\,_pF_q$ with $p > 2$ and $q > 1$. For example I get a relation between $\,_4F_3$ and $\,_3 F_2$ for a certain choice of parameters (my relation involves three different positive integers).

Do you know if there are tables of known relations between generalized hypergeometric functions? I have seen a couple on the Wikipedia page but that's all.

Thank you!

 

[jasonRF]

Besides DLMF, I would check Wolfram Mathworld online. In terms of books, Erdelyi's "Higher Transcendental Functions" volumes 1-3 are legally available online:
http://numerical.recipes/oldverswitcher.html
They probably have a few relations.

I suspect that you don't have access to a library these days, but if you do then there may be other books to look at. For example, Yudell Luke's books "The special functions and their approximations" vol 1-2, and "mathematical functions and their approximations" should have some interesting information.

jason

 

[nrqed]

Besides DLMF, I would check Wolfram Mathworld online. In terms of books, Erdelyi's "Higher Transcendental Functions" volumes 1-3 are legally available online:
http://numerical.recipes/oldverswitcher.html
They probably have a few relations.

I suspect that you don't have access to a library these days, but if you do then there may be other books to look at. For example, Yudell Luke's books "The special functions and their approximations" vol 1-2, and "mathematical functions and their approximations" should have some interesting information.

jason

Thank you again, Jason. I appreciate it.