1
$\begingroup$

Is there any simple proof of this identity

$${_4F_3}[\{\frac{1}{2}+\frac{n}{4},1+\frac{n}{4},-p,p\},\{\frac{1}{2},\frac{3}{2},\frac{1}{2}+\frac{n}{2}\},1]=\frac{\Gamma[\frac{1+n}{2}] (\frac{\cos[\frac{n \pi }{2}] \Gamma[\frac{1}{2}-\frac{n}{2}+p]}{\Gamma[\frac{3}{2}+p]}-\frac{\Gamma[-\frac{1}{2}+p]}{\Gamma[\frac{1+n}{2}+p]})}{4 \sqrt{\pi }}$$

Here $p$ and $n$ are certain positive integers. It seems that the formula is true only for some integers...

$\endgroup$
3
  • 1
    $\begingroup$ (a) What does "a simple proof" mean? (b) Do you know of a complicated proof? $\endgroup$
    – Igor Rivin
    Mar 23, 2015 at 0:54
  • $\begingroup$ The identity isn't true, as you can check with your favourite CAS, at least not for general $n,p$. Did you mean to include some restrictions on what $n$ and $p$ are? $\endgroup$ Mar 23, 2015 at 9:27
  • 1
    $\begingroup$ i assume that $p$ and $n$ are non-negative integers, right ? Using the identity written here: functions.wolfram.com/HypergeometricFunctions/Hypergeometric4F3/… (and ignoring that it claims to be true only for - in the OP's notation - $-n/2-1$ a positive integer) Mathematica gives the formula of the OP. No proof is provided, unfortunately. $\endgroup$ Mar 23, 2015 at 17:22

0

Browse other questions tagged or ask your own question.