Notation for hypergeometric functions

I know about the usual notation for referring to hypergeometric functions, such as $_pF_q$.

However, I've found something like $\boxed{_p\tilde{F}_q}$ as a part of the expression for calculating the raw moments of a random variable following a doubly noncentral $F$ distribution in this web page: http://mathworld.wolfram.com/NoncentralF-Distribution.html . I had never seen that before.

What does $_p\tilde{F}_q$ stand for?

migrated from mathematica.stackexchange.comSep 15 '17 at 13:31

This question came from our site for users of Wolfram Mathematica.

• It seems that page neglected to define that symbol. Anyway, it is a regularized hypergeometric function. – J. M. is a poor mathematician Sep 14 '17 at 7:28
• @J.M. Yes, they did not define what it is. That is why I asked it here. Would you like to post your contribution as an answer? – Vicent Sep 14 '17 at 7:30
• This is a very simple question, but I was not able to find the answer in any other way, so I think it may be useful for other people having the same doubt that me in the future. – Vicent Sep 14 '17 at 7:33
• Is that notation used anywhere other than Wolfram/Mathematica? – GEdgar Sep 15 '17 at 13:35
• @GEdgar As far as I know, yes, but I didn't know it when I asked this question. Anyway, I think it is a simple but interesting question. I suggested moving it to this site. – Vicent Sep 15 '17 at 13:37

$${}_p \tilde{F}_q\left({{a_1,\dots,a_p}\atop{b_1,\dots,b_q}}\middle|x\right)=\frac1{\prod\limits_{k=1}^q\Gamma(b_k)}{}_p F_q\left({{a_1,\dots,a_p}\atop{b_1,\dots,b_q}}\middle|x\right)$$
In the DLMF, the term "scaled" or "Olver's hypergeometric function" is used, with the notation ${}_p\mathbf{F}_q\left(\mathbf a;\mathbf b;z\right)$.