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.com Sep 15 '17 at 13:31

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

  • 1
    $\begingroup$ It seems that page neglected to define that symbol. Anyway, it is a regularized hypergeometric function. $\endgroup$ – J. M. is a poor mathematician Sep 14 '17 at 7:28
  • $\begingroup$ @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? $\endgroup$ – Vicent Sep 14 '17 at 7:30
  • $\begingroup$ 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. $\endgroup$ – Vicent Sep 14 '17 at 7:33
  • $\begingroup$ Is that notation used anywhere other than Wolfram/Mathematica? $\endgroup$ – GEdgar Sep 15 '17 at 13:35
  • $\begingroup$ @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. $\endgroup$ – Vicent Sep 15 '17 at 13:37

As I noted in the comments, the function they neglected to define in the MathWorld link is what they term the "regularized hypergeometric function",

$${}_p \tilde{F}_q\left({{a_1,\dots,a_p}\atop{b_1,\dots,b_q}}\middle|x\right)=\frac1{\prod\limits_{k=1}^q\Gamma(b_k)}{}_p F_q\left({{a_1,\dots,a_p}\atop{b_1,\dots,b_q}}\middle|x\right)$$

i.e., the usual hypergeometric function divided by the product of gamma functions evaluated at the denominator parameters.

In the DLMF, the term "scaled" or "Olver's hypergeometric function" is used, with the notation ${}_p\mathbf{F}_q\left(\mathbf a;\mathbf b;z\right)$.


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