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Is it possible to express the Clausen Hypergeometric Function $_3F_2(a,a,b;p,p;x)$ (the first two parameters and the last two are identical) in terms of the Gauss Hypergeometric Function $_2F_1()$ and Gamma function, with transformed arguments?

The reason that I would really like to do this, is because the variable $x$ is outside the range $0 \le x< 1$ in which the simple series expansion in terms of Pochhammer symbols is applicable. For the $_2F_1()$ function, I have a set of expansions with transformations for $x-$ values outside the range $0 \le x < 1$, following the paper "Computing the Hypergeometric Function" by Robert C. Forrey, Journal of Computational Physics, Vol. 137, #79, pp 79-100

Thanks

Sharat V Chandrasekhar
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  • If you match up the Mellin transforms for a single function this is equivalent to finding coefficients $\alpha,\beta,\gamma$ such that $$ \frac{\Gamma(p)\Gamma(p)\Gamma(a-s)\Gamma(a-s)\Gamma(b-a)}{\Gamma(a)\Gamma(a)\Gamma(b)\Gamma(p-s)\Gamma(p-s)}\Gamma(s) = K\frac{\Gamma(\gamma)\Gamma(\alpha-s)\Gamma(\beta-s)}{\Gamma(\alpha)\Gamma(\beta)\Gamma(\gamma-s)}\Gamma(s) $$ where $K$ is some other constant (probably in terms of the Gamma function). – Benedict W. J. Irwin Feb 18 '20 at 16:00
  • I'm afraid that I don't quite follow the logic in how this relates to the $_3F_2$ function. I'm certainly missing some crucial detail here. – Sharat V Chandrasekhar Feb 18 '20 at 16:23
  • In my problem, $x$ is real-valued. However, the hypergeometric function $_2F_1$ (and quite possibly $_3F_2$) is complex-valued for $x>1$. – Sharat V Chandrasekhar Feb 18 '20 at 16:25

1 Answers1

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I need some help with a potential answer to my question:

Digging through Wolfram, I found the following:

enter image description here where I'm assuming that the $\epsilon^{(2)}$ terms are some sort of series expansion coefficients of the $_3F_2$ function based on the parameters, and not the argument $z$. The site does not describe them in any detail. Any help here would be deeply appreciated.

Thanks

There is a result due to Choi and Hasanov that states

enter image description here

What is extremely unfortunate however, is that in my case, the element $\alpha_p < 0$ rendering this result inapplicable. Otherwise, this would've solved my problem, because it specifically reduces $_3F_2()$ to a summation of $_2F_1()$ which is exactly what I was seeking.

Sharat V Chandrasekhar
  • 1,168