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Let $_2 F_1 (a,b,c,z)$ be the hypergeometric function. As a result of some integration, I obtained the following expression $$ f(x) = \frac{\Gamma(2k)\Gamma(2m)}{\Gamma(m)^2\Gamma(k)^2} \frac{\Gamma(m+k)^2}{\Gamma(2k+2m)} \frac{_2 F_1 (2k,k+m,2k+2m,1-\frac{1}{x})}{x^{k+1}}. $$ The parameters $m$ and $k$ are positive numbers. I will have to use this result in some other integrals, namely the average bit error rate and the capacity integrals: $$ P(E) = \int_0^{+\infty} \alpha \mathrm{erfc}(\sqrt{\beta x}) f(x) dx, $$ and $$ C = \frac{1}{\ln(2)}\int_0^{+\infty} \log(1+x) f(x) dx. $$

I thought of using the Meijer G function to obtained some closed-form expression but I need to transform the argument of the hypergeometric function in $f(x)$ to be linear. Is there any relation that could help me ? and is there a relation that can simplify the product of such hypergometric function and rational function $\frac{1}{x^{k+1}}$ ?

Thank you.

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  • $\begingroup$ I found these relations for the $\mathrm{erfc(.)}$ and $\log(.)$ function with the Meijer G function. $$ \mathrm{erfc}(\sqrt{x}) = \frac{1}{\sqrt{\pi}} G_{12}^{20} \left( x \left| \begin{smallmatrix} 1 \\ 0, \frac{1}{2} \end{smallmatrix} \right.\right) $$ and $$ \log(1+x) = G_{12}^{20} \left( x \left| \begin{smallmatrix} 1,1 \\ 1,0 \end{smallmatrix} \right.\right) $$ $\endgroup$ – قيس بن فرج Sep 10 '13 at 22:58
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Here is a transformation for the $_2F_1 (a,b,c,z)$ part, that results in a form where the argument is linear in $x.\,$ With $z=1-\frac{1}{x}$ we have $\frac{z}{z-1} = 1-x$, and $1-z = \frac{1}{x}$. Therefore the first transformation from http://dlmf.nist.gov/15.8.E1 gives $$_2F_1 \left(a,b,c,1-\frac{1}{x}\right) = \, _2F_1(a,b,c,z)\\ = \,(1-z)^a\, _2F_1 \left(a,c-b,c,\frac{z}{z-1}\right)\\ = x^{-a} \,_2F_1 (a,c-b,c,1-x) $$ And for the second formula: $$_2F_1 \left(a,b,c,1-\frac{1}{x}\right) = x^{-b} \,_2F_1 (c-a,b,c,1-x) $$ You have to attention to range restrictions for $x$, but I think they are valid for $x>0$.

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I finally got what I was looking for. In line 21 of Table 8.4.49 in "Integrals and Series, vol. 3" by A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, we can find a relation between the $_2F_1$ hypergeometric function and the Meijer G function as: $$ _2F_1 \left(a,b;c;1-\frac{1}{x}\right) = B_{11} G_{22}^{22} \left( x \left| \begin{array}{c} 1,1+a+b-c \\ a,b \end{array} \right. \right). $$ By using also a Meijer G representation for $\mathrm{erfc}$ or $\log$, we can solve the above integrals mentioned in the question.

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  • $\begingroup$ $B_{11}$ is defined as: $$ B_{11} = \frac{ \Gamma(c) }{ \Gamma(a) \Gamma(b) \Gamma(c-a) \Gamma(c-b) } $$ $\endgroup$ – قيس بن فرج Sep 24 '13 at 19:54

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