If the Gauss hypergeometric function 
 $F(1, 3/2, 5/2; z^2) = 3 [\tanh^{-1} (z) –z]/z^3$  

what is the corresponding result for $F(1,15/8,23/8;z^8) $?



hyperbolic functions and Gauss hypergeometric series
There is a special form for hypergeometric functions of the form
$$ F(1,\beta,1+\beta,z) = \beta z^{-\beta}B(z,\beta,0) $$
when $\beta = p/q$ is a rational number (where $B$ is the incomplete beta function).


Write
$$ \frac{z^\beta}{\beta}F(1,\beta,1+\beta,z) = \sum\_{k\geq0} \frac{z^{\beta+k}}{\beta+k} = q\sum\_{k\geq0}\frac{(z^{1/q})^{p+k q}}{p+k q} = q\sum\_{k\geq0}\frac{(z^{1/q})^k}{k}[k\bmod q=p]. $$


The point is that you can use
$$ \log(1-x) = -\sum\_{k\geq1} \frac{x^k}{k} $$
and
$$ \sum\_{0\leq l < q} \zeta^{a l} = q[a\bmod q=0], \qquad \zeta=e^{2\pi i/q}, \quad a\in\mathbb{Z} $$
to write
$$ \sum\_{0\leq l < q} -\zeta^{-l p}\log(1-\zeta^l x) = \sum\_{k\geq1}\sum\_{0\leq l < q} \frac{\zeta^{-lp+lk}x^k}{k} = \sum\_{k\geq1} \frac{x^k}{k}q[k\bmod q=p]. $$


So the closed form is
$$ \frac{z^\beta}{\beta}F(1,\beta,1+\beta,z) = \sum\_{0\leq l < q}-\zeta^{-lp}\log(1-\zeta^l z^{1/q}), \qquad \beta=\frac pq, \quad p,q\in\mathbb{Z}. $$


I would be interested to know how this form should be generalized to irrational $\beta$.


