Expressing $\mathrm{B}(\sinh(x), \cosh(x))$ in terms of elementary functions

Question

Is it possible to express:

$\mathrm{B}(\sinh(x), \cosh(x))$

(where $\mathrm{B}$ is the beta function)

In closed form, in terms of elementary functions?

Answer 1

Here is my "NO" (sort-of) answer. Compute the series $$ \mathrm{B}(\sinh x, \cosh x) = \frac{1}{x}-{\frac {1}{6}}x-\frac{1}{12}\,{\pi }^{2}{x}^{2}+ \left( \frac{1}{2}\,\zeta \left( 3 \right) +{\frac {7}{360}} \right) {x}^{3}+O \left( {x}^{4} \right) $$ But $\zeta(3)$ is not known to occur in any elementary function. SO: if this is elementary, it will provide a new expression for $\zeta(3)$.

Answer 2

In terms of closed form, $B(sinh(x),cosh(x))$ is already a closed form. So, the question is "What do you expect ? ". Another closed form is $\Gamma(sinh(x))\Gamma(cosh(x))/\Gamma(cosh(x)+sinh(x))$. I cannot say if it is a better closed form or not. In terms of a finite number of only elementary functions, I think that it is impossible (if an integral is excluded as a kind of closed form).