Explicit solution for the Toda-model saddle-point equation

Derive the explicit solution, for general inverse temperature, of the Toda-model equation \(\log(\beta)=-\Gamma'(-s^\star)/\Gamma(-s^\star)\) determining the saddle-point parameter \(s^\star(\beta)\), in order to obtain the average potential energy beyond its low- and high-temperature asymptotic limits.

Background

In the analysis of the Toda model, the canonical partition function is expressed asymptotically in terms of a saddle-point parameter $s^\star(\beta)$. The average potential energy depends on this parameter, which is determined implicitly by an equation involving the Gamma function and its derivative.

The paper states that an explicit solution of this equation is not known in general. Only the asymptotic behavior of the average potential energy in the limits $\beta\to\infty$ and $\beta\to0$ is obtained.