Let $M_d$ be the maximum of a standard Bessel bridge of dimension $d$. A series formula for $P(M_d \le a)$ due to Gikhman and Kiefer for $d = 1,2, \ldots$ is shown to be valid for all real $d \gt 0$. Various other characterizations of the distribution of $M_d$ are given, including formulae for its Mellin transform, which is an entire function. The asymptotic distribution of $M_d$ is described both as $d$ tends to infinity and as $d$ tends to zero.
"The Law of the Maximum of a Bessel Bridge." Electron. J. Probab. 4 1 - 35, 1999. https://doi.org/10.1214/EJP.v4-52