Let $X_t$ be standard Brownian motion in $R^d$ starting at fixed $X_0$, and let $T$ be the hitting time for a sphere or concentric spherical shell. Explicit formulas, in terms of a natural Laplace-Gegenbauer transform, are obtained for the joint distribution of $T$ and $X_T$.
"Hitting Spheres with Brownian Motion." Ann. Probab. 8 (1) 164 - 169, February, 1980. https://doi.org/10.1214/aop/1176994833