  In this work we address partial wave decompositions of thermal one-point
functions in conformal field theories on $S^1 \times S^{d-1}$. With the help of
Casimir differential equations we develop efficient algorithms to compute the
relevant conformal blocks for an external field of arbitrary spin and with any
spin exchange along the thermal circle, at least in three dimensions. This is
achieved by identifying solutions to the Casimir equations with a special class
of spherical functions in the harmonic analysis of the conformal group. The
resulting blocks are then applied to study the decomposition of one-point
functions of the scalar $\phi^2$ and the stress tensor $T$ for a
three-dimensional free scalar field $\phi$. We are able to read off averaged
OPE coefficients into exchanged fields of high weight and spin for a complete
set of tensor structures. We also extract an asymptotic behaviour of conformal
blocks and use it to analyse the density of heavy-heavy-light OPE coefficients
for spinning operators, comparing it with semi-classical predictions, such as
the dimensions of operators at large charge.
