  In this note, we study the high-temperature convolution introduced in Ref.\
\cite{mergny_cconv}, between two symmetric Bernoulli distributions. We give an
analytical expression for both the Stieltjes transform and the density. This
result provides the first non-trivial expression for the high-temperature
convolution of two distributions and gives a new family of densities,
interpolating between the centered binomial distribution with number of trials
$n{=}2$ and probability of success $p{=}1/2$, and the centered and re-scaled
arcsine law.
