  The Cauchy identities play an important role in the theory of symmetric
functions. It is known that Cauchy sums for the $q$-Whittaker and the skew
Schur polynomials produce the same factorized expressions modulo a
$q$-Pochhammer symbol. We consider the sums with restrictions on the length of
the first rows for labels of both polynomials and prove an identity which
relates them. The proof is based on techniques from integrable probability: we
rewrite the identity in terms of two probability measures: the $q$-Whittaker
measure and the periodic Schur measure. The relation follows by comparing their
Fredholm determinant formulas.
