Convolutions of Goss and Pellarin $L$-series

We establish special value results of convolutions of Goss and Pellarin $L$-series attached to Drinfeld modules that take values in Tate algebras. Applying the class module formula of Demeslay to certain rigid analytic twists of one Drinfeld module by another, we extend the special value formula for the Pellarin $L$-function associated to the Carlitz module and the Anderson-Thakur function to Drinfeld modules of arbitrary rank and their rigid analytic trivializations. By way of the theory of Schur polynomials these identities take the form of specializations of convolutions of Rankin-Selberg type. These convolution $L$-series are also identified with covolumes of Stark units.