  The generating function of Stieltjes-Carlitz polynomials is a solution of
Heun's differential equation and using this relation Carlitz was the first to
get exact closed forms for some Heun functions. Similarly the associated
Stieltjes-Carlitz polynomials lead to a new differential equation which we call
associated Heun. Thanks to the link with orthogonal polynomials we are able to
deduce two integral relations connecting associated Heun functions with
different parameters and to exhibit the set of associated Heun functions which
generalize Carlitz's. Part of these results were used by the author to derive
the Stieltjes transform of the measure of orthogonality for the associated
Stieltjes-Carlitz polynomials using asymptotic analysis; here we present a new
derivation of this result.
