D-finite symmetric functions

The fruitful relation between the theory of symmetric functions and that of D-finite power series was first introduced by Goulden and Jackson in 1980, and later extended by Gessel, who stated two important results that provide closure properties of D-finite symmetric series under the scalar and inner products. These products are very important from the computational and combinatorial points of view, as a prime tool for coefficient extraction in symmetric series. Gessel presented some enumerative problems that can be better understood using his results on D-finiteness. We connect these notions with Scharf, Thibon and Wybourne's results on reduced Kronecker products. Also, we extend the necessary conditions on one of Gessel's theorems and determine some consequences in Young Tableaux enumeration.