Closed form for the lacunary series g(x) in the Lacunary Cauchy identity

Determine a closed-form expression for the lacunary series g(x) = ∑_{n ≥ 0} p^{2n} x^{p^{n}}, which appears in the Lacunary Cauchy identity for p-Schur functions associated with the group algebras KG_r and KSG_r when p is an odd prime. The goal is to express g(x) in a closed analytic form (rather than as a formal series), potentially as an elementary function or a standard special function, under the absolute convergence regime described in the paper.

Background

In Section 4, the authors introduce p-Schur functions and p-Kostka numbers for hook partitions associated with the groups G_r and SG_r. They derive a Lacunary Cauchy identity that involves a generating function g(x), defined by the series g(x) = ∑_{n ≥ 0} p^{2n} x^{p^{n}}. This g(x) plays a central role in expressing sums of products of p-Schur functions across shapes and degrees.

The authors note that g(x) is a lacunary series and that it converges absolutely, but they do not provide or identify a closed-form expression. Establishing such a closed form would clarify analytic properties of the identity and potentially connect it to known generating functions or special functions in combinatorial representation theory.