Finite closed forms for the standardized absolute third moments

Derive a finite closed-form expression for the standardized absolute third central moments \(\beta_{\mathrm{opt}}\) and \(\beta_{\mathrm{an}}\) associated, respectively, with unrestricted affine Brownian approximation and Brownian approximation by a line through the origin.

Background

The paper derives convergent exponential-integral series, together with explicit truncation bounds, for the standardized absolute third central moments $\beta_A$ of the two Brownian affine-approximation lifetimes. These formulas yield highly accurate numerical values for both the unrestricted affine and anchored models.

Despite obtaining exact infinite-series representations, the paper does not establish a finite closed form for either constant. Thus, determining whether such finite expressions exist, and deriving them if they do, remains unresolved within the treatment presented.