Closed-form resummation of the Lévy-flight series expression for Q(0, C)

Derive a closed-form expression for the first-return cost distribution Q(0, C) by resumming the series Q(0, C) = (1/√(4π C^3)) ∑_{m=1}^{∞} [(2m−2)!]/[(m−1)! (m−1)! 2^{2m−1}] exp(−m^2/(4C)) that arises for the one-dimensional random walk with Lévy-flight jump distribution f(η) = [1/(2 √(4π |η|^3))] exp(−1/(4|η|)) and cost function h(η)=|η|. Establish an exact closed form (if it exists) or a provably convergent alternative representation for Q(0, C) valid for all C>0.

Background

In the subcase x0=0, the authors consider a specific Lévy-flight model with jump distribution f(η) = [1/(2 √(4π |η|^3))] exp(−1/(4|η|)) and linear cost h(η)=|η|. Using their general framework, they obtain the Laplace transform of Q(0, C) and invert it term-by-term to produce an explicit infinite series representation for Q(0, C).

While asymptotic behaviors for small and large C are derived analytically from the series, a closed-form resummation would provide a more compact expression, enable further analytical manipulation, and potentially reveal additional structural properties (e.g., special-function identities or universality features). The authors explicitly state that they were unable to resum this series, leaving the existence and form of a closed expression unresolved.