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It is known that the moments of the maximum value of a one-dimensionalconditional Brownian motion, the three-dimensional Bessel bridge with duration1 started from the origin, are expressed using the Riemann zeta function. Weconsider a system of two Bessel bridges, in which noncolliding condition isimposed. We show that the moments of the maximum value is then expressed usingthe double Dirichlet series, or using the integrals of products of the Jacobitheta functions and its derivatives. Since the present system will be providedas a diffusion scaling limit of a version of vicious walker model, the ensembleof 2-watermelons with a wall, the dominant terms in long-time asymptotics ofmoments of height of 2-watermelons are completely determined. For the height of2-watermelons with a wall, the average value was recently studied by Fulmek bya method of enumerative combinatorics.

Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function