On a problem due to Glasser on analytically tractable moments

Glasser, in 2011, introduced a remarkable integral identity of physical interest and suggested that the evaluation <?CDATA ∫₀1/√2 k \, K²(k) \, dk = π G/4?> ∫ 0 1 / 2 k K 2 ( k ) d k = π G 4 provides the unique analytically tractable moment of K 2 on a sub-unit interval, where K denotes the complete elliptic integral of the first kind, and where <?CDATA G = 1/1² - 1/3² + 1/5² - ⋯?> G = 1 1 2 − 1 3 2 + 1 5 2 − ⋯ denotes Catalan’s constant. We show how a case of Clausen’s product identity related to Ramanujan’s series for <?CDATA 1/π?> 1 π may be applied, via an integration argument derived from our past work in fractional analysis and Fourier–Legendre theory, to show how higher moments of K 2 on the same sub-unit interval may be evaluated analytically in terms of the Γ-function. This and Glasser’s moment formula are motivated by how closely related moment formulas for powers of K arise in the study of Feynman diagrams.