On a problem due to Glasser on analytically tractable moments

Abstract

Abstract Glasser, in 2011, introduced a remarkable integral identity of physical interest and {\color{blue} suggested} that the evaluation $\int_{0}^{1/\sqrt{2}} k \, K^{2}(k) \, dk = \frac{\pi 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 $G = \frac{1}{1^2} - \frac{1}{3^2} + \frac{1}{5^2} - \cdots$ denotes Catalan's constant. We show how a case of Clausen's product identity related to Ramanujan's series for $\frac{1}{\pi}$ 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 $\Gamma$-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.