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 ∫ 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...

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Veröffentlicht in:Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2024-03, Vol.57 (12), p.12
1. Verfasser: Campbell, John M
Format: Artikel
Sprache:eng
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Zusammenfassung:Glasser, in 2011, introduced a remarkable integral identity of physical interest and suggested that the evaluation ∫ 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 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 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.
ISSN:1751-8113
1751-8121
DOI:10.1088/1751-8121/ad2e3e