Let K(k)K(k) and E(k)E(k) be the complete elliptic integrals of the first and second kind, with K′(x)=K(1−x2)K'(x)=K(\sqrt{1-x^2}) and E′(x)=E(1−x2)E'(x)=E(\sqrt{1-x^2}) the complementary versions. Wan (2012) determined the odd moments of the same-type two-product integrals ∫01xnf(x)g(x) dx\int_0^1 x^n f(x)g(x)\,dx, fg∈{K2,KE,E2,K′2,K′E′,E′2}fg\in\{K^2,KE,E^2,K'^2,K'E',E'^2\}, in terms of rationals and rational multiples of ζ(3)\zeta(3), and proved four linear relations among the five even moments that lack closed forms: ∫01E2\int_0^1 E^2, ∫01x2E2\int_0^1 x^2E^2, ∫01EK\int_0^1 EK, ∫01x2EK\int_0^1 x^2EK and ∫01x2K2\int_0^1 x^2K^2. He conjectured a fifth relation, ∫01(2K2−4EK+3E2−K′E′) dx=0\int_0^1(2K^2-4EK+3E^2-K'E')\,dx=0, which together with the four proved relations would determine all five moments through two reference constants: ∫01K2 dx\int_0^1 K^2\,dx (known only through equivalent integral representations) and ∫01K′E′ dx\int_0^1 K'E'\,dx (a very-well-poised 7F6_7F_6 hypergeometric value). His second conjecture, ∫01K′3 dx=Γ(1/4)8/(128π2)\int_0^1 K'^3\,dx=\Gamma(1/4)^8/(128\pi^2), was proved independently by Zhou (2014) and by Rogers, Wan and Zucker (2015), which also settled the inter-group rationality question of his Theorem 5; but no subsequent work — across the lattice-sum, Fourier-Legendre/fractional-calculus, newform LL-value and period-polynomial programs, through 2026 — proves or refutes the first conjecture or evaluates the five moments. The problem is to settle that identity and thereby close the determination of this moment stratum.

Background

For a modulus k∈[0,1]k\in[0,1] the complete elliptic integrals of the first and second kind are K(k)=∫0π/2(1−k2sin⁡2θ)−1/2dθK(k)=\int_0^{\pi/2}(1-k^2\sin^2\theta)^{-1/2}d\theta and E(k)=∫0π/2(1−k2sin⁡2θ)1/2dθE(k)=\int_0^{\pi/2}(1-k^2\sin^2\theta)^{1/2}d\theta, with complementary versions K′(x)=K(1−x2)K'(x)=K(\sqrt{1-x^2}), E′(x)=E(1−x2)E'(x)=E(\sqrt{1-x^2}). Moments of products of these functions — definite integrals ∫01xnf(x)g(x) dx\int_0^1 x^n f(x)g(x)\,dx with f,gf,g among K,E,K′,E′K,E,K',E' — arise as moments of the distance from the origin in planar uniform random walks, as Bessel moments in quantum-field-theoretic loop integrals such as the sunrise diagram (a standard two-loop Feynman graph), and in the evaluation of two-dimensional lattice sums (Ewald-type sums over the sites of a planar lattice); their closed-form evaluation is a classical topic of experimental mathematics, in which integer-relation tools such as PSLQ and the Inverse Symbolic Calculator are used on Bailey's tables of conjectural definite-integral identities before proofs are found. Legendre's relation EK′+E′K−KK′=π/2EK'+E'K-KK'=\pi/2 ties the complementary pair together and underlies most identities among these moments.

Wan (2012) organized this stratum. For odd nn, the moments of the six same-type pairings K2K^2, KEKE, E2E^2, K′2K'^2, K′E′K'E' and E′2E'^2 evaluate to rational combinations of 11 and ζ(3)\zeta(3), via three-term recurrences with explicitly computed ignition values (for example ∫01xK′(x)2dx=7ζ(3)/4\int_0^1 xK'(x)^2dx=7\zeta(3)/4 and ∫01x3K(x)2dx=(2+7ζ(3))/8\int_0^1 x^3K(x)^2dx=(2+7\zeta(3))/8); the odd moments of mixed pairings such as KK′KK' involve other constants instead (e.g. ∫01xK(x)K′(x)dx=π3/16\int_0^1 xK(x)K'(x)dx=\pi^3/16). The even moments not involving EE are likewise settled, e.g. ∫01KK′ dx=(π3/8)C\int_0^1 KK'\,dx=(\pi^3/8)C with C=4F3(12,12,12,12;1,1,1;1)C={}_4F_3(\frac12,\frac12,\frac12,\frac12;1,1,1;1), while ∫01K2 dx\int_0^1 K^2\,dx is characterized by the equivalent representations of his Theorem 4 (such as 12∫01K′2 dx=∫01KK′x′ dx=∫012EK/(1+x) dx\frac12\int_0^1 K'^2\,dx=\int_0^1 KK'x'\,dx=\int_0^1 2EK/(1+x)\,dx, with x′=1−x2x'=\sqrt{1-x^2}) but has no known closed form. Five even moments involving EE remain without closed forms: ∫01E2\int_0^1 E^2, ∫01x2E2\int_0^1 x^2E^2, ∫01EK\int_0^1 EK, ∫01x2EK\int_0^1 x^2EK and ∫01x2K2\int_0^1 x^2K^2. Wan proved four linear relations among them: ∫01(3E2−2EK)dx=1\int_0^1(3E^2-2EK)dx=1, ∫01(5x2E2−2x2EK)dx=1\int_0^1(5x^2E^2-2x^2EK)dx=1, ∫01((1−3x2)EK+E2−(1−x2)K2)dx=0\int_0^1((1-3x^2)EK+E^2-(1-x^2)K^2)dx=0 and ∫01(1+x2)K2dx=2∫01EK dx\int_0^1(1+x^2)K^2dx=2\int_0^1 EK\,dx (the second is printed in the paper without the x2x^2 on the EKEK term, a typo contradicted by the paper's own general recurrence). His Conjecture 1, equation (50), supplies the fifth relation ∫01(2K2−4EK+3E2−K′E′)dx=0\int_0^1(2K^2-4EK+3E^2-K'E')dx=0, and the paper observes that a proof would give enough information to solve for all five moments. His Theorem 5 splits a family of threefold-product integrals into two rationally closed groups, with the rationality of the ratio between the groups left open, tied to his Conjecture 2, ∫01K′(x)3dx=2K(1/2)4=Γ(1/4)8/(128π2)\int_0^1 K'(x)^3dx=2K(1/\sqrt2)^4=\Gamma(1/4)^8/(128\pi^2).

The subsequent literature settled the K-only part of this agenda. Zhou proved Conjecture 2 by his theory of Legendre functions and spherical rotations (Tricomi pairing of Clebsch-Gordan-type integrals, i.e. pairing theorems for integrals over products of Legendre functions, in analogy with angular-momentum coupling coefficients), obtaining the chain ∫01K′3=(10/3)∫01K3=5∫01xK3=6∫01xK K′2=Γ(1/4)8/(128π2)\int_0^1K'^3=(10/3)\int_0^1K^3=5\int_0^1xK^3=6\int_0^1xK\,K'^2=\Gamma(1/4)^8/(128\pi^2), which in particular supplies the missing rational bridge between the two groups of Theorem 5; Rogers, Wan and Zucker proved the same evaluation via the critical LL-value L(g,4)=Γ8(1/4)/(3840π2)L(g,4)=\Gamma^8(1/4)/(3840\pi^2) of the eta-product g(τ)=η4(τ)η2(2τ)η4(4τ)g(\tau)=\eta^4(\tau)\eta^2(2\tau)\eta^4(4\tau) (a weight-5 cusp form built from the Dedekind eta function η(τ)=eπiτ/12∏n≥1(1−e2πinτ)\eta(\tau)=e^{\pi i\tau/12}\prod_{n\geq1}(1-e^{2\pi i n\tau}); its critical LL-value L(g,4)L(g,4) is the value at s=4s=4 of the LL-function, i.e. the Mellin-transform Dirichlet series, of gg), and also evaluated complementary integrals such as ∫01E(k)K′(k)2dk=Γ8(1/4)/(384π2)+π3/12\int_0^1E(k)K'(k)^2dk=\Gamma^8(1/4)/(384\pi^2)+\pi^3/12; Bao recently explained these evaluations through period relations for theta products (period-polynomial identities for the modular forms attached to Jacobi theta functions). However, the programs that cracked the K-only moments do not reach the EE-containing two-product stratum: Wan and Zucker's lattice-sum paper evaluates algebraically weighted integrands (e.g. ∫01K3/k′ dk\int_0^1 K^3/k'\,dk), Campbell, D'Aurizio and Sondow's Fourier-Legendre and fractional-calculus methods target differently weighted or threefold products, Campbell's Clebsch-Gordan program handles threefold products, Au's method relates K-only moments to newform LL-values (LL-functions of normalized Hecke eigenforms), and Bao's period relations cover cubic moments of theta products. As of 2026, Conjecture 1 and the five moments remain unproved and unevaluated in the literature, though the identity is supported by high-precision numerical evidence.

Problem Statement

Let K(k)=∫0π/2(1−k2sin⁡2θ)−1/2dθK(k)=\int_0^{\pi/2}(1-k^2\sin^2\theta)^{-1/2}d\theta and E(k)=∫0π/2(1−k2sin⁡2θ)1/2dθE(k)=\int_0^{\pi/2}(1-k^2\sin^2\theta)^{1/2}d\theta denote the complete elliptic integrals of the first and second kind, and K′(x)=K(1−x2)K'(x)=K(\sqrt{1-x^2}), E′(x)=E(1−x2)E'(x)=E(\sqrt{1-x^2}) their complementary versions; all integrals below are over x∈[0,1]x\in[0,1] with respect to dxdx.

Settle Conjecture 1 of Wan (2012), equation (50) of that paper: prove or refute the identity ∫01(2K(x)2−4E(x)K(x)+3E(x)2−K′(x)E′(x)) dx=0,\int_0^1 \left(2K(x)^2-4E(x)K(x)+3E(x)^2-K'(x)E'(x)\right)\,dx = 0, and thereby close the determination of the five even moments of products of two elliptic integrals that remain without closed forms, ∫01E(x)2dx,∫01x2E(x)2dx,∫01E(x)K(x)dx,∫01x2E(x)K(x)dx,∫01x2K(x)2dx,\int_0^1 E(x)^2dx,\quad \int_0^1 x^2E(x)^2dx,\quad \int_0^1 E(x)K(x)dx,\quad \int_0^1 x^2E(x)K(x)dx,\quad \int_0^1 x^2K(x)^2dx, which are tied to the conjectured identity by the four relations already proved in the source: ∫01(3E2−2EK)dx=1\int_0^1(3E^2-2EK)dx=1, ∫01(5x2E2−2x2EK)dx=1\int_0^1(5x^2E^2-2x^2EK)dx=1, ∫01((1−3x2)EK+E2−(1−x2)K2)dx=0\int_0^1((1-3x^2)EK+E^2-(1-x^2)K^2)dx=0, ∫01(1+x2)K2dx=2∫01EK dx\int_0^1(1+x^2)K^2dx=2\int_0^1 EK\,dx. If the identity holds, the five moments are given by ∫01EK=1+2A0−D2,∫01E2=2+2A0−D3,∫01x2K2=1+A0−D,\int_0^1 EK=\frac{1+2A_0-D}{2},\quad \int_0^1 E^2=\frac{2+2A_0-D}{3},\quad \int_0^1 x^2K^2=1+A_0-D, ∫01x2EK=13(∫01EK+∫01E2−A0+∫01x2K2),∫01x2E2=1+2∫01x2EK5,\int_0^1 x^2EK=\frac{1}{3}\left(\int_0^1 EK+\int_0^1 E^2-A_0+\int_0^1 x^2K^2\right),\quad \int_0^1 x^2E^2=\frac{1+2\int_0^1x^2EK}{5}, where A0=∫01K(x)2dxA_0=\int_0^1K(x)^2dx (known only through the equivalent representations of Wan's Theorem 4, e.g. A0=12∫01K′2dx=∫012EK/(1+x) dxA_0=\frac12\int_0^1K'^2dx=\int_0^12EK/(1+x)\,dx) and D=∫01K′(x)E′(x)dxD=\int_0^1K'(x)E'(x)dx (Zudilin's very-well-poised 7F6_7F_6 evaluation, i.e. the convergent generalized hypergeometric series D=π432 7F6(−12,12,12,12,12,12,54; 1,14,1,1,1,2; 1)D=\frac{\pi^4}{32}\,{}_7F_6(-\frac12,\frac12,\frac12,\frac12,\frac12,\frac12,\frac54;\,1,\frac14,1,1,1,2;\,1)). Any of the following counts as a resolution: (a) a proof of the identity, with the resulting closed forms for the five moments; (b) closed forms for the five moments obtained by any other method — these determine A0=2∫01EK−∫01x2K2A_0=2\int_0^1EK-\int_0^1x^2K^2 through the fourth proved relation, so the conjectured identity reduces to the explicit comparison D=2A0−4∫01EK+3∫01E2D=2A_0-4\int_0^1EK+3\int_0^1E^2 and is settled by substitution; (c) a rigorous disproof, i.e. a certified demonstration that the left-hand side is nonzero. The objective is one coherent closure: completing the determination of the two-product even-moment stratum that the source paper left open, in the same sense in which the companion conjecture ∫01K′3dx=Γ(1/4)8/(128π2)\int_0^1K'^3dx=\Gamma(1/4)^8/(128\pi^2) completed the K-only stratum.

References

  1. J. G. Wan, Moments of products of elliptic integrals, Advances in Applied Mathematics 48 (2012) 121-141, DOI 10.1016/j.aam.2011.04.007, arXiv:1101.1132, https://arxiv.org/abs/1101.1132
  2. Y. Zhou, Legendre functions, spherical rotations, and multiple elliptic integrals, The Ramanujan Journal 34 (2014) 373-428, DOI 10.1007/s11139-013-9502-2, arXiv:1301.1735, https://arxiv.org/abs/1301.1735
  3. M. Rogers, J. G. Wan, I. J. Zucker, Moments of elliptic integrals and critical L-values, The Ramanujan Journal 37 (2015) 113-130, DOI 10.1007/s11139-014-9584-5, arXiv:1303.2259, https://arxiv.org/abs/1303.2259
  4. Y. Zhou, On some integrals over the product of three Legendre functions, The Ramanujan Journal 35 (2014) 311-326, DOI 10.1007/s11139-013-9504-0, arXiv:1304.1606, https://arxiv.org/abs/1304.1606
  5. J. G. Wan, I. J. Zucker, Integrals of K and E from lattice sums, The Ramanujan Journal 40 (2016) 257-278, DOI 10.1007/s11139-015-9710-z, arXiv:1410.7081, https://arxiv.org/abs/1410.7081
  6. J. M. Campbell, J. D'Aurizio, J. Sondow, On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier-Legendre expansions, Journal of Mathematical Analysis and Applications 479 (2019) 90-121, DOI 10.1016/j.jmaa.2019.06.017, arXiv:1710.03221, https://arxiv.org/abs/1710.03221
  7. J. M. Campbell, New Clebsch-Gordan-type integrals involving threefold products of complete elliptic integrals, arXiv:2302.05819 (2023, preprint), https://arxiv.org/abs/2302.05819
  8. J. M. Campbell, On a problem due to Glasser on analytically tractable moments, Journal of Physics A: Mathematical and Theoretical 57 (2024) 12LT01, DOI 10.1088/1751-8121/ad2e3e, https://doi.org/10.1088/1751-8121/ad2e3e
  9. Y. Zhou, Hilbert transforms and sum rules of Bessel moments, The Ramanujan Journal 48 (2019) 159-172, DOI 10.1007/s11139-017-9945-y, arXiv:1706.01068, https://arxiv.org/abs/1706.01068
  10. Y. Zhou, On Borwein's conjectures for planar uniform random walks, Journal of the Australian Mathematical Society 107 (2019) 392-411, DOI 10.1017/S1446788719000351, arXiv:1708.02857, https://arxiv.org/abs/1708.02857
  11. Y. Zhou, On Laporta's 4-loop sunrise formulae, The Ramanujan Journal 50 (2019) 465-503, DOI 10.1007/s11139-018-0090-z, arXiv:1801.02182, https://arxiv.org/abs/1801.02182
  12. Y. Zhou, Notes on certain binomial harmonic sums of Sun's type, The Ramanujan Journal 69, Article 83 (2026), DOI 10.1007/s11139-026-01333-5, arXiv:2503.05599, https://arxiv.org/abs/2503.05599
  13. K. C. Au, Linear relations between L-values of newforms and moments of elliptic K integral, arXiv:2509.19960 (2025, preprint), https://arxiv.org/abs/2509.19960
  14. D. Bao, Period relations for theta products and elliptic integral moments, arXiv:2608.26514 (2026, preprint), https://arxiv.org/abs/2608.26514