Hankel determinants of Eisenstein series

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dc.creatorMilne, Stephen C.
dc.date2000-09-13
dc.date2000-12-27
dc.date.accessioned2026-07-07T04:37:22Z
dc.date.available2026-07-07T04:37:22Z
dc.descriptionIn this paper we prove Garvan's conjectured formula for the square of the modular discriminant $Δ$ as a 3 by 3 Hankel determinant of classical Eisenstein series $E_{2n}$. We then obtain similar formulas involving minors of Hankel determinants for $E_{2r}Δ^m$, for $m=1,2,3$ and $r=2,3,4,5,7$, and $E_{14}Δ^4$. We next use Mathematica to discover, and then the standard structure theory of the ring of modular forms, to derive the general form of our infinite family of formulas extending the classical formula for $Δ$ and Garvan's formula for $Δ^2$. This general formula expresses the $n\times n$ Hankel determinant $\det(E_{2(i+j)}(q))_{1\leq i,j\leq n}$ as the product of $Δ^{n-1}(τ)$, a homogeneous polynomial in $E_4^3$ and $E_6^2$, and if needed, $E_4$. We also include a simple verification proof of the classical 2 by 2 Hankel determinant formula for $Δ$. This proof depends upon polynomial properties of elliptic function parameters from Jacobi's Fundamenta Nova. The modular forms approach provides a convenient explanation for the determinant identities in this paper.
dc.description13 pages. AmSTeX file. Final accepted version. To appear in Symbolic Computation, Number Theory,Special Functions, Physics and Combinatorics, F. Garvan and M. Ismail, Eds., Developments in Mathematics, Kluwer Academic Publishers, (2001)
dc.identifierhttps://arxiv.org/abs/math/0009130
dc.identifierhttp://arxiv.org/abs/math/0009130
dc.identifierin vol. 4 of Developments in Mathematics, Kluwer Academic Pub., Dordrecht, 2001, pp. 171--188.
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/59927
dc.subjectNumber Theory
dc.subjectClassical Analysis and ODEs
dc.subjectPrimary 11F11, 05A19; Secondary 33D99, 33E05
dc.titleHankel determinants of Eisenstein series
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