2000/09/30 by Stephen C. Milne · 1 citation
Mathematics · #math.NT #math.CA #msc:11F11 #msc:05A19 #msc:33D99 #msc:33E05
published as in vol. 4 of Developments in Mathematics, Kluwer Academic Pub., Dordrecht, 2001, pp. 171--188. · 13 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)
arxiv created 2000/12/27 · arxiv updated 2009/11/30
In 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 E2n. We then obtain similar formulas involving minors of Hankel determinants for E2rΔm, for m=1,2,3 and r=2,3,4,5,7, and E14Δ4. 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 Δ2. This general formula expresses the n× n Hankel determinant det(E2(i+j)(q))1≤ i,j≤ n as the product of Δn-1(τ), a homogeneous polynomial in E43 and E62, and if needed, E4. 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.