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Hankel Determinants of Eisenstein Series

S. Milne
Journal / Anthology

Year: 2000

In this paper we prove Garvan's conjectured formula for the square of the modular discriminant Delta as a 3-by-3 hankel determinant of classical Eisenstein series E_2n. We then obtain similar formulas involving minors of Hankel determinands for E_2r Delta^m, for m = 1, 2, 3 and r = 2, 3, 4, 5, 7, and E_14 Delta^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 infitine family of formulas extending the classical formula for Delta and Garvan's formula for Delta^2. This general formula epresses the n-by-n Hankel determinant det(E_2(i+j) (q)) (1<=i,j<=n) as the product f Delta^(n-1) (tau), a homogeneous polynomial in (E_4)^3 and (E_6)^2, and if needed, E_4. We also include a smple verification proof of the calssical 2-by-2 Hankel determinant formula for Delta. 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.

*Mathematics > Number Theory