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Hankel Determinants of Eisenstein Series

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Book cover Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics

Part of the book series: Developments in Mathematics ((DEVM,volume 4))

Abstract

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 E14 Δ4. 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 x n Hankel determinant det \(\left( {{E_{2\left( {i + j} \right)}}\left( q \right)} \right)1 \leqslant i,j \leqslant n\) as the product of Δn-1(τ), a homogeneous polynomial in E 34 and E 26 , 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.

Author was partially supported by National Security Agency grants MDA 904-97-1-0019 and MDA904-99-1-0003

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Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, Columbus, Ohio, 43210, USA

    Stephen C. Milne

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  1. Stephen C. Milne
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Editors and Affiliations

  1. Department of Mathematics, University of Florida, Gainesville, Florida, 32611, USA

    Frank G. Garvan

  2. Department of Mathematics, University of South Florida, Tampa, Florida, 33620, USA

    Mourad E. H. Ismail

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Milne, S.C. (2001). Hankel Determinants of Eisenstein Series. In: Garvan, F.G., Ismail, M.E.H. (eds) Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Developments in Mathematics, vol 4. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0257-5_10

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  • DOI: https://doi.org/10.1007/978-1-4613-0257-5_10

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4020-0101-7

  • Online ISBN: 978-1-4613-0257-5

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