Generalized Lambert series

  • Ronald Evans
Part of the Progress in Mathematics book series (PM, volume 138)

Abstract

Let q = exp(2πiτ) with Im τ > 0, so 0 < |q| < 1. For any positive integer n, define
$$S_n = \sum\limits_{k \ne 0} {\frac{{( - 1)^k q^{(k^2 + k)/2} }}{{(1 - q^k )^n }},}$$
where the sum is over all nonzero integers k. Malcolm Perry needed to know the modular properties of S 2, which arose in his work in quantum string theory. Ramanujan evaluated S 2 in terms of Eisenstein series. We prove a general transformation formula that enables us to evaluate each sum S n in terms of Eisenstein series, and to thus determine the modular properties of S n . Moreover, our formula yields systematic proofs of related q-series identities of Ramanujan, proofs which are considerably simpler than those in the literature.

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References

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Copyright information

© Birkhäuser Boston 1996

Authors and Affiliations

  • Ronald Evans
    • 1
  1. 1.Department of MathematicsUniversity of California at San DiegoLa JollaUSA

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