A generalization of the Lipschitz summation formula and some applications

Abstract

Abstract. The Lipschitz formula is extended to a two-variable form. While the theorem itself is of independent interest, we justify its existence further by indicating several applications in the theory of modular forms. Lipschitz's formulaAs originally conceived, the Lipschitz Summation Formula (henceforth, LSF ) gives a Fourier expansion for certain functions which arise in the theory of modular forms. More explicitly, it states that if µ ∈ Z and Re α > 1 or µ ∈ R \ Z and Re α > 0, then . In number theory, the LSF is a useful tool in solving the problem of representations of a given integer as a sum of squares [2]; this is the starting point for its connection to the theory of modular forms. Later on, H. Maass considers an analogous series, motivated by his quest for the Fourier coefficients of nonanalytic modular forms. Specifically, he looks atwhere α and β are complex numbers such that Re (α + β) > 1; this requires a more complicated LSF (see [10, pp. 209-211] for a statement and proof of the formula), which was also used by Siegel [17]. Both versions of Lipschitz's formula are incorporated in the following, due to John Hawkins, which first appears in [4]:

Twisted Eisenstein series, cotangent‐zeta sums, and quantum modular forms

“…We have chosen this method due to its explicit and tractable nature. Using Lipschitz summation [13,15], with (s) > 1, we find that…”

“…We have chosen this method due to its explicit and tractable nature. Using Lipschitz summation [13, 15], with , we find that Thus, we have that …”

On the Generalized Two Variable Eisenstein Series

“…Replacing m by −m and applying a generalization of the Lipschitz summation formula [6], for Re s > Re s 1 > 1 (Re s > 2),…”

Hyperfunctions, formal groups and generalized Lipschitz summation formulas

“…Definition 5. The delta rational function is the series expressed for any n ∈ Z, and q ∈ D by (12) δ…”

“…The future research plans include an extension of the proposed construction to the multidimensional case (see also [12] for a different generalization), as well as possible applications of the proposed Lipschitz formulae to the study of Eisenstein series and periods of modular forms.…”