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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]:

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