Transformation formulae and asymptotic expansions for double holomorphic Eisenstein series of two complex variables

Abstract

The main object of study in this paper is the double holomorphic Eisenstein series $$\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})$$ having two complex variables $$\mathbf{s}=(s_1,s_2)$$ and two parameters $$\mathbf{z}= (z_1,z_2)$$ which satisfies either $$\mathbf{z}\in (\mathfrak {H}^+)^2$$ or $$\mathbf{z}\in (\mathfrak {H}^-)^2$$ , where $$\mathfrak {H}^{\pm }$$ denotes the complex upper and lower half-planes, respectively. For $$\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})$$ , its transformation properties and asymptotic aspects are studied when the distance $$|z_2-z_1|$$ becomes both small and large under certain natural settings on the movement of $$\mathbf{z}\in (\mathfrak {H}^{\pm })^2$$ . Prior to the proofs our main results, a new parameter $$\eta $$ , which plays a pivotal role in describing our results, is introduced in connection with the difference $$z_2-z_1$$ . We then establish complete asymptotic expansions for $$\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})$$ when $$\mathbf{z}$$ moves within the poly-sector either $$(\mathfrak {H}^+)^2$$ or $$(\mathfrak {H}^-)^2$$ , so as to $$\eta \rightarrow 0$$ through $$|\arg \eta |<\pi /2$$ in the ascending order of $$\eta $$ (Theorem 1). This further leads us to show that counterpart expansions exist for $$\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})$$ in the descending order of $$\eta $$ as $$\eta \rightarrow \infty $$ through $$|\arg \eta |<\pi /2$$ (Theorem 2). Our second main formula in Theorem 2 yields a functional equation for $$\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})$$ (Corollaries 2.1, 2.2), and also reduces naturally to various expressions of $$\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})$$ in closed forms for integer lattice point $$\mathbf{s}\in \mathbb {Z}^2$$ (Corollaries 2.3–2.17). Most of these results reveal that the particular values of $$\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})$$ at $$\mathbf{s}\in \mathbb {Z}^2$$ are closely linked to Weierstras’ elliptic function, the classical Eisenstein series reformulated by Ramanujan, and the Jordan–Kronecker type functions, each associated with the bases $$2\pi (1, z_j)$$ , $$j=1,2$$ . The latter two functions were extensively utilized by Ramanujan in the course of developing his theories of Eisenstein series, elliptic functions, and theta functions. As for the methods used, crucial roles in the proofs are played by the Mellin–Barnes type integrals, manipulated with several properties of hypergeometric functions; the transference from Theorem 1 to Theorem 2 is, for instance, achieved by a connection formula for Kummer’s confluent hypergeometric functions.