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

Masanori Katsurada, Takumi Noda

Research output: Contribution to journalArticle

Abstract

The main object of study in this paper is the double holomorphic Eisenstein series (Formula presented.) having two complex variables (Formula presented.) and two parameters (Formula presented.) which satisfies either (Formula presented.) or (Formula presented.), where (Formula presented.) denotes the complex upper and lower half-planes, respectively. For (Formula presented.), its transformation properties and asymptotic aspects are studied when the distance (Formula presented.) becomes both small and large under certain natural settings on the movement of (Formula presented.). Prior to the proofs our main results, a new parameter (Formula presented.), which plays a pivotal role in describing our results, is introduced in connection with the difference (Formula presented.). We then establish complete asymptotic expansions for (Formula presented.) when (Formula presented.) moves within the poly-sector either (Formula presented.) or (Formula presented.), so as to (Formula presented.) through (Formula presented.) in the ascending order of (Formula presented.) (Theorem 1). This further leads us to show that counterpart expansions exist for (Formula presented.) in the descending order of (Formula presented.) as (Formula presented.) through (Formula presented.) (Theorem 2). Our second main formula in Theorem 2 yields a functional equation for (Formula presented.) (Corollaries 2.1, 2.2), and also reduces naturally to various expressions of (Formula presented.) in closed forms for integer lattice point (Formula presented.) (Corollaries 2.3–2.17). Most of these results reveal that the particular values of (Formula presented.) at (Formula presented.) are closely linked to Weierstraß’ elliptic function, the classical Eisenstein series reformulated by Ramanujan, and the Jordan–Kronecker type functions, each associated with the bases (Formula presented.), (Formula presented.). 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.

Original languageEnglish
Pages (from-to)1-44
Number of pages44
JournalRamanujan Journal
DOIs
Publication statusAccepted/In press - 2017 Jul 31

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Transformation Formula
Eisenstein Series
Complex Variables
Asymptotic Expansion
Theorem
Elliptic function

Keywords

  • Asymptotic expansion
  • Eisenstein series
  • Jordan–Kronecker function
  • Mellin–Barnes integral
  • Weierstraß elliptic function

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

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title = "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 (Formula presented.) having two complex variables (Formula presented.) and two parameters (Formula presented.) which satisfies either (Formula presented.) or (Formula presented.), where (Formula presented.) denotes the complex upper and lower half-planes, respectively. For (Formula presented.), its transformation properties and asymptotic aspects are studied when the distance (Formula presented.) becomes both small and large under certain natural settings on the movement of (Formula presented.). Prior to the proofs our main results, a new parameter (Formula presented.), which plays a pivotal role in describing our results, is introduced in connection with the difference (Formula presented.). We then establish complete asymptotic expansions for (Formula presented.) when (Formula presented.) moves within the poly-sector either (Formula presented.) or (Formula presented.), so as to (Formula presented.) through (Formula presented.) in the ascending order of (Formula presented.) (Theorem 1). This further leads us to show that counterpart expansions exist for (Formula presented.) in the descending order of (Formula presented.) as (Formula presented.) through (Formula presented.) (Theorem 2). Our second main formula in Theorem 2 yields a functional equation for (Formula presented.) (Corollaries 2.1, 2.2), and also reduces naturally to various expressions of (Formula presented.) in closed forms for integer lattice point (Formula presented.) (Corollaries 2.3–2.17). Most of these results reveal that the particular values of (Formula presented.) at (Formula presented.) are closely linked to Weierstra{\ss}’ elliptic function, the classical Eisenstein series reformulated by Ramanujan, and the Jordan–Kronecker type functions, each associated with the bases (Formula presented.), (Formula presented.). 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.",
keywords = "Asymptotic expansion, Eisenstein series, Jordan–Kronecker function, Mellin–Barnes integral, Weierstra{\ss} elliptic function",
author = "Masanori Katsurada and Takumi Noda",
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AU - Katsurada, Masanori

AU - Noda, Takumi

PY - 2017/7/31

Y1 - 2017/7/31

N2 - The main object of study in this paper is the double holomorphic Eisenstein series (Formula presented.) having two complex variables (Formula presented.) and two parameters (Formula presented.) which satisfies either (Formula presented.) or (Formula presented.), where (Formula presented.) denotes the complex upper and lower half-planes, respectively. For (Formula presented.), its transformation properties and asymptotic aspects are studied when the distance (Formula presented.) becomes both small and large under certain natural settings on the movement of (Formula presented.). Prior to the proofs our main results, a new parameter (Formula presented.), which plays a pivotal role in describing our results, is introduced in connection with the difference (Formula presented.). We then establish complete asymptotic expansions for (Formula presented.) when (Formula presented.) moves within the poly-sector either (Formula presented.) or (Formula presented.), so as to (Formula presented.) through (Formula presented.) in the ascending order of (Formula presented.) (Theorem 1). This further leads us to show that counterpart expansions exist for (Formula presented.) in the descending order of (Formula presented.) as (Formula presented.) through (Formula presented.) (Theorem 2). Our second main formula in Theorem 2 yields a functional equation for (Formula presented.) (Corollaries 2.1, 2.2), and also reduces naturally to various expressions of (Formula presented.) in closed forms for integer lattice point (Formula presented.) (Corollaries 2.3–2.17). Most of these results reveal that the particular values of (Formula presented.) at (Formula presented.) are closely linked to Weierstraß’ elliptic function, the classical Eisenstein series reformulated by Ramanujan, and the Jordan–Kronecker type functions, each associated with the bases (Formula presented.), (Formula presented.). 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.

AB - The main object of study in this paper is the double holomorphic Eisenstein series (Formula presented.) having two complex variables (Formula presented.) and two parameters (Formula presented.) which satisfies either (Formula presented.) or (Formula presented.), where (Formula presented.) denotes the complex upper and lower half-planes, respectively. For (Formula presented.), its transformation properties and asymptotic aspects are studied when the distance (Formula presented.) becomes both small and large under certain natural settings on the movement of (Formula presented.). Prior to the proofs our main results, a new parameter (Formula presented.), which plays a pivotal role in describing our results, is introduced in connection with the difference (Formula presented.). We then establish complete asymptotic expansions for (Formula presented.) when (Formula presented.) moves within the poly-sector either (Formula presented.) or (Formula presented.), so as to (Formula presented.) through (Formula presented.) in the ascending order of (Formula presented.) (Theorem 1). This further leads us to show that counterpart expansions exist for (Formula presented.) in the descending order of (Formula presented.) as (Formula presented.) through (Formula presented.) (Theorem 2). Our second main formula in Theorem 2 yields a functional equation for (Formula presented.) (Corollaries 2.1, 2.2), and also reduces naturally to various expressions of (Formula presented.) in closed forms for integer lattice point (Formula presented.) (Corollaries 2.3–2.17). Most of these results reveal that the particular values of (Formula presented.) at (Formula presented.) are closely linked to Weierstraß’ elliptic function, the classical Eisenstein series reformulated by Ramanujan, and the Jordan–Kronecker type functions, each associated with the bases (Formula presented.), (Formula presented.). 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.

KW - Asymptotic expansion

KW - Eisenstein series

KW - Jordan–Kronecker function

KW - Mellin–Barnes integral

KW - Weierstraß elliptic function

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