• Title/Summary/Keyword: modular functions

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MOCK THETA FUNCTIONS OF ORDER 2 AND THEIR SHADOW COMPUTATIONS

  • Kang, Soon-Yi;Swisher, Holly
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.6
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    • pp.2155-2163
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    • 2017
  • Zwegers showed that a mock theta function can be completed to form essentially a real analytic modular form of weight 1/2 by adding a period integral of a certain weight 3/2 unary theta series. This theta series is related to the holomorphic modular form called the shadow of the mock theta function. In this paper, we discuss the computation of shadows of the second order mock theta functions and show that they share the same shadow with a mock theta function which appears in the Mathieu moonshine phenomenon.

MODULAR TRANSFORMATION FORMULAE COMING FROM GENERALIZED NON-HOLOMORPHIC EISENSTEIN SERIES AND INFINITE SERIES IDENTITIES

  • Lim, Sung Geun
    • Honam Mathematical Journal
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    • v.43 no.2
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    • pp.221-237
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    • 2021
  • B. C. Berndt has found modular transformation formulae for a large class of functions coming from generalized Eisenstein series. Using those formulae, he established a lot of infinite series identities, some of which explain many infinite series identities given by Ramanujan. Continuing his work, the author proved a lot of new infinite series identities. Moreover, recently the author found transformation formulae for a class of functions coming from generalized non-holomorphic Eisenstein series. In this paper, using those formulae, we evaluate a few new infinite series identities which generalize the author's previous results.

RAMANUJAN CONTINUED FRACTIONS OF ORDER EIGHTEEN

  • Yoon Kyung Park
    • Journal of the Korean Mathematical Society
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    • v.60 no.2
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    • pp.395-406
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    • 2023
  • As an analogy of the Rogers-Ramanujan continued fraction, we define a Ramanujan continued fraction of order eighteen. There are essentially three Ramanujan continued fractions of order eighteen, and we study them using the theory of modular functions. First, we prove that they are modular functions and find the relations with the Ramanujan cubic continued fraction C(𝜏). We can then obtain that their values are algebraic numbers. Finally, we evaluate them at some imaginary quadratic quantities.

ARTIN SYMBOLS OVER IMAGINARY QUADRATIC FIELDS

  • Dong Sung Yoon
    • East Asian mathematical journal
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    • v.40 no.1
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    • pp.95-107
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    • 2024
  • Let K be an imaginary quadratic field with ring of integers 𝓞K and N be a positive integer. By K(N) we mean the ray class field of K modulo N𝓞K. In this paper, for each prime p of K relatively prime to N𝓞K we explicitly describe the action of the Artin symbol (${\frac{K_{(N)}/K}{p}}$) on special values of modular functions of level N. Furthermore, we extend the Kronecker congruence relation for the elliptic modular function j to some modular functions of higher level.

HYPERGEOMETRIC FUNCTIONS AND EICHLER INTEGRALS

  • Lim, Su-Bong
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.12 no.4
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    • pp.223-226
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    • 2008
  • Duke and Imamo$\bar{g}$lu express the Eichler integrals associated to modular forms of weight 3 in terms of generalized hypergeometric functions. We extend this result to most general modular forms of weight 3.

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SERIES RELATIONS COMING FROM CERTAIN FUNCTIONS RELATED TO GENERALIZED NON-HOLOMORPHIC EISENSTEIN SERIES

  • Lim, Sung Geun
    • Journal of the Chungcheong Mathematical Society
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    • v.34 no.2
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    • pp.139-155
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    • 2021
  • Using a modular transformation formula for a class of functions related to generalized non-holomorphic Eisenstein series, we find a new class of infinite series about identities, some of which include generalized formulae of several Berndt's results.

ARITHMETIC OF THE MODULAR FUNCTION $j_4$

  • Kim, Chang-Heon;Koo, Ja-Kyung
    • Journal of the Korean Mathematical Society
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    • v.36 no.4
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    • pp.707-723
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    • 1999
  • Since the modular curve $X(4)=\Gamma(4)/{\mathfrak{}}^*$ has genus 0, we have a field isomorphism K(X(4)){\approx}\mathcal{C}(j_{4})$ where $j_{4}(z)={\theta}_{3}(\frac{z}{2})/{\theta}_{4}(\frac{z}{2})$ is a quotient of Jacobi theta series ([9]). We derive recursion formulas for the Fourier coefficients of $j_4$ and $N(j_{4})$ (=the normalized generator), respectively. And we apply these modular functions to Thompson series and the construction of class fields.

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ON SOME MODULAR EQUATIONS AND THEIR APPLICATIONS I

  • Yi, Jinhee;Cho, Man Gi;Kim, Jeong Hwan;Lee, Seong Hoi;Yu, Jae Myung;Paek, Dae Hyun
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.3
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    • pp.761-766
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    • 2013
  • We derive several modular equations and present their proofs based on concise algebraic computations. In addition, we establish explicit relations and formulas for some parameterizations for the theta functions ${\varphi}$ and ${\psi}$ and show some applications of the modular equations to evaluations of the cubic continued fraction and the theta function ${\psi}$.