Ramanujan continued fractions of order eighteen

Yoon Kyung Park

doi:10.4134/JKMS.j220180

Journal of the
Korean Mathematical Society JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2023; 60(2): 395-406

Online first article February 23, 2023 Printed March 1, 2023

https://doi.org/10.4134/JKMS.j220180

Ramanujan continued fractions of order eighteen

Yoon Kyung Park

Seoul National University of Science and Technology

Abstract

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Abstract

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 $C(\tau)$. We can then obtain that their values are algebraic numbers. Finally, we evaluate them at some imaginary quadratic quantities.

Keywords: Ramanujan continued fraction, modular function, Klein forms

MSC numbers: 11F03, 11R04, 11G16

Supported by: This study was financially supported by NRF 2021R1F1A1055200.