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A Four-Genus Kronecker-Limit Evaluation of the Alternating Rogers-Ramanujan Continued Fraction
Abstract
We evaluate the six odd class-number-four cases left unevaluated in Ramanathan's treatment of the Rogers-Ramanujan continued fraction. Let S(q)=−R(−q), R(q)=cfracq1/51+cfracq1+cfracq21+cfracq31+⋯. For n=39,87,111,119,159,287 we determine the value of S(e−π/√(5n)) by applying the genus-character form of the Kronecker limit formula to the four ideal classes of ℚ(√(−5n)). The resulting expressions are given in terms of fundamental units of real quadratic fields. In particular, if Xn=S(e−π/√(5n)), then Xn−5+11−Xn5=5√5/Un, where the six quantities Un are displayed explicitly below. We also give the corresponding radical expressions and quartic algebraic certificates, together with numerical checks of the six evaluations.
Related
- Eta-Quotient Representations for a Three Parameter Family of Modular Functions Associated with the Rogers-Ramanujan Continued Fraction
- On Ramanujan's cubic continued fraction, II
- Logarithmic moments of the Rogers-Ramanujan continued fraction and odd zeta values
- An elementary treatment of Ramanujan's elementary continued fractions
- Cyclic Kronecker Fourier Transforms, Weber Norms, and Rogers-Ramanujan Singular Values
BibTeX
@misc{jha2026fourgenus,
title = {{A Four-Genus Kronecker-Limit Evaluation of the Alternating Rogers-Ramanujan Continued Fraction}},
author = {Sumit Kumar Jha},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10642},
eprint = {2609.10642},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10642},
}
RIS
TY - UNPB
TI - A Four-Genus Kronecker-Limit Evaluation of the Alternating Rogers-Ramanujan Continued Fraction
AU - Sumit Kumar Jha
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10642
UR - https://arxiv.org/abs/2609.10642
AB - We evaluate the six odd class-number-four cases left unevaluated in Ramanathan's treatment of the Rogers-Ramanujan continued fraction. Let S(q)=−R(−q), R(q)=cfracq^1/51+cfracq1+cfracq^21+cfracq^31+⋯. For n=39,87,111,119,159,287 we determine the value of S(e^−π/√(5n)) by applying the genus-character form of the Kronecker limit formula to the four ideal classes of ℚ(√(−5n)). The resulting expressions are given in terms of fundamental units of real quadratic fields. In particular, if X_n=S(e^−π/√(5n)), then X_n^−5+11−X_n^5=5√5/U_n, where the six quantities U_n are displayed explicitly below. We also give the corresponding radical expressions and quartic algebraic certificates, together with numerical checks of the six evaluations.
ER -