Let C(q) be the Ramanujan’s cubic continued fraction and U(q) be the continued fraction of order 12. We use the affine models on the modular curve <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$X_0(6n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>X</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>6</mn> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> over <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation> involving <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$C^3(q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>C</mi> <mn>3</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and the formulas for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$C^3(q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>C</mi> <mn>3</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$C^3(q^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>C</mi> <mn>3</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in terms of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$U(-q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo stretchy="false">(</mo> <mo>-</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> to establish the existence of the modular equations between <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$U(-q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo stretchy="false">(</mo> <mo>-</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$C(q^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and between <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$U(-q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo stretchy="false">(</mo> <mo>-</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$U(-q^{2n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo stretchy="false">(</mo> <mo>-</mo> <msup> <mi>q</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for any positive integer n, extending the results of Srivatsa Kumar, Vidya and Mahadeva Naika et. al. We also explicitly give the modular equations between <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$U(-q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo stretchy="false">(</mo> <mo>-</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$C(q^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>C</mi> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for certain values of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$n\le 14$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≤</mo> <mn>14</mn> </mrow> </math></EquationSource> </InlineEquation> and between <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$U(-q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo stretchy="false">(</mo> <mo>-</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$U(-q^{2n})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo stretchy="false">(</mo> <mo>-</mo> <msup> <mi>q</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$n\le 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≤</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. We apply the methods of Lee and Park and some properties of <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\eta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>-quotients to present our results.

Modular equations between certain continued fractions

期刊论文
发表日期：

2026年4月6日

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Russelle Guadalupe

摘要

Let C(q) be the Ramanujan’s cubic continued fraction and U(q) be the continued fraction of order 12. We use the affine models on the modular curve $X_0(6n)$ $X_{0} (6 n)$ over $\mathbb {Q}$ $Q$ involving $C^3(q)$ $C^{3} (q)$ and the formulas for $C^3(q)$ $C^{3} (q)$ and $C^3(q^2)$ $C^{3} (q^{2})$ in terms of $U(-q)$ $U (- q)$ to establish the existence of the modular equations between $U(-q)$ $U (- q)$ and $C(q^n)$ $C (q^{n})$ and between $U(-q)$ $U (- q)$ and $U(-q^{2n})$ $U (- q^{2 n})$ for any positive integer n, extending the results of Srivatsa Kumar, Vidya and Mahadeva Naika et. al. We also explicitly give the modular equations between $U(-q)$ $U (- q)$ and $C(q^n)$ $C (q^{n})$ for certain values of $n\le 14$ $n \leq 14$ and between $U(-q)$ $U (- q)$ and $U(-q^{2n})$ $U (- q^{2 n})$ for $n\le 3$ $n \leq 3$ . We apply the methods of Lee and Park and some properties of $\eta $ $η$ -quotients to present our results.