<p>We investigate a hypergeometric transformation formula originating in Legendre’s classical work on elliptic integrals. Though noted by Kummer in 1836, this transformation was first explicitly recorded in contemporary literature in 2018. We extend the validity range of this relation from its originally stated unit interval to the entire non-negative real axis, providing a rigorous justification through convergence analysis and identifying a hidden inversion symmetry in the transformation. This extension leads to an appropriate generalization of the original transformation formula. We establish an explicit connection between the studied hypergeometric transformation and singular values <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$K(k_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of complete elliptic integrals of the first kind. This enables us to derive alternative expressions for singular moduli <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$k_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>k</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> and their associated values <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$K(k_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for various integers <i>n</i>. Notably, we provide an explicit expression for the singular modulus <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$k_{75}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>k</mi> <mn>75</mn> </msub> </math></EquationSource> </InlineEquation> absent from comprehensive reference tables and correct a published misprint for the case <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n=24$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>24</mn> </mrow> </math></EquationSource> </InlineEquation>. Working in the reverse direction, we derive new explicit evaluations of Gauss hypergeometric functions <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$_2F_1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mn>2</mn> <mrow /> </mmultiscripts> <msub> <mi>F</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> with various parameter sets and arguments, including a closed form for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$_2F_1(\frac{1}{6},\frac{2}{3};1;-1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mn>2</mn> <mrow /> </mmultiscripts> <msub> <mi>F</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> <mo>,</mo> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> <mo>;</mo> <mn>1</mn> <mo>;</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Finally, we explore connections with Ramanujan’s hypergeometric transformations, revealing previously unrecognized relationships between these classical results.</p>

Legendre’s elliptic integral, hypergeometric transformation formulas, singular values and explicit evaluations

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发表日期：

2026年5月5日

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D. Ritelli,
M. A. Shpot,
H. M. Srivastava

摘要

We investigate a hypergeometric transformation formula originating in Legendre’s classical work on elliptic integrals. Though noted by Kummer in 1836, this transformation was first explicitly recorded in contemporary literature in 2018. We extend the validity range of this relation from its originally stated unit interval to the entire non-negative real axis, providing a rigorous justification through convergence analysis and identifying a hidden inversion symmetry in the transformation. This extension leads to an appropriate generalization of the original transformation formula. We establish an explicit connection between the studied hypergeometric transformation and singular values $K(k_n)$ $K (k_{n})$ of complete elliptic integrals of the first kind. This enables us to derive alternative expressions for singular moduli $k_n$ $k_{n}$ and their associated values $K(k_n)$ $K (k_{n})$ for various integers n. Notably, we provide an explicit expression for the singular modulus $k_{75}$ $k_{75}$ absent from comprehensive reference tables and correct a published misprint for the case $n=24$ $n = 24$ . Working in the reverse direction, we derive new explicit evaluations of Gauss hypergeometric functions $_2F_1$ $_{2} F_{1}$ with various parameter sets and arguments, including a closed form for $_2F_1(\frac{1}{6},\frac{2}{3};1;-1)$ $_{2} F_{1} (\frac{1}{6}, \frac{2}{3}; 1; - 1)$ . Finally, we explore connections with Ramanujan’s hypergeometric transformations, revealing previously unrecognized relationships between these classical results.