We compute Fourier transforms of functions expressed as a ratio of one of the Jacobi elliptic functions divided by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\sinh (\pi x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>sinh</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\cosh (\pi x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>cosh</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In many cases, the resulting Fourier transform remains within the same class of functions. Applying the Mellin transform, we obtain sixteen Eisenstein-type series <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\zeta _{j,l}(s,\tau )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ζ</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>l</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, for which we establish several results: analytic continuation with respect to the variable s, a functional equation connecting <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\zeta _{j,l}(s,\tau )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ζ</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>l</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\zeta _{l,j}(1-s,-1/\tau )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ζ</mi> <mrow> <mi>l</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>s</mi> <mo>,</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and explicit expressions for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\zeta _{j,l}(s,\tau )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ζ</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>l</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> when s runs through a sequence of positive even or odd integers.

Fourier transform pairs and Eisenstein-type series related to Jacobi elliptic functions

期刊论文
发表日期：

2026年3月7日

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Peng-Cheng Hang,
Alexey Kuznetsov

摘要

We compute Fourier transforms of functions expressed as a ratio of one of the Jacobi elliptic functions divided by $\sinh (\pi x)$ $sinh (π x)$ or $\cosh (\pi x)$ $cosh (π x)$ . In many cases, the resulting Fourier transform remains within the same class of functions. Applying the Mellin transform, we obtain sixteen Eisenstein-type series $\zeta _{j,l}(s,\tau )$ $ζ_{j, l} (s, τ)$ , for which we establish several results: analytic continuation with respect to the variable s, a functional equation connecting $\zeta _{j,l}(s,\tau )$ $ζ_{j, l} (s, τ)$ and $\zeta _{l,j}(1-s,-1/\tau )$ $ζ_{l, j} (1 - s, - 1 / τ)$ , and explicit expressions for $\zeta _{j,l}(s,\tau )$ $ζ_{j, l} (s, τ)$ when s runs through a sequence of positive even or odd integers.