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<p>Recently, we have considered inverse applications of the generalized Littlewood theorem about integrals of the logarithm of analytic functions, and using it proved and reproved a number of transformation and <i>n</i>-tuple product rules for different elliptic functions. Here we continue this approach, and establish additional rules of this type for elliptic theta – functions and Weierstrass sigma-functions. Then we apply it to establish such rules for certain ratios of the elliptic theta functions, viz. the derivative of the Weierstrass <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\wp\)</EquationSource> </InlineEquation>–function and Jacobi elliptic functions. The most part of these rules are known (but some apparently do required more details), but some, to the best of our knowledge, are new. We also presented some corollaries of the established rules, such as e.g. the relation <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\delta }_{2}\left(\tau \right)=4-4\sum_{k=2}^{\infty }{\delta }_{2k}\left(\tau \right){2}^{-2k}\)</EquationSource> </InlineEquation> between different Eisenstein sums (see the text for additional conditions), or equalities like <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\({\delta }_{2}\left(2i\right)=\frac{{\Gamma }^{4}\left(1/4\right)}{32\pi }+\frac{\pi }{2}\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\sum_{k=1}^{l}\wp \left(k\tau ,\left(2l+1\right)\tau \right)=\frac{1}{2}\left[{\delta }_{2}\left(\tau \right)-\left(2l+1\right){\delta }_{2}\left(\left(2l+1\right)\tau \right)\right]\)</EquationSource> </InlineEquation>, etc.</p>

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On some additional n-tuple product rules for elliptic functions and their corollaries

  • 期刊论文

  • 发表日期：

    2026年5月22日

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  • Sergey K. Sekatskii

摘要

Recently, we have considered inverse applications of the generalized Littlewood theorem about integrals of the logarithm of analytic functions, and using it proved and reproved a number of transformation and n-tuple product rules for different elliptic functions. Here we continue this approach, and establish additional rules of this type for elliptic theta – functions and Weierstrass sigma-functions. Then we apply it to establish such rules for certain ratios of the elliptic theta functions, viz. the derivative of the Weierstrass \(\wp\) –function and Jacobi elliptic functions. The most part of these rules are known (but some apparently do required more details), but some, to the best of our knowledge, are new. We also presented some corollaries of the established rules, such as e.g. the relation \({\delta }_{2}\left(\tau \right)=4-4\sum_{k=2}^{\infty }{\delta }_{2k}\left(\tau \right){2}^{-2k}\) between different Eisenstein sums (see the text for additional conditions), or equalities like \({\delta }_{2}\left(2i\right)=\frac{{\Gamma }^{4}\left(1/4\right)}{32\pi }+\frac{\pi }{2}\) , \(\sum_{k=1}^{l}\wp \left(k\tau ,\left(2l+1\right)\tau \right)=\frac{1}{2}\left[{\delta }_{2}\left(\tau \right)-\left(2l+1\right){\delta }_{2}\left(\left(2l+1\right)\tau \right)\right]\) , etc.