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<p>We provide a method for determining explicit symbolic evaluations for infinite families of continued fractions with quadratic partial numerators and constant partial denominators. In this manner, we obtain continued fraction identities generalizing results due to Ramanujan and Stieltjes. Our method relies on Nörlund’s formula together with generalizations of classical hypergeometric series identities due to Rakha and Rathie. We also introduce and apply a related technique concerning linear forms for continued fractions to build on the work of Dougherty-Bliss and Zeilberger related to “The Ramanujan Machine.”</p>

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Applications of Nörlund’s formula and linear forms for continued fractions

  • 期刊论文

  • 发表日期：

    2025年2月20日

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  • John Campbell,
  • Kwang-Wu Chen

摘要

We provide a method for determining explicit symbolic evaluations for infinite families of continued fractions with quadratic partial numerators and constant partial denominators. In this manner, we obtain continued fraction identities generalizing results due to Ramanujan and Stieltjes. Our method relies on Nörlund’s formula together with generalizations of classical hypergeometric series identities due to Rakha and Rathie. We also introduce and apply a related technique concerning linear forms for continued fractions to build on the work of Dougherty-Bliss and Zeilberger related to “The Ramanujan Machine.”