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<p>A two-term functional equation for an infinite series involving the digamma function and a logarithmic factor is derived. A modular relation on p. 220 of Ramanujan’s Lost Notebook as well as a corresponding recent result for the derivative of Deninger’s function are two main ingredients in its derivation. An interesting integral <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1125_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathscr {H}(x)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, which is of independent interest, plays a prominent role in our functional equation. Several alternative representations for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1125_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathscr {H}(x)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">H</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are obtained.</p>

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On a function of Ramanujan twisted by a logarithm

  • 期刊论文

  • 发表日期：

    2025年5月23日

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  • Atul Dixit,
  • Sumukha Sathyanarayana,
  • N. Guru Sharan

摘要

A two-term functional equation for an infinite series involving the digamma function and a logarithmic factor is derived. A modular relation on p. 220 of Ramanujan’s Lost Notebook as well as a corresponding recent result for the derivative of Deninger’s function are two main ingredients in its derivation. An interesting integral \(\mathscr {H}(x)\) H ( x ) , which is of independent interest, plays a prominent role in our functional equation. Several alternative representations for \(\mathscr {H}(x)\) H ( x ) are obtained.