We introduce an analytic function <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1045_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="113" /> </InlineMediaObject> <EquationSource Format="TEX">$\Psi (s_1,\ldots ,s_d;w)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>s</mi> <mi>d</mi> </msub> <mo>;</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> that interpolates truncated multiple zeta functions <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1045_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="100" /> </InlineMediaObject> <EquationSource Format="TEX">$\zeta _N(s_1,\ldots ,s_d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ζ</mi> <mi>N</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>s</mi> <mi>d</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We represent this interpolant as a Mellin transform of a function <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1045_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="113" /> </InlineMediaObject> <EquationSource Format="TEX">$G(q_1,\ldots ,q_d;w)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>q</mi> <mi>d</mi> </msub> <mo>;</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and, using this expression, give the analytic continuation. Further, the harmonic product relations for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1045_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\Psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ψ</mi> </math></EquationSource> </InlineEquation> and G are established via relevant Hopf algebra structures, and some properties of the function G are provided.

Interpolant of truncated multiple zeta functions

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2025年3月4日

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Kentaro Ihara,
Yayoi Nakamura,
Shuji Yamamoto

摘要

We introduce an analytic function $\Psi (s_1,\ldots ,s_d;w)$ $Ψ (s_{1}, \dots, s_{d}; w)$ that interpolates truncated multiple zeta functions $\zeta _N(s_1,\ldots ,s_d)$ $ζ_{N} (s_{1}, \dots, s_{d})$ . We represent this interpolant as a Mellin transform of a function $G(q_1,\ldots ,q_d;w)$ $G (q_{1}, \dots, q_{d}; w)$ and, using this expression, give the analytic continuation. Further, the harmonic product relations for $\Psi $ $Ψ$ and G are established via relevant Hopf algebra structures, and some properties of the function G are provided.