<p>The aim of this paper is to discuss the transcendence over a field <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_660_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {K}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">K</mi> </math></EquationSource> </InlineEquation>, of certain infinite product of the form <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_660_Article_IEq2.gif" Format="GIF" Height="48" Rendition="HTML" Resolution="72" Type="Linedraw" Width="101" /> </InlineMediaObject> <EquationSource Format="TEX">$\displaystyle \prod _{i=0}^{\infty }(1-X^{-\lambda _{i}})$</EquationSource> <EquationSource Format="MATHML"><math> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <munderover> <mo>∏</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>∞</mi> </munderover> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>X</mi> <mrow> <mo>-</mo> <msub> <mi>λ</mi> <mi>i</mi> </msub> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </mstyle> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_660_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$(\lambda _{i})_{i\ge 0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>i</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> is an increasing sequence of positive integers, throughout the value of their irrationality measures.</p>

Transcendence of some infinite products in the field of power series

期刊论文
发表日期：

2025年6月6日

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Farah Abbes,
Khalil Ayadi

摘要

The aim of this paper is to discuss the transcendence over a field $\mathbb {K}$ $K$ , of certain infinite product of the form $\displaystyle \prod _{i=0}^{\infty }(1-X^{-\lambda _{i}})$ $\prod_{i = 0}^{\infty} (1 - X^{- λ_{i}})$ , where $(\lambda _{i})_{i\ge 0}$ ${(λ_{i})}_{i \geq 0}$ is an increasing sequence of positive integers, throughout the value of their irrationality measures.