We introduce the partition function <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_992_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$p_m(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mi>m</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which counts the number of partitions of n having more parts larger than m compared to parts less than or equal to m. We show that the generating function for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_992_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$p_m (n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mi>m</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is related to the q-Laguerre polynomial. Using relations among q-Laguerre polynomials, we investigate partition identities and inequalities for the partition function <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_992_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$p_{d, \alpha , \beta } (n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mrow> <mi>d</mi> <mo>,</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which includes <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_992_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$p_m (n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mi>m</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for a special case.

q-Laguerre polynomials and partitions with inequality conditions

期刊论文
发表日期：

2025年2月5日

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Byungchan Kim,
Eunmi Kim

摘要

We introduce the partition function $p_m(n)$ $p_{m} (n)$ , which counts the number of partitions of n having more parts larger than m compared to parts less than or equal to m. We show that the generating function for $p_m (n)$ $p_{m} (n)$ is related to the q-Laguerre polynomial. Using relations among q-Laguerre polynomials, we investigate partition identities and inequalities for the partition function $p_{d, \alpha , \beta } (n)$ $p_{d, α, β} (n)$ , which includes $p_m (n)$ $p_{m} (n)$ for a special case.