Let f(t) be a characteristic function. The question on infinite divisibility of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7984_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="161" /> </InlineMediaObject> <EquationSource Format="TEX">$g_{2k}(t)\!\!=\!\!f^{(2k)}(t)/\!f^{(2k)}(0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>g</mi> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="-0.166667em" /> <mspace width="-0.166667em" /> <mo>=</mo> <mspace width="-0.166667em" /> <mspace width="-0.166667em" /> <msup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mspace width="-0.166667em" /> <msup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is considered. The condition for that function not to be infinitely divisible is given. Some examples of infinite divisibility of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7984_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$g_{2k}(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>g</mi> <mrow> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are presented. Bibliography: 2 titles.

ON A PROBLEM OF INFINITE DIVISIBILITY

期刊论文
发表日期：

2025年9月1日

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L. B. Klebanov

摘要

Let f(t) be a characteristic function. The question on infinite divisibility of $g_{2k}(t)\!\!=\!\!f^{(2k)}(t)/\!f^{(2k)}(0)$ $g_{2 k} (t) = f^{(2 k)} (t) / f^{(2 k)} (0)$ is considered. The condition for that function not to be infinitely divisible is given. Some examples of infinite divisibility of $g_{2k}(t)$ $g_{2 k} (t)$ are presented. Bibliography: 2 titles.