Wronskian[z^s KelvinBer[a z^r], z^s KelvinBei[a z^r], z^s KelvinKer[a z^r], z^s KelvinKei[a z^r], z] == (-a^4) r^6 z^(-6 + 4 r + 4 s)

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <mi> W </mi> <mi> z </mi> </msub> <mo> ( </mo> <mrow> <mrow> <msup> <mi> z </mi> <mi> s </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> ber </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <msup> <mi> z </mi> <mi> s </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> bei </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <msup> <mi> z </mi> <mi> s </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> ker </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <msup> <mi> z </mi> <mi> s </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> kei </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> </mrow> <mo> &#8290; </mo> <msup> <mi> r </mi> <mn> 6 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> r </mi> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> s </mi> </mrow> <mo> - </mo> <mn> 6 </mn> </mrow> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> W </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> s </ci> </apply> <apply> <ci> KelvinBer </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> s </ci> </apply> <apply> <ci> KelvinBei </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> s </ci> </apply> <apply> <ci> KelvinKer </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> s </ci> </apply> <apply> <ci> KelvinKei </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <power /> <ci> r </ci> <cn type='integer'> 6 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> r </ci> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <ci> s </ci> </apply> <cn type='integer'> -6 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02