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variants of this functions
HypergeometricPFQ




Mathematica Notation

Traditional Notation









Hypergeometric Functions > HypergeometricPFQ[{a1,...,ap},{b1,...,bq},z] > Differential equations > Ordinary linear differential equations and wronskians > For the direct function itself > Representation of fundamental system solutions near point z==0 for p<=q+1 in the general case





http://functions.wolfram.com/07.31.13.0006.01









  


  








Input Form





Wronskian[HypergeometricPFQRegularized[{Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z], z^(1 - Subscript[b, 1]) HypergeometricPFQRegularized[ {1 + Subscript[a, 1] - Subscript[b, 1], \[Ellipsis], 1 + Subscript[a, p] - Subscript[b, 1]}, {2 - Subscript[b, 1], 1 + Subscript[b, 2] - Subscript[b, 1], \[Ellipsis], 1 + Subscript[b, q] - Subscript[b, 1]}, z], \[Ellipsis], z^(1 - Subscript[b, k]) HypergeometricPFQRegularized[ {1 + Subscript[a, 1] - Subscript[b, k], \[Ellipsis], 1 + Subscript[a, p] - Subscript[b, k]}, {2 - Subscript[b, k], 1 + Subscript[b, 1] - Subscript[b, k], \[Ellipsis], 1 + Subscript[b, k - 1] - Subscript[b, k], 1 + Subscript[b, k + 1] - Subscript[b, k], \[Ellipsis], 1 + Subscript[b, q] - Subscript[b, k]}, z], \[Ellipsis], z^(1 - Subscript[b, q]) HypergeometricPFQRegularized[ {1 + Subscript[a, 1] - Subscript[b, q], \[Ellipsis], 1 + Subscript[a, p] - Subscript[b, q]}, {2 - Subscript[b, q], 1 + Subscript[b, 1] - Subscript[b, q], \[Ellipsis], 1 + Subscript[b, q - 1] - Subscript[b, q]}, z], z] == (Product[Sin[Pi Subscript[b, k]], {k, 1, q}] Product[Sin[Pi (Subscript[b, j] - Subscript[b, k])], {k, 1, q}, {j, 1, k - 1}] z^(-((q (q - 1))/2) - Sum[Subscript[b, k], {k, 1, q}]) (KroneckerDelta[p, q + 1] (1 - z)^(-q + Sum[Subscript[b, k], {k, 1, q}] - Sum[Subscript[a, l], {l, 1, q + 1}]) + E^z KroneckerDelta[p, q] + UnitStep[q - p - 1]))/Pi^((q (1 + q))/2)








Standard Form





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MathML Form







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</mo> <mrow> <mrow> <msup> <mi> &#960; </mi> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> q </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mn> 2 </mn> </mfrac> <mtext> </mtext> </mrow> </msup> <mo> ( </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mi> j </mi> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> q </mi> </mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> p </mi> <mo> , </mo> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> </msub> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> q </mi> </mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> l </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <msub> <mi> a </mi> <mi> l </mi> </msub> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mi> z </mi> </msup> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> p </mi> <mo> , </mo> <mi> q </mi> </mrow> </msub> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> p </mi> </mrow> <mo> + </mo> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





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Date Added to functions.wolfram.com (modification date)





2007-05-02