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variants of this functions
MeijerG






Mathematica Notation

Traditional Notation









Hypergeometric Functions > MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z] > Integral transforms > Hankel transforms





http://functions.wolfram.com/07.34.22.0008.01









  


  










Input Form





HankelTransform[t^(\[Alpha] - 1) MeijerG[{{Subscript[a, 1], \[Ellipsis], Subscript[a, n]}, {Subscript[a, n + 1], \[Ellipsis], Subscript[a, p]}}, {{Subscript[b, 1], \[Ellipsis], Subscript[b, m]}, {Subscript[b, m + 1], \[Ellipsis], Subscript[b, q]}}, \[Omega] t^((2 l)/k)], {t, \[Nu]}, z] == ((k^\[Mu] (2 l)^(\[Alpha] - 1/2))/(z^\[Alpha] (2 Pi)^(SuperStar[c] (k - 1)))) MeijerG[{{(1 - (\[Alpha] + \[Nu] + 1/2)/2)/l, \[Ellipsis], (l - (\[Alpha] + \[Nu] + 1/2)/2)/l, Subscript[a, 1]/k, \[Ellipsis], (Subscript[a, 1] + k - 1)/k, \[Ellipsis], Subscript[a, n]/k, \[Ellipsis], (Subscript[a, n] + k - 1)/k}, {Subscript[a, n + 1]/k, \[Ellipsis], (Subscript[a, n + 1] + k - 1)/k, \[Ellipsis], Subscript[a, p]/k, \[Ellipsis], (Subscript[a, p] + k - 1)/k, (1 - (\[Alpha] - \[Nu] + 1/2)/2)/l, \[Ellipsis], (l - (\[Alpha] - \[Nu] + 1/2)/2)/l}}, {{Subscript[b, 1]/k, \[Ellipsis], (Subscript[b, 1] + k - 1)/k, \[Ellipsis], Subscript[b, m]/k, \[Ellipsis], (Subscript[b, m] + k - 1)/ k}, {Subscript[b, m + 1]/k, \[Ellipsis], (Subscript[b, m + 1] + k - 1)/ k, \[Ellipsis], Subscript[b, q]/k, \[Ellipsis], (Subscript[b, q] + k - 1)/k}}, (\[Omega]^k (2 l)^(2 l))/ (z^(2 l) k^(k (q - p)))] /; Element[k, Integers] && k > 0 && Element[l, Integers] && l > 0 && GCD[k, l] == 1 && (\[DoubleStruckCapitalC]\[DoubleStruckCapitalC] /; \[Alpha] -> (2 \[Alpha] + 1)/4 && \[Sigma] == z^2/4 && s == 1 && t == u == 0 && v == 2 && Subscript[d, 1] == -Subscript[d, 2] == \[Nu]/2)










Standard Form





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MathML Form







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<mfrac> <mrow> <mi> l </mi> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> &#945; </mi> <mo> + </mo> <mi> &#957; </mi> <mo> + </mo> <mrow> <mn> 1 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mn> 2 </mn> </mrow> </mrow> <mi> l </mi> </mfrac> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mi> n </mi> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mi> n </mi> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mfrac> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> &#8230; 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/ </mo> <mn> 2 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mn> 2 </mn> </mrow> </mrow> <mi> l </mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mi> m </mi> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mi> m </mi> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <msub> <mi> b </mi> <mi> q </mi> </msub> <mi> k </mi> </mfrac> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[RowBox[List[RowBox[List[&quot;k&quot;, &quot; &quot;, &quot;p&quot;]], &quot;+&quot;, RowBox[List[&quot;2&quot;, &quot;l&quot;]]]], &quot;,&quot;, RowBox[List[&quot;k&quot;, &quot; &quot;, &quot;q&quot;]]]], RowBox[List[RowBox[List[&quot;k&quot;, &quot; &quot;, &quot;m&quot;]], &quot;,&quot;, RowBox[List[RowBox[List[&quot;k&quot;, &quot; &quot;, &quot;n&quot;]], &quot;+&quot;, &quot;l&quot;]]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[FractionBox[RowBox[List[SuperscriptBox[&quot;\[Omega]&quot;, &quot;k&quot;], &quot; &quot;, SuperscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;l&quot;]], &quot;)&quot;]], RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;l&quot;]]]]], RowBox[List[SuperscriptBox[&quot;z&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;l&quot;]]], &quot; &quot;, SuperscriptBox[&quot;k&quot;, RowBox[List[&quot;k&quot;, &quot; &quot;, RowBox[List[&quot;(&quot;, RowBox[List[&quot;q&quot;, &quot;-&quot;, &quot;p&quot;]], &quot;)&quot;]]]]]]]], MeijerG, Rule[Editable, True]], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[RowBox[List[TagBox[FractionBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, RowBox[List[RowBox[List[&quot;(&quot;, RowBox[List[&quot;\[Alpha]&quot;, &quot;+&quot;, &quot;\[Nu]&quot;, &quot;+&quot;, RowBox[List[&quot;1&quot;, &quot;/&quot;, &quot;2&quot;]]]], &quot;)&quot;]], &quot;/&quot;, 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TagBox[FractionBox[SubscriptBox[&quot;a&quot;, &quot;p&quot;], &quot;k&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;p&quot;], &quot;+&quot;, &quot;k&quot;, &quot;-&quot;, &quot;1&quot;]], &quot;k&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, RowBox[List[RowBox[List[&quot;(&quot;, RowBox[List[&quot;\[Alpha]&quot;, &quot;-&quot;, &quot;\[Nu]&quot;, &quot;+&quot;, RowBox[List[&quot;1&quot;, &quot;/&quot;, &quot;2&quot;]]]], &quot;)&quot;]], &quot;/&quot;, &quot;2&quot;]]]], &quot;l&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;l&quot;, &quot;-&quot;, RowBox[List[RowBox[List[&quot;(&quot;, RowBox[List[&quot;\[Alpha]&quot;, &quot;-&quot;, 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&quot;-&quot;, &quot;1&quot;]], &quot;k&quot;], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> l </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> , </mo> <mi> l </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#8450;&#8450; </mi> <mo> /; </mo> <mi> &#945; </mi> </mrow> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#945; </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> &#8743; </mo> <mrow> <mi> &#963; </mi> <mo> &#10869; </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <mi> s </mi> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> t </mi> <mo> &#10869; </mo> <mi> u </mi> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> v </mi> <mo> &#10869; </mo> <mn> 2 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> d </mi> <mn> 1 </mn> </msub> <mo> &#10869; </mo> <mrow> <mo> - </mo> <msub> <mi> d </mi> <mn> 2 </mn> </msub> </mrow> <mo> &#10869; </mo> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HankelTransform", "[", RowBox[List[RowBox[List[SuperscriptBox["t_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", "n_"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["a_", RowBox[List["n_", "+", "1"]]], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", "p_"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", "m_"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", RowBox[List["m_", "+", "1"]]], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", "q_"]]], "}"]]]], "}"]], ",", RowBox[List["\[Omega]_", " ", SuperscriptBox["t_", FractionBox[RowBox[List["2", " ", "l_"]], "k"]]]]]], "]"]]]], ",", RowBox[List["{", RowBox[List["t_", ",", "\[Nu]_"]], 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RowBox[List[FractionBox[SubscriptBox["bb", "1"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["bb", "1"], "+", "k", "-", "1"]], "k"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["bb", "m"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["bb", "m"], "+", "k", "-", "1"]], "k"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[SubscriptBox["b", RowBox[List["m", "+", "1"]]], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", RowBox[List["m", "+", "1"]]], "+", "k", "-", "1"]], "k"], ",", "\[Ellipsis]", ",", FractionBox[SubscriptBox["bb", "q"], "k"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["bb", "q"], "+", "k", "-", "1"]], "k"]]], "}"]]]], "}"]], ",", FractionBox[RowBox[List[SuperscriptBox["\[Omega]", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "l"]], ")"]], RowBox[List["2", " ", "l"]]]]], RowBox[List[SuperscriptBox["z", RowBox[List["2", " ", "l"]]], " ", 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Date Added to functions.wolfram.com (modification date)





2001-10-29