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Hypergeometric Functions
MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,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 in the general case
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http://functions.wolfram.com/07.34.13.0004.01
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(-1)^(m + n - p) z Fold[Function[{f, l}, z D[f, z] + (1 - Subscript[a, l])
f], w[z], {1, \[Ellipsis], p}] -
Fold[Function[{f, k}, z D[f, z] - Subscript[b, k] f], w[z],
{1, \[Ellipsis], q}] == 0 /;
w[z] == Sum[Subscript[c, k] z^Subscript[b, k] HypergeometricPFQ[
{1 + Subscript[b, k] - Subscript[a, 1], \[Ellipsis],
1 + Subscript[b, k] - Subscript[a, p]},
{1 + Subscript[b, k] - Subscript[b, 1], \[Ellipsis],
1 + Subscript[b, k] - Subscript[b, k - 1], 1 + Subscript[b, k] -
Subscript[b, k + 1], \[Ellipsis], 1 + Subscript[b, k] -
Subscript[b, q]}, (-1)^(p - m - n) z], {k, 1, q}] &&
(p < q || (p == q && Abs[z] < 1)) && ForAll[{j, k},
Element[{j, k}, Integers] && j != k && 1 <= j <= q && 1 <= k <= q,
!Element[Subscript[b, j] - Subscript[b, k], Integers]]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "+", "n", "-", "p"]]], " ", "z", " ", RowBox[List["Fold", "[", RowBox[List[RowBox[List["Function", "[", RowBox[List[RowBox[List["{", RowBox[List["f", ",", "l"]], "}"]], ",", RowBox[List[RowBox[List["z", " ", RowBox[List[SubscriptBox["\[PartialD]", "z"], "f"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["a", "l"]]], ")"]], " ", "f"]]]]]], "]"]], ",", RowBox[List["w", "[", "z", "]"]], ",", RowBox[List["{", RowBox[List["1", ",", "\[Ellipsis]", ",", "p"]], "}"]]]], "]"]]]], "-", " ", RowBox[List["Fold", "[", RowBox[List[RowBox[List["Function", "[", RowBox[List[RowBox[List["{", RowBox[List["f", ",", "k"]], "}"]], ",", RowBox[List[RowBox[List["z", " ", RowBox[List[SubscriptBox["\[PartialD]", "z"], "f"]]]], "-", RowBox[List[SubscriptBox["b", "k"], " ", "f"]]]]]], "]"]], ",", RowBox[List["w", "[", "z", "]"]], ",", RowBox[List["{", RowBox[List["1", ",", "\[Ellipsis]", ",", "q"]], "}"]]]], "]"]]]], "\[Equal]", "0"]], " ", "/;", RowBox[List[RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "q"], RowBox[List[SubscriptBox["c", "k"], SuperscriptBox["z", SubscriptBox["b", "k"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["a", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["a", "p"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["b", RowBox[List["k", "-", "1"]]]]], ",", RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["b", RowBox[List["k", "+", "1"]]]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["b", "q"]]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "-", "m", "-", "n"]]], " ", "z"]]]], "]"]]]]]]]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["p", "<", "q"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["p", "\[Equal]", "q"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]]]], ")"]]]], ")"]], "\[And]", RowBox[List[SubscriptBox["\[ForAll]", RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], ",", RowBox[List[RowBox[List[RowBox[List["{", RowBox[List["j", ",", "k"]], "}"]], "\[Element]", "Integers"]], "\[And]", RowBox[List["j", "\[NotEqual]", "k"]], "\[And]", RowBox[List["1", "\[LessEqual]", "j", "\[LessEqual]", "q"]], "\[And]", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "q"]]]]]]], RowBox[List["(", "\[InvisibleSpace]", RowBox[List["Not", "[", RowBox[List[RowBox[List[SubscriptBox["b", "j"], "-", SubscriptBox["b", "k"]]], "\[Element]", "Integers"]], "]"]], ")"]]]]]]]]]]
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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> <mtext> </mtext> <mrow> <mrow> <mstyle scriptlevel='0'> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mi> p </mi> </mrow> </msup> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> l </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> p </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mfrac> <mi> d </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> l </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> - </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mfrac> <mi> d </mi> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mfrac> </mrow> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> w </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mstyle> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> w </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <mrow> <msub> <mi> c </mi> <mi> k </mi> </msub> <mo> ⁢ </mo> <msup> <mi> z </mi> <msub> <mi> b </mi> <mi> k </mi> </msub> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mi> p </mi> </msub> <msub> <mi> F </mi> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> + </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> + </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> </mrow> <mo> ; </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "p"], SubscriptBox["F", RowBox[List["q", "-", "1"]]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"], "+", SubscriptBox["b", "k"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "p"], "+", SubscriptBox["b", "k"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"], "+", SubscriptBox["b", "k"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", RowBox[List["k", "-", "1"]]], "+", SubscriptBox["b", "k"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", RowBox[List["k", "+", "1"]]], "+", SubscriptBox["b", "k"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "q"], "+", SubscriptBox["b", "k"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "-", "m", "-", "n"]]], " ", "z"]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> p </mi> <mo> < </mo> <mi> q </mi> </mrow> <mo> ∨ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> p </mi> <mo>  </mo> <mi> q </mi> </mrow> <mo> ∧ </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <msub> <mo> ∀ </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mi> j </mi> <mo> , </mo> <mi> k </mi> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mrow> <mrow> <mo> { </mo> <mrow> <mi> j </mi> <mo> , </mo> <mi> k </mi> </mrow> <mo> } </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> j </mi> <mo> ≠ </mo> <mi> k </mi> </mrow> <mo> ∧ </mo> <mrow> <mn> 1 </mn> <mo> ≤ </mo> <mi> j </mi> <mo> ≤ </mo> <mi> q </mi> </mrow> <mo> ∧ </mo> <mrow> <mn> 1 </mn> <mo> ≤ </mo> <mi> k </mi> <mo> ≤ </mo> <mi> q </mi> </mrow> </mrow> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> b </mi> <mi> j </mi> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> k </mi> </msub> </mrow> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> m </ci> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> </apply> </apply> <ci> z </ci> <apply> <product /> <bvar> <ci> l </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> p </ci> </uplimit> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <times /> <ci> d </ci> <apply> <power /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> l </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> q </ci> </uplimit> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <times /> <ci> d </ci> <apply> <power /> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> w </ci> <ci> z </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> q </ci> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> k </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> <ci> … </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> p </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> </list> <list> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> <ci> … </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> <ci> … </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> q </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> </list> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <or /> <apply> <lt /> <ci> p </ci> <ci> q </ci> </apply> <apply> <and /> <apply> <eq /> <ci> p </ci> <ci> q </ci> </apply> <apply> <lt /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <forall /> <bvar> <list> <ci> j </ci> <ci> k </ci> </list> </bvar> <bvar> <apply> <and /> <apply> <in /> <list> <ci> j </ci> <ci> k </ci> </list> <integers /> </apply> <apply> <neq /> <ci> j </ci> <ci> k </ci> </apply> <apply> <leq /> <cn type='integer'> 1 </cn> <ci> j </ci> <ci> q </ci> </apply> <apply> <leq /> <cn type='integer'> 1 </cn> <ci> k </ci> <ci> q </ci> </apply> </apply> </bvar> <apply> <notin /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> </apply> </apply> <integers /> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z,r] | |
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