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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/07.34.13.0005.01
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Wronskian[z^Subscript[b, 1] HypergeometricPFQ[
{1 + Subscript[b, 1] - Subscript[a, 1], \[Ellipsis],
1 + Subscript[b, 1] - Subscript[a, p]},
{1 + Subscript[b, 1] - Subscript[b, 2], \[Ellipsis],
1 + Subscript[b, 1] - Subscript[b, q]}, (-1)^(p - m - n) z],
\[Ellipsis], 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], \[Ellipsis],
z^Subscript[b, q] HypergeometricPFQ[{1 + Subscript[b, q] - Subscript[a, 1],
\[Ellipsis], 1 + Subscript[b, q] - Subscript[a, p]},
{1 + Subscript[b, q] - Subscript[b, 1], \[Ellipsis],
1 + Subscript[b, q] - Subscript[b, q - 1]}, (-1)^(p - m - n) z], z] ==
((-Product[Subscript[b, j] - Subscript[b, k], {k, 1, q}, {j, 1, k - 1}])
(KroneckerDelta[p, q] (1 - (-1)^(m + n - q) z)^
(-q - Sum[Subscript[b, k], {k, 1, q}] + Sum[Subscript[a, l],
{l, 1, q}]) + Exp[(-(-1)^(m + n - q)) z] KroneckerDelta[p, q - 1] +
UnitStep[q - p - 2]))/z^((q (q - 1))/2 - Sum[Subscript[b, k], {k, 1, q}])
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Cell[BoxData[RowBox[List[RowBox[List["Wronskian", "[", RowBox[List[RowBox[List[SuperscriptBox["z", SubscriptBox["b", "1"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["b", "1"], "-", SubscriptBox["a", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", "1"], "-", SubscriptBox["a", "p"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["b", "1"], "-", SubscriptBox["b", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", "1"], "-", SubscriptBox["b", "q"]]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "-", "m", "-", "n"]]], " ", "z"]]]], "]"]]]], ",", "\[Ellipsis]", ",", RowBox[List[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"]]]], "]"]]]], ",", "\[Ellipsis]", ",", RowBox[List[SuperscriptBox["z", SubscriptBox["b", "q"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["b", "q"], "-", SubscriptBox["a", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", "q"], "-", SubscriptBox["a", "p"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["b", "q"], "-", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", "q"], "-", SubscriptBox["b", RowBox[List["q", "-", "1"]]]]]]], "}"]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "-", "m", "-", "n"]]], " ", "z"]]]], "]"]]]], ",", " ", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["(", " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "q"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], RowBox[List["k", "-", "1"]]], RowBox[List["(", RowBox[List[SubscriptBox["b", "j"], "-", SubscriptBox["b", "k"]]], ")"]]]]]], ")"]]]], SuperscriptBox["z", RowBox[List["-", RowBox[List["(", " ", RowBox[List[FractionBox[RowBox[List["q", RowBox[List["(", RowBox[List["q", "-", "1"]], ")"]]]], "2"], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "q"], SubscriptBox["b", "k"]]]]], ")"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List["p", ",", "q"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "+", "n", "-", "q"]]], " ", "z"]]]], ")"]], RowBox[List["(", " ", RowBox[List[RowBox[List["-", "q"]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "q"], SubscriptBox["b", "k"]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["l", "=", "1"]], "q"], SubscriptBox["a", "l"]]]]], ")"]]]]], "+", RowBox[List[RowBox[List["Exp", "[", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "+", "n", "-", "q"]]]]], " ", "z"]], "]"]], RowBox[List["KroneckerDelta", "[", RowBox[List["p", ",", RowBox[List["q", "-", "1"]]]], "]"]]]], "+", RowBox[List["UnitStep", "[", RowBox[List["q", "-", "p", "-", "2"]], "]"]]]], ")"]]]]]]]]
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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> <msub> <mi> b </mi> <mn> 1 </mn> </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> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> b </mi> <mn> 1 </mn> </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> <mn> 1 </mn> </msub> </mrow> </mrow> <mo> ; </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> </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[RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"], "+", SubscriptBox["b", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "p"], "+", SubscriptBox["b", "1"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", RowBox[List["1", "+", SubscriptBox["b", "1"], "-", SubscriptBox["b", "2"]]]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", "1"], "-", SubscriptBox["b", "q"]]], ",", 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> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <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> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msup> <mi> z </mi> <msub> <mi> b </mi> <mi> q </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> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> b </mi> <mi> q </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> q </mi> </msub> </mrow> </mrow> <mo> ; </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> </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> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </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[RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"], "+", SubscriptBox["b", "q"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "p"], "+", SubscriptBox["b", "q"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", RowBox[List["1", "+", SubscriptBox["b", "q"], "-", SubscriptBox["b", "1"]]]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", "q"], "-", SubscriptBox["b", RowBox[List["q", "-", "1"]]]]], ",", 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> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <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> </mrow> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mi> q </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <munderover> <mo> ∑ </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> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> p </mi> <mo> , </mo> <mi> q </mi> </mrow> </msub> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <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> q </mi> </mrow> </msup> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> q </mi> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> l </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <msub> <mi> a </mi> <mi> l </mi> </msub> </mrow> <mo> - </mo> <mrow> <munderover> <mo> ∑ </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> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <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> q </mi> </mrow> </msup> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <msub> <semantics> <mi> δ </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> </mrow> <mo> + </mo> <mrow> <semantics> <mi> θ </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mi> p </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <msub> <mi> W </mi> <mi> z </mi> </msub> <mo> ( </mo> <mrow> <mrow> <msup> <mi> z </mi> <msub> <mi> b </mi> <mn> 1 </mn> </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> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> b </mi> <mn> 1 </mn> </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> <mn> 1 </mn> </msub> </mrow> </mrow> <mo> ; </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> </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[RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"], "+", SubscriptBox["b", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "p"], "+", SubscriptBox["b", "1"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", RowBox[List["1", "+", SubscriptBox["b", "1"], "-", SubscriptBox["b", "2"]]]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", "1"], "-", SubscriptBox["b", "q"]]], ",", 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> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <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> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msup> <mi> z </mi> <msub> <mi> b </mi> <mi> q </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> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> b </mi> <mi> q </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> q </mi> </msub> </mrow> </mrow> <mo> ; </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> </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> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </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[RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"], "+", SubscriptBox["b", "q"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "p"], "+", SubscriptBox["b", "q"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", RowBox[List["1", "+", SubscriptBox["b", "q"], "-", SubscriptBox["b", "1"]]]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", "q"], "-", SubscriptBox["b", RowBox[List["q", "-", "1"]]]]], ",", 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> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <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> </mrow> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mi> q </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <munderover> <mo> ∑ </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> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> p </mi> <mo> , </mo> <mi> q </mi> </mrow> </msub> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <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> q </mi> </mrow> </msup> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> q </mi> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> l </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <msub> <mi> a </mi> <mi> l </mi> </msub> </mrow> <mo> - </mo> <mrow> <munderover> <mo> ∑ </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> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <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> q </mi> </mrow> </msup> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <msub> <semantics> <mi> δ </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> </mrow> <mo> + </mo> <mrow> <semantics> <mi> θ </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mi> p </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
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