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 | | http://functions.wolfram.com/07.22.25.0001.01 | 
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 | | Limit[(1/Gamma[Subscript[b, 1]]) HypergeometricPFQ[{Subscript[a, 1]}, 
     {Subscript[b, 1], Subscript[b, 2]}, z], Subscript[b, 1] -> -n] == 
  z^(n + 1) Pochhammer[Subscript[a, 1], n + 1] HypergeometricPFQRegularized[
    {Subscript[a, 1] + n + 1}, {n + 2, Subscript[b, 2] + n + 1}, z] /; 
 Element[n, Integers] && n >= 0 | 
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 | | Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Limit", "[", " ", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["Gamma", "[", SubscriptBox["b", "1"], "]"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", SubscriptBox["a", "1"], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"]]], "}"]], ",", "z"]], "]"]]]], ",", RowBox[List[SubscriptBox["b", "1"], "\[Rule]", RowBox[List["-", "n"]]]]]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["z", RowBox[List["n", "+", "1"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "1"], ",", RowBox[List["n", "+", "1"]]]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], "+", "n", "+", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["n", "+", "2"]], ",", RowBox[List[SubscriptBox["b", "2"], "+", "n", "+", "1"]]]], "}"]], ",", "z"]], "]"]]]]]], " ", "/;", " ", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]] | 
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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>  <mrow>  <munder>  <mi> lim </mi>  <mrow>  <msub>  <mi> b </mi>  <mn> 1 </mn>  </msub>  <semantics>  <mo> → </mo>  <annotation encoding='Mathematica'> "\[Rule]" </annotation>  </semantics>  <mrow>  <mo> - </mo>  <mi> n </mi>  </mrow>  </mrow>  </munder>  <mo> ⁢ </mo>  <mtext>   </mtext>  <mfrac>  <semantics>  <mrow>  <mrow>  <msub>  <mo>   </mo>  <mn> 1 </mn>  </msub>  <msub>  <mi> F </mi>  <mn> 2 </mn>  </msub>  </mrow>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> ; </mo>  <mrow>  <msub>  <mi> b </mi>  <mn> 1 </mn>  </msub>  <mo> , </mo>  <msub>  <mi> b </mi>  <mn> 2 </mn>  </msub>  </mrow>  <mo> ; </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["z", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation>  </semantics>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <msub>  <mi> b </mi>  <mn> 1 </mn>  </msub>  <mo> ) </mo>  </mrow>  </mfrac>  </mrow>  <mo> ⩵ </mo>  <mrow>  <msup>  <mi> z </mi>  <mrow>  <mi> n </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msup>  <mo> ⁢ </mo>  <semantics>  <msub>  <mrow>  <mo> ( </mo>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> ) </mo>  </mrow>  <mrow>  <mi> n </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msub>  <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "1"], ")"]], RowBox[List["n", "+", "1"]]], Pochhammer] </annotation>  </semantics>  <mo> ⁢ </mo>  <semantics>  <mrow>  <mrow>  <msub>  <mo>   </mo>  <mn> 1 </mn>  </msub>  <msub>  <mover>  <mi> F </mi>  <mo> ~ </mo>  </mover>  <mn> 2 </mn>  </msub>  </mrow>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mi> n </mi>  <mo> + </mo>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ; </mo>  <mrow>  <mrow>  <mi> n </mi>  <mo> + </mo>  <mn> 2 </mn>  </mrow>  <mo> , </mo>  <mrow>  <mi> n </mi>  <mo> + </mo>  <msub>  <mi> b </mi>  <mn> 2 </mn>  </msub>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </mrow>  <mo> ; </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List["n", "+", SubscriptBox["a", "1"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["n", "+", "2"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", SubscriptBox["b", "2"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox["z", HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation>  </semantics>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <mi> n </mi>  <mo> ∈ </mo>  <mi> ℕ </mi>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <apply>  <ci> Condition </ci>  <apply>  <eq />  <apply>  <limit />  <bvar>  <apply>  <ci> Subscript </ci>  <ci> b </ci>  <cn type='integer'> 1 </cn>  </apply>  </bvar>  <condition>  <apply>  <tendsto />  <apply>  <ci> Subscript </ci>  <ci> b </ci>  <cn type='integer'> 1 </cn>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> n </ci>  </apply>  </apply>  </condition>  <apply>  <times />  <apply>  <ci> HypergeometricPFQ </ci>  <list>  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <cn type='integer'> 1 </cn>  </apply>  </list>  <list>  <apply>  <ci> Subscript </ci>  <ci> b </ci>  <cn type='integer'> 1 </cn>  </apply>  <apply>  <ci> Subscript </ci>  <ci> b </ci>  <cn type='integer'> 2 </cn>  </apply>  </list>  <ci> z </ci>  </apply>  <apply>  <power />  <apply>  <ci> Gamma </ci>  <apply>  <ci> Subscript </ci>  <ci> b </ci>  <cn type='integer'> 1 </cn>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  <apply>  <times />  <apply>  <power />  <ci> z </ci>  <apply>  <plus />  <ci> n </ci>  <cn type='integer'> 1 </cn>  </apply>  </apply>  <apply>  <ci> Pochhammer </ci>  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <cn type='integer'> 1 </cn>  </apply>  <apply>  <plus />  <ci> n </ci>  <cn type='integer'> 1 </cn>  </apply>  </apply>  <apply>  <ci> HypergeometricPFQRegularized </ci>  <list>  <apply>  <plus />  <ci> n </ci>  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <cn type='integer'> 1 </cn>  </apply>  <cn type='integer'> 1 </cn>  </apply>  </list>  <list>  <apply>  <plus />  <ci> n </ci>  <cn type='integer'> 2 </cn>  </apply>  <apply>  <plus />  <ci> n </ci>  <apply>  <ci> Subscript </ci>  <ci> b </ci>  <cn type='integer'> 2 </cn>  </apply>  <cn type='integer'> 1 </cn>  </apply>  </list>  <ci> z </ci>  </apply>  </apply>  </apply>  <apply>  <in />  <ci> n </ci>  <ci> ℕ </ci>  </apply>  </apply>  </annotation-xml>  </semantics>  </math> 
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 | | Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Limit", "[", RowBox[List[FractionBox[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", SubscriptBox["a_", "1"], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", SubscriptBox["b_", "2"]]], "}"]], ",", "z_"]], "]"]], RowBox[List["Gamma", "[", SubscriptBox["b_", "1"], "]"]]], ",", RowBox[List[SubscriptBox["b_", "1"], "\[Rule]", RowBox[List["-", "n_"]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["n", "+", "1"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["aa", "1"], ",", RowBox[List["n", "+", "1"]]]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "1"], "+", "n", "+", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["n", "+", "2"]], ",", RowBox[List[SubscriptBox["bb", "2"], "+", "n", "+", "1"]]]], "}"]], ",", "z"]], "]"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]] | 
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 | | Date Added to functions.wolfram.com (modification date) | 
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 | | HypergeometricPFQ[{},{},z] |  | HypergeometricPFQ[{},{b},z] |  | HypergeometricPFQ[{a},{},z] |  | HypergeometricPFQ[{a},{b},z] |  | HypergeometricPFQ[{a1,a2},{b1},z] |  | HypergeometricPFQ[{a1,a2},{b1,b2},z] |  | HypergeometricPFQ[{a1,a2},{b1,b2,b3},z] |  | HypergeometricPFQ[{a1,a2,a3},{b1,b2},z] |  | HypergeometricPFQ[{a1,a2,a3,a4},{b1,b2,b3},z] |  | HypergeometricPFQ[{a1,a2,a3,a4,a5},{b1,b2,b3,b4},z] |  | HypergeometricPFQ[{a1,a2,a3,a4,a5,a6},{b1,b2,b3,b4,b5},z] |  | HypergeometricPFQ[{a1,...,ap},{b1,...,bq},z] |  |  | 
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