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 | | http://functions.wolfram.com/07.32.03.0019.01 | 
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 | | HypergeometricPFQRegularized[{Subscript[a, 1], Subscript[a, 2], \[Ellipsis], 
    Subscript[a, q + 1]}, {1 + Subscript[a, 1], 1 + Subscript[a, 2], 
    \[Ellipsis], 1 + Subscript[a, q]}, z] == (1/Gamma[m]^(q + 1)) 
   Sum[StirlingS1[m, k] LerchPhi[z, q - k + 1, m], {k, 0, m}] /; 
 Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, q + 1] == 
   m && Element[m, Integers] && m > 0 | 
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 | | Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["q", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", "q"]]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["Gamma", "[", "m", "]"]], RowBox[List["q", "+", "1"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], " ", RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["LerchPhi", "[", RowBox[List["z", ",", RowBox[List["q", "-", "k", "+", "1"]], ",", "m"]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "\[Equal]", SubscriptBox["a", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["q", "+", "1"]]], "\[Equal]", "m"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "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>  <semantics>  <mrow>  <mrow>  <msub>  <mo>   </mo>  <mrow>  <mi> q </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msub>  <msub>  <mover>  <mi> F </mi>  <mo> ~ </mo>  </mover>  <mi> q </mi>  </msub>  </mrow>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> , </mo>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  <mo> , </mo>  <mo> … </mo>  <mo> , </mo>  <msub>  <mi> a </mi>  <mrow>  <mi> q </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msub>  </mrow>  <mo> ; </mo>  <mrow>  <mrow>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> , </mo>  <mrow>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> , </mo>  <mo> … </mo>  <mo> , </mo>  <mrow>  <msub>  <mi> a </mi>  <mi> q </mi>  </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[RowBox[List["q", "+", "1"]], TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", RowBox[List["q", "+", "1"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "1"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "q"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox["z", HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQRegularized] </annotation>  </semantics>  <mo> ⩵ </mo>  <mrow>  <mfrac>  <mn> 1 </mn>  <msup>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mi> m </mi>  <mo> ) </mo>  </mrow>  <mrow>  <mi> q </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msup>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <munderover>  <mo> ∑ </mo>  <mrow>  <mi> k </mi>  <mo> = </mo>  <mn> 0 </mn>  </mrow>  <mi> m </mi>  </munderover>  <mrow>  <msubsup>  <semantics>  <mi> S </mi>  <annotation encoding='Mathematica'> TagBox["S", StirlingS1] </annotation>  </semantics>  <mi> m </mi>  <mrow>  <mo> ( </mo>  <mi> k </mi>  <mo> ) </mo>  </mrow>  </msubsup>  <mo> ⁢ </mo>  <semantics>  <mrow>  <mi> Φ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> z </mi>  <mo> , </mo>  <mrow>  <mi> q </mi>  <mo> - </mo>  <mi> k </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> , </mo>  <mi> m </mi>  </mrow>  <mo> ) </mo>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List["\[CapitalPhi]", "(", RowBox[List[TagBox["z", Rule[Editable, True]], ",", TagBox[RowBox[List["q", "-", "k", "+", "1"]], Rule[Editable, True]], ",", TagBox["m", Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2, $CellContext`e3], LerchPhi[$CellContext`e1, $CellContext`e2, $CellContext`e3]]]] </annotation>  </semantics>  </mrow>  </mrow>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <mrow>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> ⩵ </mo>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  <mo> ⩵ </mo>  <mo> … </mo>  <mo> ⩵ </mo>  <msub>  <mi> a </mi>  <mrow>  <mi> q </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msub>  <mo> ⩵ </mo>  <mi> m </mi>  </mrow>  <mo> ∧ </mo>  <mrow>  <mi> m </mi>  <mo> ∈ </mo>  <msup>  <mi> ℕ </mi>  <mo> + </mo>  </msup>  </mrow>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <apply>  <ci> Condition </ci>  <apply>  <eq />  <apply>  <ci> HypergeometricPFQRegularized </ci>  <list>  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <cn type='integer'> 1 </cn>  </apply>  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <cn type='integer'> 2 </cn>  </apply>  <ci> … </ci>  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <apply>  <plus />  <ci> q </ci>  <cn type='integer'> 1 </cn>  </apply>  </apply>  </list>  <list>  <apply>  <plus />  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <cn type='integer'> 1 </cn>  </apply>  <cn type='integer'> 1 </cn>  </apply>  <apply>  <plus />  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <cn type='integer'> 2 </cn>  </apply>  <cn type='integer'> 1 </cn>  </apply>  <ci> … </ci>  <apply>  <plus />  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <ci> q </ci>  </apply>  <cn type='integer'> 1 </cn>  </apply>  </list>  <ci> z </ci>  </apply>  <apply>  <times />  <apply>  <times />  <cn type='integer'> 1 </cn>  <apply>  <power />  <apply>  <power />  <apply>  <ci> Gamma </ci>  <ci> m </ci>  </apply>  <apply>  <plus />  <ci> q </ci>  <cn type='integer'> 1 </cn>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <sum />  <bvar>  <ci> k </ci>  </bvar>  <lowlimit>  <cn type='integer'> 0 </cn>  </lowlimit>  <uplimit>  <ci> m </ci>  </uplimit>  <apply>  <times />  <apply>  <ci> StirlingS1 </ci>  <ci> m </ci>  <ci> k </ci>  </apply>  <apply>  <ci> LerchPhi </ci>  <ci> z </ci>  <apply>  <plus />  <ci> q </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> k </ci>  </apply>  <cn type='integer'> 1 </cn>  </apply>  <ci> m </ci>  </apply>  </apply>  </apply>  </apply>  </apply>  <apply>  <and />  <apply>  <eq />  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <cn type='integer'> 1 </cn>  </apply>  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <cn type='integer'> 2 </cn>  </apply>  <ci> … </ci>  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <apply>  <plus />  <ci> q </ci>  <cn type='integer'> 1 </cn>  </apply>  </apply>  <ci> m </ci>  </apply>  <apply>  <in />  <ci> m </ci>  <apply>  <ci> SuperPlus </ci>  <ci> ℕ </ci>  </apply>  </apply>  </apply>  </apply>  </annotation-xml>  </semantics>  </math> 
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 | | Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a_", "1"], ",", SubscriptBox["a_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", RowBox[List["q_", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a_", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["a_", "2"]]], ",", "\[Ellipsis]_", ",", RowBox[List["1", "+", SubscriptBox["a_", "q_"]]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["LerchPhi", "[", RowBox[List["z", ",", RowBox[List["q", "-", "k", "+", "1"]], ",", "m"]], "]"]]]]]], SuperscriptBox[RowBox[List["Gamma", "[", "m", "]"]], RowBox[List["q", "+", "1"]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "\[Equal]", SubscriptBox["a", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["q", "+", "1"]]], "\[Equal]", "m"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]] | 
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 | | Date Added to functions.wolfram.com (modification date) | 
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