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http://functions.wolfram.com/06.17.26.0003.01
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HarmonicNumber[z, r] == HypergeometricPFQ[{1, Subscript[a, 1],
Subscript[a, 2], \[Ellipsis], Subscript[a, r]},
{1 + Subscript[a, 1], 1 + Subscript[a, 2], \[Ellipsis],
1 + Subscript[a, r]}, 1] -
HypergeometricPFQ[{1, Subscript[b, 1], Subscript[b, 2], \[Ellipsis],
Subscript[b, r]}, {1 + Subscript[b, 1], 1 + Subscript[b, 2],
\[Ellipsis], 1 + Subscript[b, r]}, 1]/(z + 1)^r /;
Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, r] == 1 &&
Subscript[b, 1] == Subscript[b, 2] == \[Ellipsis] == Subscript[b, r] ==
z + 1 && Element[r - 1, Integers] && r - 1 > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HarmonicNumber", "[", RowBox[List["z", ",", "r"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "r"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", "r"]]]]], "}"]], ",", "1"]], "]"]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["-", "r"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "r"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["b", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["b", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", "r"]]]]], "}"]], ",", "1"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "\[Equal]", SubscriptBox["a", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", "r"], "\[Equal]", "1"]], "\[And]", RowBox[List[SubscriptBox["b", "1"], "\[Equal]", SubscriptBox["b", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["b", "r"], "\[Equal]", RowBox[List["z", "+", "1"]]]], "\[And]", RowBox[List[RowBox[List["r", "-", "1"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["r", "-", "1"]], ">", "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> <msubsup> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mi> z </mi> <mrow> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ⩵ </mo> <mtext> </mtext> <mrow> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mrow> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <msub> <mi> F </mi> <mi> r </mi> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <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> <mi> r </mi> </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> r </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["r", "+", "1"]], TraditionalForm]], SubscriptBox["F", "r"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "r"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "1"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "r"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> r </mi> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mrow> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <msub> <mi> F </mi> <mi> r </mi> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> r </mi> </msub> </mrow> <mo> ; </mo> <mrow> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mi> r </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["r", "+", "1"]], TraditionalForm]], SubscriptBox["F", "r"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "r"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["b", "1"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", "r"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox["1", 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> <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> <mi> r </mi> </msub> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> ⩵ </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> ⩵ </mo> <mo> … </mo> <mo> ⩵ </mo> <msub> <mi> b </mi> <mi> r </mi> </msub> <mo> ⩵ </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <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> HarmonicNumber </ci> <ci> z </ci> <ci> r </ci> </apply> <apply> <plus /> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='integer'> 1 </cn> <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> <ci> r </ci> </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> r </ci> </apply> <cn type='integer'> 1 </cn> </apply> </list> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> r </ci> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='integer'> 1 </cn> <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> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> r </ci> </apply> </list> <list> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <ci> … </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> r </ci> </apply> <cn type='integer'> 1 </cn> </apply> </list> <cn type='integer'> 1 </cn> </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> <ci> r </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <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> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> r </ci> </apply> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <in /> <apply> <plus /> <ci> r </ci> <cn type='integer'> -1 </cn> </apply> <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["HarmonicNumber", "[", RowBox[List["z_", ",", "r_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["Join", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]], ",", RowBox[List["Table", "[", RowBox[List["1", ",", RowBox[List["{", RowBox[List["K$1", ",", "2", ",", "r"]], "}"]]]], "]"]]]], "]"]], ",", RowBox[List["Join", "[", RowBox[List[RowBox[List["{", "2", "}"]], ",", RowBox[List["Table", "[", RowBox[List["2", ",", RowBox[List["{", RowBox[List["K$1", ",", "2", ",", "r"]], "}"]]]], "]"]]]], "]"]], ",", "1"]], "]"]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["-", "r"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["Join", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "z"]]]], "}"]], ",", RowBox[List["Table", "[", RowBox[List[RowBox[List["1", "+", "z"]], ",", RowBox[List["{", RowBox[List["K$1", ",", "2", ",", "r"]], "}"]]]], "]"]]]], "]"]], ",", RowBox[List["Join", "[", RowBox[List[RowBox[List["{", RowBox[List["2", "+", "z"]], "}"]], ",", RowBox[List["Table", "[", RowBox[List[RowBox[List["2", "+", "z"]], ",", RowBox[List["{", RowBox[List["K$1", ",", "2", ",", "r"]], "}"]]]], "]"]]]], "]"]], ",", "1"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["r", "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "r"]], ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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