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variants of this functions
HarmonicNumber




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > HarmonicNumber[z,r] > Representations through more general functions > Through hypergeometric functions > Involving pFq





http://functions.wolfram.com/06.17.26.0003.01









  


  








Input Form





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








Standard Form





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MathML Form







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</mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> &#10869; </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> &#10869; </mo> <mo> &#8230; </mo> <mo> &#10869; </mo> <msub> <mi> a </mi> <mi> r </mi> </msub> <mo> &#10869; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> &#10869; </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> &#10869; </mo> <mo> &#8230; </mo> <mo> &#10869; </mo> <msub> <mi> b </mi> <mi> r </mi> </msub> <mo> &#10869; </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &#8712; </mo> <msup> <mi> &#8469; </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> &#8230; </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> &#8230; </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> &#8230; </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> &#8230; </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> &#8230; </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> &#8230; </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> &#8469; </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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"]]]]]]]]]]


Date Added to functions.wolfram.com (modification date)





2001-10-29