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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 Meijer G > Classical cases for the direct function itself





http://functions.wolfram.com/06.17.26.0004.01









  


  








Input Form





HarmonicNumber[z, r] == MeijerG[{{0, 1 - Subscript[a, 1], \[Ellipsis], 1 - Subscript[a, r]}, {}}, {{0}, {-Subscript[a, 1], \[Ellipsis], -Subscript[a, r]}}, -1] - MeijerG[{{0, 1 - Subscript[b, 1], \[Ellipsis], 1 - Subscript[b, r]}, {}}, {{0}, {-Subscript[b, 1], \[Ellipsis], -Subscript[b, r]}}, -1] /; 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'> <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> &#10869; </mo> <mtext> </mtext> <mrow> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mrow> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mrow> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> </msubsup> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> r </mi> </msub> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> , </mo> <mrow> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mo> - </mo> <msub> <mi> a </mi> <mi> r </mi> </msub> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[RowBox[List[&quot;r&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;,&quot;, RowBox[List[&quot;r&quot;, &quot;+&quot;, &quot;1&quot;]]]], RowBox[List[&quot;1&quot;, &quot;,&quot;, RowBox[List[&quot;r&quot;, &quot;+&quot;, &quot;1&quot;]]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[RowBox[List[&quot;-&quot;, &quot;1&quot;]], MeijerG, Rule[Editable, True]], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[RowBox[List[TagBox[&quot;0&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;a&quot;, &quot;1&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;a&quot;, &quot;r&quot;]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[&quot;0&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;-&quot;, SubscriptBox[&quot;a&quot;, &quot;1&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;-&quot;, SubscriptBox[&quot;a&quot;, &quot;r&quot;]]], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> - </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mrow> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mrow> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> </msubsup> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mi> r </mi> </msub> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> , </mo> <mrow> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mo> - </mo> <msub> <mi> b </mi> <mi> r </mi> </msub> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[RowBox[List[&quot;r&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;,&quot;, RowBox[List[&quot;r&quot;, &quot;+&quot;, &quot;1&quot;]]]], RowBox[List[&quot;1&quot;, &quot;,&quot;, RowBox[List[&quot;r&quot;, &quot;+&quot;, &quot;1&quot;]]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[RowBox[List[&quot;-&quot;, &quot;1&quot;]], MeijerG, Rule[Editable, True]], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[RowBox[List[TagBox[&quot;0&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;1&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;r&quot;]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[&quot;0&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;1&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;\[Ellipsis]&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;-&quot;, SubscriptBox[&quot;b&quot;, &quot;r&quot;]]], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> </mrow> </mrow> <mo> /; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HarmonicNumber", "[", RowBox[List["z_", ",", "r_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["1", "-", SubscriptBox["a", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["a", "r"]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "0", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", SubscriptBox["a", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["-", SubscriptBox["a", "r"]]]]], "}"]]]], "}"]], ",", RowBox[List["-", "1"]]]], "]"]], "-", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["1", "-", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["b", "r"]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "0", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["-", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["-", SubscriptBox["b", "r"]]]]], "}"]]]], "}"]], ",", RowBox[List["-", "1"]]]], "]"]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "\[Equal]", SubscriptBox["a", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", "r"], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["b", "1"], "\[Equal]", SubscriptBox["b", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["b", "r"], "\[Equal]", RowBox[List["z", "+", "1"]]]], "&&", RowBox[List[RowBox[List["r", "-", "1"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["r", "-", "1"]], ">", "0"]]]]]]]]]]


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





2001-10-29