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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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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





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