PolyGamma[2, z] == -(1/z^3) - 1/z^2 - 1/(2 z^3 (z + ContinueFraction[{((1 + Floor[k/2]) Floor[(1 + k)/2] Floor[(3 + k)/2])/(3 - (-1)^k + 2 k), z}, {k, 1, Infinity}])) /; Re[z] > 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["2", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", SuperscriptBox["z", "3"]]]], "-", FractionBox["1", SuperscriptBox["z", "2"]], "-", RowBox[List["1", "/", RowBox[List["(", RowBox[List["2", " ", SuperscriptBox["z", "3"], " ", RowBox[List["(", RowBox[List["z", "+", RowBox[List["ContinueFraction", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["Floor", "[", FractionBox["k", "2"], "]"]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "k"]], "2"], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["3", "+", "k"]], "2"], "]"]]]], RowBox[List["3", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], "+", RowBox[List["2", " ", "k"]]]]], ",", "z"]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "1", ",", InterpretationBox["\[Infinity]", DirectedInfinity[1]]]], "}"]]]], "]"]]]], ")"]]]], ")"]]]]]]]], "/;", " ", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mfrac> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <msubsup> <mrow> <msub> <mi> &#922; </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mi> k </mi> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> + </mo> <mn> 3 </mn> </mrow> </mfrac> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mn> 1 </mn> <mi> &#8734; </mi> </msubsup> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> PolyGamma </ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <ci> z </ci> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <apply> <ci> Subscript </ci> <ci> &#922; </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <floor /> <apply> <times /> <ci> k </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <infinity /> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyGamma", "[", RowBox[List["2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", SuperscriptBox["z", "3"]]]], "-", FractionBox["1", SuperscriptBox["z", "2"]], "-", FractionBox["1", RowBox[List["2", " ", SuperscriptBox["z", "3"], " ", RowBox[List["(", RowBox[List["z", "+", RowBox[List["ContinueFraction", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["Floor", "[", FractionBox["k", "2"], "]"]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "k"]], "2"], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["3", "+", "k"]], "2"], "]"]]]], RowBox[List["3", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], "+", RowBox[List["2", " ", "k"]]]]], ",", "z"]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "1", ",", "\[Infinity]"]], "}"]]]], "]"]]]], ")"]]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]]]

2001-10-29