Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Pochhammer






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Pochhammer[a,n] > Series representations > Generalized power series > Expansions of (-m+Epsilon)n at Epsilon==0/;nZ && n<=m && mN





http://functions.wolfram.com/06.10.06.0027.01









  


  










Input Form





Pochhammer[-m + \[Epsilon], n] \[Proportional] (((-1)^n m!)/(m - n)!) (1 + \[Epsilon] (-PolyGamma[1 + m] + PolyGamma[1 + m - n]) + (1/2) \[Epsilon]^2 (PolyGamma[1 + m]^2 - 2 PolyGamma[1 + m] PolyGamma[1 + m - n] + PolyGamma[1 + m - n]^2 + PolyGamma[1, 1 + m] - PolyGamma[1, 1 + m - n]) - (1/6) \[Epsilon]^3 (PolyGamma[1 + m]^3 - 3 PolyGamma[1 + m]^2 PolyGamma[1 + m - n] - PolyGamma[1 + m - n]^3 - 3 PolyGamma[1 + m - n] (PolyGamma[1, 1 + m] - PolyGamma[1, 1 + m - n]) + 3 PolyGamma[1 + m] (PolyGamma[1 + m - n]^2 + PolyGamma[1, 1 + m] - PolyGamma[1, 1 + m - n]) + PolyGamma[2, 1 + m] - PolyGamma[2, 1 + m - n]) + (1/24) \[Epsilon]^4 (PolyGamma[1 + m]^4 - 4 PolyGamma[1 + m]^3 PolyGamma[1 + m - n] + PolyGamma[1 + m - n]^4 + 3 PolyGamma[1, 1 + m]^2 + 6 PolyGamma[1 + m - n]^2 (PolyGamma[1, 1 + m] - PolyGamma[1, 1 + m - n]) + 6 PolyGamma[1 + m]^2 (PolyGamma[1 + m - n]^2 + PolyGamma[1, 1 + m] - PolyGamma[1, 1 + m - n]) - 6 PolyGamma[1, 1 + m] PolyGamma[1, 1 + m - n] + 3 PolyGamma[1, 1 + m - n]^2 - 4 PolyGamma[1 + m - n] (PolyGamma[2, 1 + m] - PolyGamma[2, 1 + m - n]) - 4 PolyGamma[1 + m] (PolyGamma[1 + m - n]^3 + 3 PolyGamma[1 + m - n] (PolyGamma[1, 1 + m] - PolyGamma[1, 1 + m - n]) - PolyGamma[2, 1 + m] + PolyGamma[2, 1 + m - n]) + PolyGamma[3, 1 + m] - PolyGamma[3, 1 + m - n]) + O[\[Epsilon]^5]) /; Element[n, Integers] && n <= m && Element[m, Integers] && m >= 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "m"]], "+", "\[Epsilon]"]], ",", "n"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["m", "!"]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "n"]], ")"]], "!"]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[Epsilon]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m"]], "]"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["\[Epsilon]", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m"]], "]"]], "2"], "-", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]]]], "+", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]], "2"], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m", "-", "n"]]]], "]"]]]], ")"]]]], "-", RowBox[List[FractionBox["1", "6"], " ", SuperscriptBox["\[Epsilon]", "3"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m"]], "]"]], "3"], "-", RowBox[List["3", " ", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m"]], "]"]], "2"], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]]]], "-", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]], "3"], "-", RowBox[List["3", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m", "-", "n"]]]], "]"]]]], ")"]]]], "+", RowBox[List["3", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m"]], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]], "2"], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m", "-", "n"]]]], "]"]]]], ")"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List["2", ",", RowBox[List["1", "+", "m"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["2", ",", RowBox[List["1", "+", "m", "-", "n"]]]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "24"], " ", SuperscriptBox["\[Epsilon]", "4"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m"]], "]"]], "4"], "-", RowBox[List["4", " ", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m"]], "]"]], "3"], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]]]], "+", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]], "4"], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m"]]]], "]"]], "2"]]], "+", RowBox[List["6", " ", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m", "-", "n"]]]], "]"]]]], ")"]]]], "+", RowBox[List["6", " ", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]], "2"], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m", "-", "n"]]]], "]"]]]], ")"]]]], "-", RowBox[List["6", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m"]]]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m", "-", "n"]]]], "]"]]]], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m", "-", "n"]]]], "]"]], "2"]]], "-", RowBox[List["4", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["2", ",", RowBox[List["1", "+", "m"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["2", ",", RowBox[List["1", "+", "m", "-", "n"]]]], "]"]]]], ")"]]]], "-", RowBox[List["4", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m"]], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]], "3"], "+", RowBox[List["3", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m", "-", "n"]]]], "]"]]]], ")"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List["2", ",", RowBox[List["1", "+", "m"]]]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["2", ",", RowBox[List["1", "+", "m", "-", "n"]]]], "]"]]]], ")"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List["3", ",", RowBox[List["1", "+", "m"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["3", ",", RowBox[List["1", "+", "m", "-", "n"]]]], "]"]]]], ")"]]]], "+", RowBox[List["O", "[", SuperscriptBox["\[Epsilon]", "5"], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[LessEqual]", "m"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> &#1013; </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;\[Epsilon]&quot;, &quot;-&quot;, &quot;m&quot;]], &quot;)&quot;]], &quot;n&quot;], Pochhammer] </annotation> </semantics> <mo> &#8733; </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> m </mi> <mo> ! </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#1013; </mi> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#1013; </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <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> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <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> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#1013; </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 24 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <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> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <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> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <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> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <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> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#1013; </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> &#1013; </mi> <mn> 5 </mn> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8804; </mo> <mi> m </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> &#1013; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <ci> n </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <factorial /> <ci> m </ci> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <ci> &#1013; </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <power /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> &#1013; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 6 </cn> <apply> <plus /> <apply> <power /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <plus /> <apply> <power /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> &#1013; </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 24 </cn> <apply> <plus /> <apply> <power /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <plus /> <apply> <power /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <apply> <power /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <plus /> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> PolyGamma </ci> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <cn type='integer'> 3 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 3 </cn> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> &#1013; </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <ci> &#1013; </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <leq /> <ci> n </ci> <ci> m </ci> </apply> <apply> <in /> <ci> m </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "m_"]], "+", "\[Epsilon]_"]], ",", "n_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["m", "!"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[Epsilon]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m"]], "]"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["\[Epsilon]", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m"]], "]"]], "2"], "-", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]]]], "+", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]], "2"], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m", "-", "n"]]]], "]"]]]], ")"]]]], "-", RowBox[List[FractionBox["1", "6"], " ", SuperscriptBox["\[Epsilon]", "3"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m"]], "]"]], "3"], "-", RowBox[List["3", " ", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m"]], "]"]], "2"], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]]]], "-", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]], "3"], "-", RowBox[List["3", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m", "-", "n"]]]], "]"]]]], ")"]]]], "+", RowBox[List["3", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m"]], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]], "2"], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m", "-", "n"]]]], "]"]]]], ")"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List["2", ",", RowBox[List["1", "+", "m"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["2", ",", RowBox[List["1", "+", "m", "-", "n"]]]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "24"], " ", SuperscriptBox["\[Epsilon]", "4"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m"]], "]"]], "4"], "-", RowBox[List["4", " ", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m"]], "]"]], "3"], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]]]], "+", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]], "4"], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m"]]]], "]"]], "2"]]], "+", RowBox[List["6", " ", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m", "-", "n"]]]], "]"]]]], ")"]]]], "+", RowBox[List["6", " ", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]], "2"], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m", "-", "n"]]]], "]"]]]], ")"]]]], "-", RowBox[List["6", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m"]]]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m", "-", "n"]]]], "]"]]]], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m", "-", "n"]]]], "]"]], "2"]]], "-", RowBox[List["4", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["2", ",", RowBox[List["1", "+", "m"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["2", ",", RowBox[List["1", "+", "m", "-", "n"]]]], "]"]]]], ")"]]]], "-", RowBox[List["4", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m"]], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]], "3"], "+", RowBox[List["3", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "m", "-", "n"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "m", "-", "n"]]]], "]"]]]], ")"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List["2", ",", RowBox[List["1", "+", "m"]]]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["2", ",", RowBox[List["1", "+", "m", "-", "n"]]]], "]"]]]], ")"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List["3", ",", RowBox[List["1", "+", "m"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["3", ",", RowBox[List["1", "+", "m", "-", "n"]]]], "]"]]]], ")"]]]], "+", SuperscriptBox[RowBox[List["O", "[", "\[Epsilon]", "]"]], "5"]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "n"]], ")"]], "!"]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[LessEqual]", "m"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02





© 1998-2014 Wolfram Research, Inc.