D[Pochhammer[a, n], {a, \[Alpha]}] == Sum[(((-1)^(k + n) StirlingS1[n, k] k!)/Gamma[k - \[Alpha] + 1]) a^(k - \[Alpha]), {k, 0, n}] /; Element[n, Integers] && n >= 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["a", ",", "\[Alpha]"]], "}"]]], RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "n"]], "]"]]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", "n"]]], RowBox[List["StirlingS1", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["k", "!"]], " "]], RowBox[List["Gamma", "[", RowBox[List["k", "-", "\[Alpha]", "+", "1"]], "]"]]], " ", SuperscriptBox["a", RowBox[List["k", "-", "\[Alpha]"]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> &#945; </mi> </msup> <semantics> <msub> <mrow> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, &quot;a&quot;, &quot;)&quot;]], &quot;n&quot;], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> a </mi> <mi> &#945; </mi> </msup> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> </mrow> </msup> <mo> &#8290; </mo> <msubsup> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox[&quot;S&quot;, StirlingS1] </annotation> </semantics> <mi> n </mi> <mrow> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </msubsup> <mo> &#8290; </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> a </mi> <mrow> <mi> k </mi> <mo> - </mo> <mi> &#945; </mi> </mrow> </msup> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> a </ci> <degree> <ci> &#945; </ci> </degree> </bvar> <apply> <ci> Pochhammer </ci> <ci> a </ci> <ci> n </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> k </ci> <ci> n </ci> </apply> </apply> <apply> <ci> StirlingS1 </ci> <ci> n </ci> <ci> k </ci> </apply> <apply> <factorial /> <ci> k </ci> </apply> <apply> <power /> <ci> a </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["a_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["Pochhammer", "[", RowBox[List["a_", ",", "n_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", "n"]]], " ", RowBox[List["StirlingS1", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]], ")"]], " ", SuperscriptBox["a", RowBox[List["k", "-", "\[Alpha]"]]]]], RowBox[List["Gamma", "[", RowBox[List["k", "-", "\[Alpha]", "+", "1"]], "]"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

2001-10-29