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variants of this functions
NorlundB




Mathematica Notation

Traditional Notation









Polynomials > NorlundB[n,α,z] > Series representations > Generalized power series > Expansions at alpha==infinity





http://functions.wolfram.com/05.17.06.0022.01









  


  








Input Form





NorlundB[n, \[Alpha], z] \[Proportional] z^n UnitStep[n] + z^n Sum[(Binomial[n, k] \[Alpha]^k (1 + 2^k Sum[StirlingS1[1 + k, r] Sum[(-1)^(j + k) j^(r - k) Binomial[k, j] Subscript[p, j, k] (1/\[Alpha]), {j, 1, k}], {r, 1, -1 + k}] - 2^k Sum[StirlingS1[1 + k, r] Sum[(-1)^(j + k) j^(1 + r - k) Binomial[k, j] Subscript[p, j, k] (1/\[Alpha]^2), {j, 1, k}], {r, 1, -2 + k}] + \[Ellipsis]))/ (-2 z)^k, {k, 0, n}] /; (Abs[\[Alpha]] -> Infinity) && Subscript[p, j, 0] == 1 && Subscript[p, j, k] == (1/k) Sum[(j m - k + m) Subscript[a, m] Subscript[p, j, k - m], {m, 1, k}] && Subscript[a, k] == 1/(k + 1)! && Element[k, Integers] && k >= 0 && n > 0








Standard Form





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MathML Form







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</mo> <msub> <mi> p </mi> <mrow> <mi> j </mi> <mo> , </mo> <mi> k </mi> </mrow> </msub> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#945; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> p </mi> <mrow> <mi> j </mi> <mo> , </mo> <mn> 0 </mn> </mrow> </msub> <mo> &#63449; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> p </mi> <mrow> <mi> j </mi> <mo> , </mo> <mi> k </mi> </mrow> </msub> <mo> &#63449; </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> k </mi> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> m </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> j </mi> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> + </mo> <mi> m </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> a </mi> <mi> m </mi> </msub> <mo> &#8290; </mo> <msub> <mi> p </mi> <mrow> <mi> j </mi> <mo> , </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> m </mi> </mrow> </mrow> </msub> </mrow> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> &#63449; </mo> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <mi> k </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> NorlundB </ci> <ci> n </ci> <ci> &#945; </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> UnitStep </ci> <ci> n </ci> </apply> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <ci> n </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <ci> &#945; </ci> <ci> k </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <sum /> <bvar> <ci> r </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> StirlingS1 </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <ci> r </ci> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> j </ci> <ci> k </ci> </apply> </apply> <apply> <power /> <ci> j </ci> <apply> <plus /> <ci> r </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <power /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Binomial </ci> <ci> k </ci> <ci> j </ci> </apply> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <sum /> <bvar> <ci> r </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> k </ci> <cn type='integer'> -2 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> StirlingS1 </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <ci> r </ci> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> j </ci> <ci> k </ci> </apply> </apply> <apply> <power /> <ci> j </ci> <apply> <plus /> <ci> r </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <ci> &#945; 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</ci> </apply> <apply> <gt /> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["NorlundB", "[", RowBox[List["n_", ",", "\[Alpha]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["z", "n"], " ", RowBox[List["UnitStep", "[", "n", "]"]]]], "+", RowBox[List[SuperscriptBox["z", "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], " ", "z"]], ")"]], RowBox[List["-", "k"]]], " ", SuperscriptBox["\[Alpha]", "k"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[SuperscriptBox["2", "k"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "1"]], RowBox[List[RowBox[List["-", "1"]], "+", "k"]]], RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["1", "+", "k"]], ",", "r"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "k"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "k"]]], " ", SuperscriptBox["j", RowBox[List["r", "-", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", SubscriptBox["p", RowBox[List["j", ",", "k"]]]]], "\[Alpha]"]]]]]]]]], "-", RowBox[List[SuperscriptBox["2", "k"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "1"]], RowBox[List[RowBox[List["-", "2"]], "+", "k"]]], RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["1", "+", "k"]], ",", "r"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "k"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "k"]]], " ", SuperscriptBox["j", RowBox[List["1", "+", "r", "-", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", SubscriptBox["p", RowBox[List["j", ",", "k"]]]]], SuperscriptBox["\[Alpha]", "2"]]]]]]]]]], "+", "\[Ellipsis]"]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Alpha]", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "0"]]], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "k"]]], "\[Equal]", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "k"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["j", " ", "m"]], "-", "k", "+", "m"]], ")"]], " ", SubscriptBox["a", "m"], " ", SubscriptBox["p", RowBox[List["j", ",", RowBox[List["k", "-", "m"]]]]]]]]], "k"]]], "&&", RowBox[List[SubscriptBox["a", "k"], "\[Equal]", FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "!"]]]]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", "\[GreaterEqual]", "0"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]


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





2007-05-02