Norlund polynomials: Specific values (formula 05.17.03.0016)

NorlundB

$Mathematica Notation$

http://functions.wolfram.com/05.17.03.0016.01

Input Form

NorlundB[10, \[Alpha], z] == z^10 - 5 z^9 \[Alpha] - 15 z^7 (-1 + \[Alpha]) \[Alpha]^2 + (15/4) z^8 \[Alpha] (-1 + 3 \[Alpha]) - (21/8) z^5 \[Alpha]^2 (2 + 5 \[Alpha] - 10 \[Alpha]^2 + 3 \[Alpha]^3) + (7/8) z^6 \[Alpha] (2 + 5 \[Alpha] - 30 \[Alpha]^2 + 15 \[Alpha]^3) - (5/48) z^3 \[Alpha]^2 (-16 - 42 \[Alpha] + 7 \[Alpha]^2 + 105 \[Alpha]^3 - 63 \[Alpha]^4 + 9 \[Alpha]^5) + (5/96) z^4 \[Alpha] (-16 - 42 \[Alpha] + 91 \[Alpha]^2 + 315 \[Alpha]^3 - 315 \[Alpha]^4 + 63 \[Alpha]^5) - (1/768) z \[Alpha]^2 (144 + 404 \[Alpha] + 100 \[Alpha]^2 - 665 \[Alpha]^3 - 448 \[Alpha]^4 + 630 \[Alpha]^5 - 180 \[Alpha]^6 + 15 \[Alpha]^7) + (1/768) z^2 \[Alpha] (144 + 404 \[Alpha] - 540 \[Alpha]^2 - 2345 \[Alpha]^3 - 840 \[Alpha]^4 + 3150 \[Alpha]^5 - 1260 \[Alpha]^6 + 135 \[Alpha]^7) + (1/101376) (\[Alpha] (-768 - 2288 \[Alpha] + 2068 \[Alpha]^2 + 11792 \[Alpha]^3 + 8195 \[Alpha]^4 - 8085 \[Alpha]^5 - 8778 \[Alpha]^6 + 6930 \[Alpha]^7 - 1485 \[Alpha]^8 + 99 \[Alpha]^9))

Standard Form

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MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msubsup> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox["B", NorlundB] </annotation> </semantics> <mn> 10 </mn> <mrow> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mi> z </mi> <mn> 10 </mn> </msup> <mo> - </mo> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mi> α </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 9 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 15 </mn> <mn> 4 </mn> </mfrac> <mo> ⁢ </mo> <mi> α </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> α </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 15 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 7 </mn> <mn> 8 </mn> </mfrac> <mo> ⁢ </mo> <mi> α </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 15 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 30 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mi> α </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 21 </mn> <mn> 8 </mn> </mfrac> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 10 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 5 </mn> <mo> ⁢ </mo> <mi> α </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 5 </mn> <mn> 96 </mn> </mfrac> <mo> ⁢ </mo> <mi> α </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 63 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 315 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 315 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 91 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 42 </mn> <mo> ⁢ </mo> <mi> α </mi> </mrow> <mo> - </mo> <mn> 16 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 5 </mn> <mn> 48 </mn> </mfrac> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 9 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 63 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 105 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 42 </mn> <mo> ⁢ </mo> <mi> α </mi> </mrow> <mo> - </mo> <mn> 16 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 768 </mn> </mfrac> <mo> ⁢ </mo> <mi> α </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 135 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 7 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 1260 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3150 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 840 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2345 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 540 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 404 </mn> <mo> ⁢ </mo> <mi> α </mi> </mrow> <mo> + </mo> <mn> 144 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 768 </mn> </mfrac> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 15 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 7 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 180 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 630 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 5 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 448 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 665 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 100 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 404 </mn> <mo> ⁢ </mo> <mi> α </mi> </mrow> <mo> + </mo> <mn> 144 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 101376 </mn> </mfrac> <mo> ⁢ </mo> <mi> α </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 99 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 9 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 1485 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 6930 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 7 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 8778 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 8085 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8195 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 11792 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2068 </mn> <mo> ⁢ </mo> <msup> <mi> α </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2288 </mn> <mo> ⁢ </mo> <mi> α </mi> </mrow> <mo> - </mo> <mn> 768 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> NorlundB </ci> <cn type='integer'> 10 </cn> <ci> α </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 10 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5 </cn> <ci> α </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 15 <sep /> 4 </cn> <ci> α </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> α </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 15 </cn> <apply> <plus /> <ci> α </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> α </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 7 <sep /> 8 </cn> <ci> α </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 15 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 30 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 5 </cn> <ci> α </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 21 <sep /> 8 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 10 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 5 </cn> <ci> α </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 5 <sep /> 96 </cn> <ci> α </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 63 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 315 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 315 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 91 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 42 </cn> <ci> α </ci> </apply> </apply> <cn type='integer'> -16 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 5 <sep /> 48 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 63 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 105 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 42 </cn> <ci> α </ci> </apply> </apply> <cn type='integer'> -16 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 768 </cn> <ci> α </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 135 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1260 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 3150 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 840 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2345 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 540 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 404 </cn> <ci> α </ci> </apply> <cn type='integer'> 144 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 768 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 15 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 180 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 630 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 448 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 665 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 100 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 404 </cn> <ci> α </ci> </apply> <cn type='integer'> 144 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 101376 </cn> <ci> α </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 99 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1485 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 8 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 6930 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8778 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 8085 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 8195 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 11792 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2068 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2288 </cn> <ci> α </ci> </apply> </apply> <cn type='integer'> -768 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2007-05-02

NorlundB[n,z]