Generalized Laguerre polynomials: Summation (formula 05.08.23.0014)

LaguerreL

$Mathematica Notation$

Polynomials

LaguerreL[n,lambda,z]

Summation

Infinite summation

http://functions.wolfram.com/05.08.23.0014.01

Input Form

Sum[((t (1 - t)^b E^((a z t)/(1 - t)))^k LaguerreL[k, c + b k, z (1 + a k)])/ ((m - c - b k - k) Pochhammer[(m - c + q - b k - k)/q, p]), {k, 0, Infinity}] == (((1 - t)^(m - c) E^((z t)/(t - 1)))/p!) Sum[(((-t)^k (1 - t)^(j q - k) Pochhammer[-p, j] Pochhammer[-m - j q, k])/ (k! j! (m - c + j q - b k - k))) Hypergeometric1F1[1, (c - m + b + 1 + b k + k - j q)/(b + 1), (z t (b + 1 + a m - a c + a j q))/((1 - t) (b + 1))], {j, 0, p}, {k, 0, m + q j}] /; \[LeftBracketingBar] t (1 - t)^b E^((a z t)/(1 - t)) \[RightBracketingBar] < 1 && Element[m, Integers] && m >= 0 && Element[p, Integers] && p >= 0 && Element[q, Integers] && q >= 0

Standard Form

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MathML Form

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</mi> <mo> ⁢ </mo> <mi> q </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["1", Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox[RowBox[List[RowBox[List["k", " ", "b"]], "+", "b", "+", "c", "+", "k", "-", "m", "-", RowBox[List["j", " ", "q"]], "+", "1"]], RowBox[List["b", "+", "1"]]], 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</mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <apply> <times /> <ci> t </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <ci> b </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> a </ci> <ci> z </ci> <ci> t </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <ci> k </ci> </apply> <apply> <ci> LaguerreL </ci> <ci> k </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <ci> b </ci> <ci> k </ci> </apply> </apply> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> 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</apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <ci> m </ci> <apply> <times /> <ci> j </ci> <ci> q </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric1F1 </ci> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> k </ci> <ci> b </ci> </apply> <ci> b </ci> <ci> c </ci> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> j </ci> <ci> q </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> z </ci> <ci> t </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> c </ci> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> m </ci> </apply> <apply> <times /> <ci> a </ci> <ci> j </ci> <ci> q </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <apply> <abs /> <apply> <times /> <ci> t </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <ci> b </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> a </ci> <ci> z </ci> <ci> t </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> <apply> <in /> <ci> p </ci> <integers /> </apply> <apply> <in /> <ci> q </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2002-12-18

LaguerreL[n,z]
LaguerreL[nu,z]
LaguerreL[nu,mu,z]