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http://functions.wolfram.com/05.08.21.0009.01
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Integrate[(t^(\[Alpha] - 1) LaguerreL[m, \[Lambda], a t]
LaguerreL[n, \[Beta], b t])/E^(p t), {t, 0, Infinity}] ==
((Gamma[\[Alpha]] Pochhammer[\[Lambda] + 1, m] Pochhammer[\[Beta] + 1, n])/
(p^\[Alpha] (m! n!))) Sum[((Pochhammer[-m, j] Pochhammer[\[Alpha], j])/
(Pochhammer[\[Lambda] + 1, j] j!)) (a/p)^j
Sum[((Pochhammer[-n, k] Pochhammer[\[Alpha] + j, k])/
(Pochhammer[\[Beta] + 1, k] k!)) (b/p)^k, {k, 0, n}], {j, 0, m}] /;
Re[\[Alpha]] > 0 && Re[p] > 0 && Element[m, Integers] && m >= 0 &&
Element[n, Integers] && n >= 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t", RowBox[List["\[Alpha]", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "p"]], " ", "t"]]], " ", RowBox[List["LaguerreL", "[", RowBox[List["m", ",", "\[Lambda]", ",", RowBox[List["a", " ", "t"]]]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List["n", ",", "\[Beta]", ",", RowBox[List["b", " ", "t"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "\[Alpha]", "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Lambda]", "+", "1"]], ",", "m"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Beta]", "+", "1"]], ",", "n"]], "]"]], " ", SuperscriptBox["p", RowBox[List["-", "\[Alpha]"]]]]], RowBox[List[RowBox[List["m", "!"]], RowBox[List["n", "!"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "m"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "m"]], ",", "j"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List["\[Alpha]", ",", "j"]], "]"]], " "]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Lambda]", "+", "1"]], ",", "j"]], "]"]], " ", RowBox[List["j", "!"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["a", "p"], ")"]], "j"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "n"]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Alpha]", "+", "j"]], ",", "k"]], "]"]], " "]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Beta]", "+", "1"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["b", "p"], ")"]], "k"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "p", "]"]], ">", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mi> ∞ </mi> </msubsup> <mrow> <mrow> <msup> <mi> t </mi> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> p </mi> </mrow> <mo> ⁢ </mo> <mi> t </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msubsup> <mi> L </mi> <mi> m </mi> <mi> λ </mi> </msubsup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <mi> L </mi> <mi> n </mi> <mi> β </mi> </msubsup> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Lambda]", "+", "1"]], ")"]], "m"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> β </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Beta]", "+", "1"]], ")"]], "n"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mi> p </mi> <mrow> <mo> - </mo> <mi> α </mi> </mrow> </msup> </mrow> <mrow> <mrow> <mi> m </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], "j"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", "\[Alpha]", ")"]], "j"], Pochhammer] </annotation> </semantics> </mrow> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Lambda]", "+", "1"]], ")"]], "j"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> a </mi> <mi> p </mi> </mfrac> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "n"]], ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <mi> α </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["j", "+", "\[Alpha]"]], ")"]], "k"], Pochhammer] </annotation> </semantics> </mrow> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> β </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Beta]", "+", "1"]], ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> b </mi> <mi> p </mi> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> p </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> m </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <ci> t </ci> <apply> <plus /> <ci> α </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> <ci> t </ci> </apply> </apply> <apply> <ci> LaguerreL </ci> <ci> m </ci> <ci> λ </ci> <apply> <times /> <ci> a </ci> <ci> t </ci> </apply> </apply> <apply> <ci> LaguerreL </ci> <ci> n </ci> <ci> β </ci> <apply> <times /> <ci> b </ci> <ci> t </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> α </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> <ci> m </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> β </ci> <cn type='integer'> 1 </cn> </apply> <ci> n </ci> </apply> <apply> <power /> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> m </ci> </apply> <apply> <factorial /> <ci> n </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <ci> j </ci> </apply> <apply> <ci> Pochhammer </ci> <ci> α </ci> <ci> j </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> <ci> j </ci> </apply> <apply> <factorial /> <ci> j </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> j </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> j </ci> <ci> α </ci> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> β </ci> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> </apply> <apply> <factorial /> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <real /> <ci> α </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <real /> <ci> p </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <in /> <ci> m </ci> <ci> ℕ </ci> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "p_"]], " ", "t_"]]], " ", RowBox[List["LaguerreL", "[", RowBox[List["m_", ",", "\[Lambda]_", ",", RowBox[List["a_", " ", "t_"]]]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List["n_", ",", "\[Beta]_", ",", RowBox[List["b_", " ", "t_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "\[Alpha]", "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Lambda]", "+", "1"]], ",", "m"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Beta]", "+", "1"]], ",", "n"]], "]"]], " ", SuperscriptBox["p", RowBox[List["-", "\[Alpha]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "m"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Alpha]", ",", "j"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["a", "p"], ")"]], "j"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "n"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Alpha]", "+", "j"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["b", "p"], ")"]], "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Beta]", "+", "1"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Lambda]", "+", "1"]], ",", "j"]], "]"]], " ", RowBox[List["j", "!"]]]]]]]]], RowBox[List[RowBox[List["m", "!"]], " ", RowBox[List["n", "!"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", "p", "]"]], ">", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
