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 LaguerreL

 http://functions.wolfram.com/05.08.21.0010.01

 Input Form

 Integrate[(t^(\[Alpha] - 1) LaguerreL[m, \[Lambda], p t] LaguerreL[n, \[Beta], p t])/E^(p t), {t, 0, Infinity}] == ((Gamma[\[Alpha]] Gamma[n - \[Alpha] + \[Beta] + 1] Gamma[m + \[Lambda] + 1])/(p^\[Alpha] (m! n! Gamma[-\[Alpha] + \[Beta] + 1] Gamma[\[Lambda] + 1]))) HypergeometricPFQ[{-m, \[Alpha], \[Alpha] - \[Beta]}, {-n + \[Alpha] - \[Beta], \[Lambda] + 1}, 1] /; Re[\[Alpha]] > 0 && Re[p] > 0 && Element[m, Integers] && m >= 0 && Element[n, Integers] && n >= 0

 Standard Form

 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["p", " ", "t"]]]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List["n", ",", "\[Beta]", ",", RowBox[List["p", " ", "t"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["p", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", "\[Alpha]", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["n", "-", "\[Alpha]", "+", "\[Beta]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["m", "+", "\[Lambda]", "+", "1"]], "]"]]]], RowBox[List[RowBox[List["m", "!"]], " ", RowBox[List["n", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], "+", "\[Beta]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "1"]], "]"]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "m"]], ",", "\[Alpha]", ",", RowBox[List["\[Alpha]", "-", "\[Beta]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "+", "\[Alpha]", "-", "\[Beta]"]], ",", RowBox[List["\[Lambda]", "+", "1"]]]], "}"]], ",", "1"]], "]"]]]]]], "/;", 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"]]]]]]]]

 MathML Form

 0 t α - 1 - p t L m λ ( p t ) L n β ( p t ) t p - α Γ ( α ) Γ ( n - α + β + 1 ) Γ ( m + λ + 1 ) m ! n ! Γ ( 1 - α + β ) Γ ( λ + 1 ) 3 F 2 ( - m , α , α - β ; - n + α - β , λ + 1 ; 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "m"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Alpha]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Alpha]", "-", "\[Beta]"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[RowBox[List["-", "n"]], "+", "\[Alpha]", "-", "\[Beta]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Lambda]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] /; Re ( α ) > 0 Re ( p ) > 0 m TagBox["\[DoubleStruckCapitalN]", Function[Integers]] n Condition t 0 t α -1 -1 p t LaguerreL m λ p t LaguerreL n β p t p -1 α Gamma α Gamma n -1 α β 1 Gamma m λ 1 m n Gamma 1 -1 α β Gamma λ 1 -1 HypergeometricPFQ -1 m α α -1 β -1 n α -1 β λ 1 1 α 0 p 0 m n [/itex]

 Rule Form

 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["p_", " ", "t_"]]]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List["n_", ",", "\[Beta]_", ",", RowBox[List["p_", " ", "t_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["p", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", "\[Alpha]", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["n", "-", "\[Alpha]", "+", "\[Beta]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["m", "+", "\[Lambda]", "+", "1"]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "m"]], ",", "\[Alpha]", ",", RowBox[List["\[Alpha]", "-", "\[Beta]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "+", "\[Alpha]", "-", "\[Beta]"]], ",", RowBox[List["\[Lambda]", "+", "1"]]]], "}"]], ",", "1"]], "]"]]]], RowBox[List[RowBox[List["m", "!"]], " ", RowBox[List["n", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "\[Alpha]"]], "+", "\[Beta]", "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Lambda]", "+", "1"]], "]"]]]]], "/;", 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"]]]]]]]]]]

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

 2007-05-02