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 LaguerreL

 http://functions.wolfram.com/05.08.21.0009.01

 Input Form

 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

 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["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"]]]]]]]]

 MathML Form

 0 t α - 1 - p t L m λ ( a t ) L n β ( b t ) t Γ ( α ) ( λ + 1 ) m TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Lambda]", "+", "1"]], ")"]], "m"], Pochhammer] ( β + 1 ) n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Beta]", "+", "1"]], ")"]], "n"], Pochhammer] p - α m ! n ! j = 0 m ( - m ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], "j"], Pochhammer] ( α ) j TagBox[SubscriptBox[RowBox[List["(", "\[Alpha]", ")"]], "j"], Pochhammer] ( λ + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Lambda]", "+", "1"]], ")"]], "j"], Pochhammer] j ! ( a p ) j k = 0 n ( - n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "n"]], ")"]], "k"], Pochhammer] ( j + α ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["j", "+", "\[Alpha]"]], ")"]], "k"], Pochhammer] ( β + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Beta]", "+", "1"]], ")"]], "k"], Pochhammer] k ! ( b p ) k /; Re ( α ) > 0 Re ( p ) > 0 m n Condition t 0 t α -1 -1 p t LaguerreL m λ a t LaguerreL n β b t Gamma α Pochhammer λ 1 m Pochhammer β 1 n p -1 α m n -1 j 0 m Pochhammer -1 m j Pochhammer α j Pochhammer λ 1 j j -1 a p -1 j k 0 n Pochhammer -1 n k Pochhammer j α k Pochhammer β 1 k k -1 b p -1 k 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["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"]]]]]]]]]]

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

 2007-05-02