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 LaguerreL

 http://functions.wolfram.com/05.08.23.0013.01

 Input Form

 Sum[((t (1 - t)^\[Mu] E^((a z t)/(1 - t)))^k/(k + 1)) LaguerreL[k, \[Lambda] + \[Mu] k, z (1 + a k)], {k, 0, Infinity}] == (1/(t (\[Lambda] - \[Mu]))) ((E^(((-z) t + z - a z)/(1 - t)) (Hypergeometric1F1[\[Lambda] - \[Mu], \[Lambda] - \[Mu] + 1, (z (a - 1))/(1 - t)] - (1 - t)^(\[Lambda] - \[Mu]) Hypergeometric1F1[\[Lambda] - \[Mu], \[Lambda] - \[Mu] + 1, z (a - 1)]))/(1 - t)^\[Lambda]) /; Abs[t (1 - t)^\[Mu] E^((a z t)/(1 - t))] < 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["t", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "t"]], ")"]], "\[Mu]"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["a", " ", "z", " ", "t"]], RowBox[List["1", "-", "t"]]]]]], ")"]], "k"], " "]], RowBox[List["k", "+", "1"]]], RowBox[List["LaguerreL", "[", RowBox[List["k", ",", RowBox[List["\[Lambda]", "+", RowBox[List["\[Mu]", " ", "k"]]]], ",", RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["a", " ", "k"]]]], ")"]]]]]], "]"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["t", " ", RowBox[List["(", RowBox[List["\[Lambda]", "-", "\[Mu]"]], ")"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "t"]], ")"]], RowBox[List["-", "\[Lambda]"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "z"]], " ", "t"]], "+", "z", "-", RowBox[List["a", " ", "z"]]]], RowBox[List["1", "-", "t"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Hypergeometric1F1", "[", RowBox[List[RowBox[List["\[Lambda]", "-", "\[Mu]"]], ",", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], ",", FractionBox[RowBox[List["z", " ", RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]]]], RowBox[List["1", "-", "t"]]]]], "]"]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "t"]], ")"]], RowBox[List["\[Lambda]", "-", "\[Mu]"]]], " ", RowBox[List["Hypergeometric1F1", "[", RowBox[List[RowBox[List["\[Lambda]", "-", "\[Mu]"]], ",", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], ",", RowBox[List["z", " ", RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]]]]]], "]"]]]]]], ")"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["t", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "t"]], ")"]], "\[Mu]"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["a", " ", "z", " ", "t"]], RowBox[List["1", "-", "t"]]]]]], "]"]], "<", "1"]]]]]]

 MathML Form

 k = 0 ( t ( 1 - t ) μ a z t 1 - t ) k k + 1 L k λ + k μ ( z ( a k + 1 ) ) 1 t ( λ - μ ) ( ( 1 - t ) - λ - a z + z + t ( - z ) 1 - t ( 1 F 1 ( λ - μ ; λ - μ + 1 ; z ( a - 1 ) 1 - t ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List["\[Lambda]", "-", "\[Mu]"]], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["z", " ", RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]]]], RowBox[List["1", "-", "t"]]], Hypergeometric1F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1] - ( 1 - t ) λ - μ 1 F 1 ( λ - μ ; λ - μ + 1 ; z ( a - 1 ) ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List["\[Lambda]", "-", "\[Mu]"]], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[RowBox[List["z", " ", RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]]]], Hypergeometric1F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1] ) ) /; "\[LeftBracketingBar]" t ( 1 - t ) μ a z t 1 - t "\[RightBracketingBar]" < 1 Condition k 0 t 1 -1 t μ a z t 1 -1 t -1 k k 1 -1 LaguerreL k λ k μ z a k 1 1 t λ -1 μ -1 1 -1 t -1 λ -1 a z z t -1 z 1 -1 t -1 Hypergeometric1F1 λ -1 μ λ -1 μ 1 z a -1 1 -1 t -1 -1 1 -1 t λ -1 μ Hypergeometric1F1 λ -1 μ λ -1 μ 1 z a -1 t 1 -1 t μ a z t 1 -1 t -1 1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["t_", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "t_"]], ")"]], "\[Mu]_"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["a_", " ", "z_", " ", "t_"]], RowBox[List["1", "-", "t_"]]]]]], ")"]], "k_"], " ", RowBox[List["LaguerreL", "[", RowBox[List["k_", ",", RowBox[List["\[Lambda]_", "+", RowBox[List["\[Mu]_", " ", "k_"]]]], ",", RowBox[List["z_", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["a_", " ", "k_"]]]], ")"]]]]]], "]"]]]], RowBox[List["k_", "+", "1"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "t"]], ")"]], RowBox[List["-", "\[Lambda]"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "z"]], " ", "t"]], "+", "z", "-", RowBox[List["a", " ", "z"]]]], RowBox[List["1", "-", "t"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Hypergeometric1F1", "[", RowBox[List[RowBox[List["\[Lambda]", "-", "\[Mu]"]], ",", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], ",", FractionBox[RowBox[List["z", " ", RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]]]], RowBox[List["1", "-", "t"]]]]], "]"]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "t"]], ")"]], RowBox[List["\[Lambda]", "-", "\[Mu]"]]], " ", RowBox[List["Hypergeometric1F1", "[", RowBox[List[RowBox[List["\[Lambda]", "-", "\[Mu]"]], ",", RowBox[List["\[Lambda]", "-", "\[Mu]", "+", "1"]], ",", RowBox[List["z", " ", RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]]]]]], "]"]]]]]], ")"]]]], RowBox[List["t", " ", RowBox[List["(", RowBox[List["\[Lambda]", "-", "\[Mu]"]], ")"]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["t", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "t"]], ")"]], "\[Mu]"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["a", " ", "z", " ", "t"]], RowBox[List["1", "-", "t"]]]]]], "]"]], "<", "1"]]]]]]]]

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

 2002-12-18