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 LaguerreL

 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

 Cell[BoxData[RowBox[List[RowBox[List[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"]], ")"]], "b"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["a", " ", "z", " ", "t"]], RowBox[List["1", "-", "t"]]]]]], ")"]], "k"], " ", RowBox[List["LaguerreL", "[", RowBox[List["k", ",", RowBox[List["c", "+", RowBox[List["b", " ", "k"]]]], ",", RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["a", " ", "k"]]]], ")"]]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "c", "-", RowBox[List["b", " ", "k"]], "-", "k"]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["m", "-", "c", "+", "q", "-", RowBox[List["b", " ", "k"]], "-", "k"]], "q"], ",", "p"]], "]"]]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "t"]], ")"]], RowBox[List["m", "-", "c"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["z", " ", "t"]], RowBox[List["t", "-", "1"]]]]]], RowBox[List["p", "!"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "p"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["m", "+", RowBox[List["q", " ", "j"]]]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "t"]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "t"]], ")"]], RowBox[List[RowBox[List["j", " ", "q"]], "-", "k"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "p"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "m"]], "-", RowBox[List["j", " ", "q"]]]], ",", "k"]], "]"]], " "]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["j", "!"]], " ", RowBox[List["(", RowBox[List["m", "-", "c", "+", RowBox[List["j", " ", "q"]], "-", RowBox[List["b", " ", "k"]], "-", "k"]], ")"]]]]], RowBox[List["Hypergeometric1F1", "[", RowBox[List["1", ",", FractionBox[RowBox[List["c", "-", "m", "+", "b", "+", "1", "+", RowBox[List["b", " ", "k"]], "+", "k", "-", RowBox[List["j", " ", "q"]]]], RowBox[List["b", "+", "1"]]], ",", FractionBox[RowBox[List["z", " ", "t", " ", RowBox[List["(", RowBox[List["b", "+", "1", "+", RowBox[List["a", " ", "m"]], "-", RowBox[List["a", " ", "c"]], "+", RowBox[List["a", " ", "j", " ", "q"]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "-", "t"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", "1"]], ")"]]]]]]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["\[LeftBracketingBar]", RowBox[List["t", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "t"]], ")"]], "b"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["a", " ", "z", " ", "t"]], RowBox[List["1", "-", "t"]]]]]], "\[RightBracketingBar]"]], "<", "1"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["p", "\[Element]", "Integers"]], "\[And]", RowBox[List["p", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["q", "\[Element]", "Integers"]], "\[And]", RowBox[List["q", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 k = 0 ( t ( 1 - t ) b a z t 1 - t ) k L k c + b k ( z ( a k + 1 ) ) ( - c - b k - k + m ) ( - c - b k - k + m + q q ) p TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List[RowBox[List["-", "c"]], "-", RowBox[List["b", " ", "k"]], "-", "k", "+", "m", "+", "q"]], "q"], ")"]], "p"], Pochhammer] ( 1 - t ) m - c z t t - 1 p ! j = 0 p k = 0 m + j q ( - t ) k ( 1 - t ) j q - k ( - p ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "p"]], ")"]], "j"], Pochhammer] ( - m - j q ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "m"]], "-", RowBox[List["j", " ", "q"]]]], ")"]], "k"], Pochhammer] k ! j ! ( - c - b k - k + m + j q ) 1 F 1 ( 1 ; k b + b + c + k - m - j q + 1 b + 1 ; z t ( b - a c + a m + a j q + 1 ) ( 1 - t ) ( b + 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["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"]]], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[FractionBox[RowBox[List["z", " ", "t", " ", RowBox[List["(", RowBox[List["b", "-", RowBox[List["a", " ", "c"]], "+", RowBox[List["a", " ", "m"]], "+", RowBox[List["a", " ", "j", " ", "q"]], "+", "1"]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "-", "t"]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", "1"]], ")"]]]]], Hypergeometric1F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1] /; "\[LeftBracketingBar]" t ( 1 - t ) b a z t 1 - t "\[RightBracketingBar]" < 1 m TagBox["\[DoubleStruckCapitalN]", Function[Integers]] p TagBox["\[DoubleStruckCapitalN]", Function[Integers]] q TagBox[TagBox["\[DoubleStruckCapitalN]", Function[Integers]], Function[Integers]] Condition k 0 t 1 -1 t b a z t 1 -1 t -1 k LaguerreL k c b k z a k 1 -1 c -1 b k -1 k m Pochhammer -1 c -1 b k -1 k m q q -1 p -1 1 -1 t m -1 c z t t -1 -1 p -1 k 0 m j q j 0 p -1 t k 1 -1 t j q -1 k Pochhammer -1 p j Pochhammer -1 m -1 j q k k j -1 c -1 b k -1 k m j q -1 Hypergeometric1F1 1 k b b c k -1 m -1 j q 1 b 1 -1 z t b -1 a c a m a j q 1 1 -1 t b 1 -1 t 1 -1 t b a z t 1 -1 t -1 1 m p q [/itex]

 Rule Form

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

 2002-12-18