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 Hypergeometric0F1

 http://functions.wolfram.com/07.17.03.0003.01

 Input Form

 Hypergeometric0F1[b, z] == ((z^((1/2) (1 - b - Abs[b - 1])) Gamma[-(1/3)] Gamma[b])/(2 3^(5/6) Gamma[1 - Abs[b - 1]])) ((Sqrt[3] Sign[b - 1] AiryAi[3^(2/3) z^(1/3)] - AiryBi[3^(2/3) z^(1/3)]) Sum[((Abs[b - 1] - 1/3 - k)! (-z)^k)/(k! (Abs[b - 1] - 1/3 - 2 k)! Pochhammer[1/3, k] Pochhammer[1 - Abs[b - 1], k]), {k, 0, Abs[b - 1] - 1/3}] + 3^(1/6) z^(1/3) (Sqrt[3] AiryBiPrime[3^(2/3) z^(1/3)] - 3 Sign[b - 1] AiryAiPrime[3^(2/3) z^(1/3)]) Sum[((Abs[b - 1] - 4/3 - k)! (-z)^k)/ (k! Pochhammer[4/3, k] (Abs[b - 1] - 4/3 - 2 k)! Pochhammer[1 - Abs[b - 1], k]), {k, 0, Abs[b - 1] - 4/3}]) /; Element[Abs[b - 1] + 2/3, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Hypergeometric0F1", "[", RowBox[List["b", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "b", "-", RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]]]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["1", "3"]]], "]"]], " ", RowBox[List["Gamma", "[", "b", "]"]], " "]], RowBox[List["2", " ", SuperscriptBox["3", RowBox[List["5", "/", "6"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]]]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["3"], " ", RowBox[List["Sign", "[", RowBox[List["b", "-", "1"]], "]"]], RowBox[List["AiryAi", "[", RowBox[List[SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["z", RowBox[List["1", "/", "3"]]]]], "]"]]]], "-", RowBox[List["AiryBi", "[", RowBox[List[SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["z", RowBox[List["1", "/", "3"]]]]], "]"]]]], " ", ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]], "-", FractionBox["1", "3"]]]], " ", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]], "-", FractionBox["1", "3"], "-", "k"]], ")"]], "!"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "k"]]], "/", RowBox[List["(", RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]], "-", FractionBox["1", "3"], "-", RowBox[List["2", " ", "k"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "3"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]]]], ",", "k"]], "]"]]]], ")"]]]]]]]], "+", RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "6"]]], " ", SuperscriptBox["z", RowBox[List["1", "/", "3"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["3"], " ", RowBox[List["AiryBiPrime", "[", RowBox[List[SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["z", RowBox[List["1", "/", "3"]]]]], "]"]]]], "-", RowBox[List["3", " ", RowBox[List["Sign", "[", RowBox[List["b", "-", "1"]], "]"]], RowBox[List["AiryAiPrime", "[", RowBox[List[SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["z", RowBox[List["1", "/", "3"]]]]], "]"]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]], "-", FractionBox["4", "3"]]]], " ", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]], "-", FractionBox["4", "3"], "-", "k"]], ")"]], "!"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "k"]]], "/", RowBox[List["(", RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["4", "3"], ",", "k"]], "]"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]], "-", FractionBox["4", "3"], "-", RowBox[List["2", " ", "k"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]]]], ",", "k"]], "]"]]]], ")"]]]]]]]]]], ")"]]]]]], "/;", RowBox[List["Element", "[", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]], "+", FractionBox["2", "3"]]], ",", "Integers"]], "]"]]]]]]

 MathML Form

 0 F 1 ( ; b ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["0", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["b", Hypergeometric0F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric0F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric0F1] z 1 2 ( - b - "\[LeftBracketingBar]" b - 1 "\[RightBracketingBar]" + 1 ) Γ ( - 1 3 ) Γ ( b ) 2 3 5 / 6 Γ ( 1 - "\[LeftBracketingBar]" b - 1 "\[RightBracketingBar]" ) ( 3 6 z 3 ( 3 Bi ( 3 2 / 3 z 3 ) - 3 sgn ( b - 1 ) Ai ( 3 2 / 3 z 3 ) ) k = 0 "\[LeftBracketingBar]" b - 1 "\[RightBracketingBar]" - 4 3 ( "\[LeftBracketingBar]" b - 1 "\[RightBracketingBar]" - k - 4 3 ) ! ( - z ) k k ! ( 4 3 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["4", "3"], ")"]], "k"], Pochhammer] ( "\[LeftBracketingBar]" b - 1 "\[RightBracketingBar]" - 2 k - 4 3 ) ! ( 1 - b - 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[LeftBracketingBar]", RowBox[List["b", "-", "1"]], "\[RightBracketingBar]"]]]], ")"]], "k"], Pochhammer] + ( 3 sgn ( b - 1 ) Ai ( 3 2 / 3 z 3 ) - Bi ( 3 2 / 3 z 3 ) ) k = 0 "\[LeftBracketingBar]" b - 1 "\[RightBracketingBar]" - 1 3 ( "\[LeftBracketingBar]" b - 1 "\[RightBracketingBar]" - k - 1 3 ) ! ( - z ) k k ! ( "\[LeftBracketingBar]" b - 1 "\[RightBracketingBar]" - 2 k - 1 3 ) ! ( 1 3 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "3"], ")"]], "k"], Pochhammer] ( 1 - b - 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[LeftBracketingBar]", RowBox[List["b", "-", "1"]], "\[RightBracketingBar]"]]]], ")"]], "k"], Pochhammer] ) /; "\[LeftBracketingBar]" b - 1 "\[RightBracketingBar]" + 2 3 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition Hypergeometric0F1 b z z 1 2 -1 b -1 b -1 1 Gamma -1 1 3 Gamma b 2 3 5 6 Gamma 1 -1 b -1 -1 3 1 6 z 1 3 3 1 2 AiryBiPrime 3 2 3 z 1 3 -1 3 Sign b -1 AiryAiPrime 3 2 3 z 1 3 k 0 b -1 -1 4 3 b -1 -1 k -1 4 3 -1 z k k Pochhammer 4 3 k b -1 -1 2 k -1 4 3 Pochhammer 1 -1 b -1 k -1 3 1 2 Sign b -1 AiryAi 3 2 3 z 1 3 -1 AiryBi 3 2 3 z 1 3 k 0 b -1 -1 1 3 b -1 -1 k -1 1 3 -1 z k k b -1 -1 2 k -1 1 3 Pochhammer 1 3 k Pochhammer 1 -1 b -1 k -1 b -1 2 3 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric0F1", "[", RowBox[List["b_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "b", "-", RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]]]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["1", "3"]]], "]"]], " ", RowBox[List["Gamma", "[", "b", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["3"], " ", RowBox[List["Sign", "[", RowBox[List["b", "-", "1"]], "]"]], " ", RowBox[List["AiryAi", "[", RowBox[List[SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["z", RowBox[List["1", "/", "3"]]]]], "]"]]]], "-", RowBox[List["AiryBi", "[", RowBox[List[SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["z", RowBox[List["1", "/", "3"]]]]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]], "-", FractionBox["1", "3"]]]], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]], "-", FractionBox["1", "3"], "-", "k"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]], "-", FractionBox["1", "3"], "-", RowBox[List["2", " ", "k"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "3"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]]]], ",", "k"]], "]"]]]]]]]]], "+", RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "6"]]], " ", SuperscriptBox["z", RowBox[List["1", "/", "3"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["3"], " ", RowBox[List["AiryBiPrime", "[", RowBox[List[SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["z", RowBox[List["1", "/", "3"]]]]], "]"]]]], "-", RowBox[List["3", " ", RowBox[List["Sign", "[", RowBox[List["b", "-", "1"]], "]"]], " ", RowBox[List["AiryAiPrime", "[", RowBox[List[SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["z", RowBox[List["1", "/", "3"]]]]], "]"]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]], "-", FractionBox["4", "3"]]]], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]], "-", FractionBox["4", "3"], "-", "k"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["4", "3"], ",", "k"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]], "-", FractionBox["4", "3"], "-", RowBox[List["2", " ", "k"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]]]], ",", "k"]], "]"]]]]]]]]]]], ")"]]]], RowBox[List["2", " ", SuperscriptBox["3", RowBox[List["5", "/", "6"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]]]], "]"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]], "+", FractionBox["2", "3"]]], "\[Element]", "Integers"]]]]]]]]

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

 2001-10-29

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