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 AiryAi

 http://functions.wolfram.com/03.05.06.0035.01

 Input Form

 AiryAi[z] == Subscript[F, Infinity][z] /; Subscript[F, n][z] == (1/(3^(2/3) Gamma[2/3])) Sum[(1/(Pochhammer[2/3, k] k!)) (z^3/9)^k, {k, 0, n}] - (z/(3^(1/3) Gamma[1/3])) Sum[(1/(Pochhammer[4/3, k] k!)) (z^3/9)^k, {k, 0, n}] == AiryAi[z] - (1/(3^(2/3) Gamma[2/3])) ((z^3/9)^(n + 1)/((n + 1)! Pochhammer[2/3, n + 1])) HypergeometricPFQ[{1}, {n + 2, n + 5/3}, z^3/9] + (z/(3^(1/3) Gamma[1/3])) ((z^3/9)^(n + 1)/ ((n + 1)! Pochhammer[4/3, n + 1])) HypergeometricPFQ[{1}, {n + 2, n + 7/3}, z^3/9] && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["AiryAi", "[", "z", "]"]], "\[Equal]", RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", "z", "]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List[" ", RowBox[List[SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", RowBox[List["Gamma", "[", FractionBox["2", "3"], "]"]]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox["1", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["2", "3"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]], SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "3"], "9"], ")"]], "k"]]]]]]], "-", RowBox[List[FractionBox[RowBox[List["z", " "]], RowBox[List[" ", RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "3"]]], " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox["1", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["4", "3"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]], SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "3"], "9"], ")"]], "k"]]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["AiryAi", "[", "z", "]"]], "-", RowBox[List[FractionBox["1", RowBox[List[" ", RowBox[List[SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", RowBox[List["Gamma", "[", FractionBox["2", "3"], "]"]]]]]]], FractionBox[SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "3"], "9"], ")"]], RowBox[List["n", "+", "1"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["2", "3"], ",", RowBox[List["n", "+", "1"]]]], "]"]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["n", "+", "2"]], ",", RowBox[List["n", "+", FractionBox["5", "3"]]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "3"], "9"]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["z", " "]], RowBox[List[" ", RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "3"]]], " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]]]]]]], FractionBox[SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "3"], "9"], ")"]], RowBox[List["n", "+", "1"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["4", "3"], ",", RowBox[List["n", "+", "1"]]]], "]"]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["n", "+", "2"]], ",", RowBox[List["n", "+", FractionBox["7", "3"]]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "3"], "9"]]], "]"]]]]]]]], StyleBox[")", Rule[FontWeight, "Plain"]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]], ")"]]]]]]

 MathML Form

 Ai ( z ) F ( z ) /; ( ( F n ( z ) 1 3 2 / 3 Γ ( 2 3 ) k = 0 n ( z 3 9 ) k ( 2 3 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], "k"], Pochhammer] k ! - z 3 3 Γ ( 1 3 ) k = 0 n ( z 3 9 ) k ( 4 3 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["4", "3"], ")"]], "k"], Pochhammer] k ! Ai ( z ) - 1 3 2 / 3 Γ ( 2 3 ) ( n + 1 ) ! ( 2 3 ) n + 1 TagBox[SubscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List["n", "+", "1"]]], Pochhammer] ( z 3 9 ) n + 1 1 F 2 ( 1 ; n + 2 , n + 5 3 ; z 3 9 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["n", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", FractionBox["5", "3"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "3"], "9"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + z ( z 3 9 ) n + 1 3 3 Γ ( 1 3 ) ( n + 1 ) ! ( 4 3 ) n + 1 TagBox[SubscriptBox[RowBox[List["(", FractionBox["4", "3"], ")"]], RowBox[List["n", "+", "1"]]], Pochhammer] 1 F 2 ( 1 ; n + 2 , n + 7 3 ; z 3 9 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["n", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", FractionBox["7", "3"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "3"], "9"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] ) Condition AiryAi z Subscript F z Subscript F n z 1 3 2 3 Gamma 2 3 -1 k 0 n z 3 9 -1 k Pochhammer 2 3 k k -1 -1 z 3 1 3 Gamma 1 3 -1 k 0 n z 3 9 -1 k Pochhammer 4 3 k k -1 AiryAi z -1 1 3 2 3 Gamma 2 3 n 1 Pochhammer 2 3 n 1 -1 z 3 9 -1 n 1 HypergeometricPFQ 1 n 2 n 5 3 z 3 9 -1 z z 3 9 -1 n 1 3 1 3 Gamma 1 3 n 1 Pochhammer 4 3 n 1 -1 HypergeometricPFQ 1 n 2 n 7 3 z 3 9 -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["AiryAi", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", "z", "]"]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "3"], "9"], ")"]], "k"], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["2", "3"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]], RowBox[List[SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", RowBox[List["Gamma", "[", FractionBox["2", "3"], "]"]]]]], "-", FractionBox[RowBox[List["z", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "3"], "9"], ")"]], "k"], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["4", "3"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "3"]]], " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]]]]]]], "\[Equal]", RowBox[List[RowBox[List["AiryAi", "[", "z", "]"]], "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "3"], "9"], ")"]], RowBox[List["n", "+", "1"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["n", "+", "2"]], ",", RowBox[List["n", "+", FractionBox["5", "3"]]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "3"], "9"]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", RowBox[List["Gamma", "[", FractionBox["2", "3"], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["2", "3"], ",", RowBox[List["n", "+", "1"]]]], "]"]]]], ")"]]]]], "+", FractionBox[RowBox[List["z", " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "3"], "9"], ")"]], RowBox[List["n", "+", "1"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["n", "+", "2"]], ",", RowBox[List["n", "+", FractionBox["7", "3"]]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "3"], "9"]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["3", RowBox[List["1", "/", "3"]]], " ", RowBox[List["Gamma", "[", FractionBox["1", "3"], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["4", "3"], ",", RowBox[List["n", "+", "1"]]]], "]"]]]], ")"]]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02