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 AiryBi

 http://functions.wolfram.com/03.06.20.0005.01

 Input Form

 D[AiryBi[z], {z, n}] == (AiryBi[z]/2) KroneckerDelta[n] + (1/(z^n 2)) (Sum[(((-1)^(j + k - 1) i (k - i - 1)! (3 k - 3 i - 1) Pochhammer[3 k - 3 j - n + 1, n] Pochhammer[-(1/3), k])/ (i! j! (k - j)! (k - 2 i)! Pochhammer[2/3, i] Pochhammer[4/3 - k, i])) (-(z^3/9))^i, {k, 0, n}, {j, 0, k}, {i, 0, k - 1}] - Sum[(((-1)^(j + k - 1) (k - i)! (1 - 3 j + 3 k) Pochhammer[3 k - 3 j - n + 2, n - 1] Pochhammer[1/3, k])/ (i! j! (k - j)! (k - 2 i)! Pochhammer[1/3, i] Pochhammer[2/3 - k, i])) (-(z^3/9))^i, {k, 0, n}, {j, 0, k}, {i, 0, k}]) AiryBi[z] + (z^(1 - n)/2) (Sum[(((-1)^(j + k - 1) (k - i - 1)! (1 - 3 j + 3 k) Pochhammer[3 k - 3 j - n + 2, n - 1] Pochhammer[1/3, k])/ (i! j! (k - j)! (k - 2 i - 1)! Pochhammer[4/3, i] Pochhammer[2/3 - k, i])) (-(z^3/9))^i, {k, 0, n}, {j, 0, k}, {i, 0, k - 1}] - Sum[(((-1)^(j + k - 1) (k - i - 1)! Pochhammer[3 k - 3 j - n + 1, n] Pochhammer[-(1/3), k])/ (i! j! (k - j)! (k - 2 i - 1)! Pochhammer[2/3, i] Pochhammer[4/3 - k, i])) (-(z^3/9))^i, {k, 0, n}, {j, 0, k}, {i, 0, k - 1}]) AiryBiPrime[z] /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["AiryBi", "[", "z", "]"]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["AiryBi", "[", "z", "]"]], "2"], RowBox[List["KroneckerDelta", "[", "n", "]"]]]], "+", RowBox[List[FractionBox[SuperscriptBox["z", RowBox[List["-", "n"]]], "2"], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["k", "-", "1"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "k", "-", "1"]]], " ", "i", " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "i", "-", "1"]], ")"]], "!"]], RowBox[List["(", RowBox[List[RowBox[List["3", " ", "k"]], "-", RowBox[List["3", "i"]], "-", "1"]], ")"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["3", " ", "k"]], "-", RowBox[List["3", " ", "j"]], "-", "n", "+", "1"]], ",", "n"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], ",", "k"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["i", "!"]], RowBox[List["j", "!"]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", RowBox[List["2", "i"]]]], ")"]], "!"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["2", "3"], ",", "i"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["4", "3"], "-", "k"]], ",", "i"]], "]"]]]], " ", ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox["z", "3"], "9"]]], ")"]], "i"]]]]]]]]], "-", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "k"], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "k", "-", "1"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "i"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", "j"]], "+", RowBox[List["3", " ", "k"]]]], ")"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["3", " ", "k"]], "-", RowBox[List["3", " ", "j"]], "-", "n", "+", "2"]], ",", RowBox[List["n", "-", "1"]]]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "3"], ",", "k"]], "]"]]]], " ", ")"]], "/", " ", RowBox[List[RowBox[List["i", "!"]], " ", RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", RowBox[List["2", "i"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "3"], ",", "i"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["2", "3"], "-", "k"]], ",", "i"]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox["z", "3"], "9"]]], ")"]], "i"]]]]]]]]]]], ")"]], " ", RowBox[List["AiryBi", "[", "z", "]"]]]], "+", RowBox[List[FractionBox[SuperscriptBox["z", RowBox[List["1", "-", "n"]]], "2"], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["k", "-", "1"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "k", "-", "1"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "i", "-", "1"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", "j"]], "+", RowBox[List["3", " ", "k"]]]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["3", " ", "k"]], "-", RowBox[List["3", " ", "j"]], "-", "n", "+", "2"]], ",", RowBox[List["n", "-", "1"]]]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "3"], ",", "k"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["i", "!"]], " ", RowBox[List["j", "!"]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", RowBox[List["2", "i"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["4", "3"], ",", "i"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["2", "3"], "-", "k"]], ",", "i"]], "]"]]]], ")"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox["z", "3"], "9"]]], ")"]], "i"]]]]]]]]], " ", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["k", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "k", "-", "1"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "i", "-", "1"]], ")"]], "!"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["3", " ", "k"]], "-", RowBox[List["3", " ", "j"]], "-", "n", "+", "1"]], ",", "n"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["i", "!"]], " ", RowBox[List["j", "!"]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", RowBox[List["2", "i"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["2", "3"], ",", "i"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["4", "3"], "-", "k"]], ",", "i"]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox["z", "3"], "9"]]], ")"]], "i"]]]]]]]]]]], ")"]], RowBox[List["AiryBiPrime", "[", "z", "]"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 n Bi ( z ) z n 1 2 Bi ( z ) δ KroneckerDelta n + 1 2 z - n ( k = 0 n j = 0 k i = 0 k - 1 ( - 1 ) j + k - 1 i ( - i + k - 1 ) ! ( - 3 i + 3 k - 1 ) ( - 3 j + 3 k - n + 1 ) n TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", "j"]], "+", RowBox[List["3", " ", "k"]], "-", "n", "+", "1"]], ")"]], "n"], Pochhammer] ( - 1 3 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "3"]]], ")"]], "k"], Pochhammer] i ! j ! ( k - j ) ! ( k - 2 i ) ! ( 2 3 ) i TagBox[SubscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], "i"], Pochhammer] ( 4 3 - k ) i TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["4", "3"], "-", "k"]], ")"]], "i"], Pochhammer] ( - z 3 9 ) i - k = 0 n j = 0 k i = 0 k ( - 1 ) j + k - 1 ( k - i ) ! ( - 3 j + 3 k + 1 ) ( - 3 j + 3 k - n + 2 ) n - 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", "j"]], "+", RowBox[List["3", " ", "k"]], "-", "n", "+", "2"]], ")"]], RowBox[List["n", "-", "1"]]], Pochhammer] ( 1 3 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "3"], ")"]], "k"], Pochhammer] i ! j ! ( k - j ) ! ( k - 2 i ) ! ( 1 3 ) i TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "3"], ")"]], "i"], Pochhammer] ( 2 3 - k ) i TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["2", "3"], "-", "k"]], ")"]], "i"], Pochhammer] ( - z 3 9 ) i ) Bi ( z ) + 1 2 z 1 - n ( k = 0 n j = 0 k i = 0 k - 1 ( - 1 ) j + k - 1 ( - i + k - 1 ) ! ( - 3 j + 3 k + 1 ) ( - 3 j + 3 k - n + 2 ) n - 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", "j"]], "+", RowBox[List["3", " ", "k"]], "-", "n", "+", "2"]], ")"]], RowBox[List["n", "-", "1"]]], Pochhammer] ( 1 3 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "3"], ")"]], "k"], Pochhammer] i ! j ! ( k - j ) ! ( - 2 i + k - 1 ) ! ( 4 3 ) i TagBox[SubscriptBox[RowBox[List["(", FractionBox["4", "3"], ")"]], "i"], Pochhammer] ( 2 3 - k ) i TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["2", "3"], "-", "k"]], ")"]], "i"], Pochhammer] ( - z 3 9 ) i - k = 0 n j = 0 k i = 0 k - 1 ( - 1 ) j + k - 1 ( - i + k - 1 ) ! ( - 3 j + 3 k - n + 1 ) n TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", "j"]], "+", RowBox[List["3", " ", "k"]], "-", "n", "+", "1"]], ")"]], "n"], Pochhammer] ( - 1 3 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "3"]]], ")"]], "k"], Pochhammer] i ! j ! ( k - j ) ! ( - 2 i + k - 1 ) ! ( 2 3 ) i TagBox[SubscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], "i"], Pochhammer] ( 4 3 - k ) i TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["4", "3"], "-", "k"]], ")"]], "i"], Pochhammer] ( - z 3 9 ) i ) Bi ( z ) /; n Condition z n AiryBi z 1 2 AiryBi z KroneckerDelta n 1 2 z -1 n i 0 k -1 j 0 k k 0 n -1 j k -1 i -1 i k -1 -3 i 3 k -1 Pochhammer -3 j 3 k -1 n 1 n Pochhammer -1 1 3 k i j k -1 j k -1 2 i Pochhammer 2 3 i Pochhammer 4 3 -1 k i -1 -1 z 3 9 -1 i -1 i 0 k j 0 k k 0 n -1 j k -1 k -1 i -3 j 3 k 1 Pochhammer -3 j 3 k -1 n 2 n -1 Pochhammer 1 3 k i j k -1 j k -1 2 i Pochhammer 1 3 i Pochhammer 2 3 -1 k i -1 -1 z 3 9 -1 i AiryBi z 1 2 z 1 -1 n i 0 k -1 j 0 k k 0 n -1 j k -1 -1 i k -1 -3 j 3 k 1 Pochhammer -3 j 3 k -1 n 2 n -1 Pochhammer 1 3 k i j k -1 j -2 i k -1 Pochhammer 4 3 i Pochhammer 2 3 -1 k i -1 -1 z 3 9 -1 i -1 i 0 k -1 j 0 k k 0 n -1 j k -1 -1 i k -1 Pochhammer -3 j 3 k -1 n 1 n Pochhammer -1 1 3 k i j k -1 j -2 i k -1 Pochhammer 2 3 i Pochhammer 4 3 -1 k i -1 -1 z 3 9 -1 i AiryBiPrime z n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["AiryBi", "[", "z_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["AiryBi", "[", "z", "]"]], " ", RowBox[List["KroneckerDelta", "[", "n", "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["z", RowBox[List["-", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["k", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "k", "-", "1"]]], " ", "i", " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "i", "-", 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RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "k", "-", "1"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "i"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", "j"]], "+", RowBox[List["3", " ", "k"]]]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["3", " ", "k"]], "-", RowBox[List["3", " ", "j"]], "-", "n", "+", "2"]], ",", RowBox[List["n", "-", "1"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "3"], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox["z", "3"], "9"]]], ")"]], "i"]]], 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RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["2", "3"], "-", "k"]], ",", "i"]], "]"]]]]]]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["k", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "k", "-", "1"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "i", "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["3", " ", "k"]], "-", RowBox[List["3", " ", "j"]], "-", "n", "+", "1"]], ",", "n"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox["z", "3"], "9"]]], ")"]], "i"]]], RowBox[List[RowBox[List["i", "!"]], " ", RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", RowBox[List["2", " ", "i"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["2", "3"], ",", "i"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["4", "3"], "-", "k"]], ",", "i"]], "]"]]]]]]]]]]]]], ")"]], " ", RowBox[List["AiryBiPrime", "[", "z", "]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02