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 HermiteH

 http://functions.wolfram.com/05.01.06.0020.01

 Input Form

 HermiteH[n, z] \[Proportional] ((((-1)^Floor[n/2] 2^n)/Sqrt[Pi]) E^(z^2/2) Floor[n/2]! (Cos[((1 - (-1)^n) Pi)/4 - 2 z Sqrt[(2 n + 1)/4]] + Sum[(Sum[((-1)^(j + r + s) 2^(2 j - 2 k + s) (1 - (-1)^n/2)^s z^(2 (-j + k - 2 r - s)) Subscript[A, 2 (k - j - r - s)] Cos[Pi (((-1)^n - 1)/4 + j - k + r + s) + 2 z Sqrt[(2 n + 1)/4]] NorlundB[j, 1 - (-1)^n/2, 1 - (-1)^n/2] Pochhammer[(-1)^n/2, j] Pochhammer[(1 - (-1)^n)/4 - j + k - s, s] Pochhammer[ (1 - (-1)^n)/4 - j + k - r - s, r] Pochhammer[(3 - (-1)^n)/4 - j + k - r - s, r] Pochhammer[(1 + (-1)^n)/4 + j - k + r + s, r] Pochhammer[(3 + (-1)^n)/4 + j - k + r + s, r])/ (j! s! Pochhammer[1/2, r]), {j, 0, k}, {r, 0, k - j}, {s, 0, k - j - r}] - (2/z^2) Sum[((-1)^(j + r + s) 2^(2 j - 2 k + s) (1 - (-1)^n/2)^s z^(2 (-j + k - 2 r - s)) Subscript[A, 2 (k - j - r - s) - 1] NorlundB[j, 1 - (-1)^n/2, 1 - (-1)^n/2] Pochhammer[(-1)^n/2, j] Pochhammer[(1 - (-1)^n)/4 - j + k - s, s] Pochhammer[(1 - (-1)^n)/4 - j + k - r - s, r] Pochhammer[(3 - (-1)^n)/4 - j + k - r - s, r] Pochhammer[((-1)^n + 1)/4 + j - k + r + s, 1 + r] Pochhammer[(3 + (-1)^n)/4 + j - k + r + s, 1 + r] Sin[Pi (((-1)^n + 1)/4 + j - k + r + s) + 2 z Sqrt[(2 n + 1)/4]])/ (j! s! Pochhammer[3/2, r]), {j, 0, k - 1}, {r, 0, k - j - 1}, {s, 0, k - j - r - 1}])/Floor[n/2]^k, {k, 1, Infinity}] + (z/(2 Sqrt[Floor[n/2]])) Sum[((((-1)^(j + r + s) 2^(2 j - 2 k + s))/ (j! s! Pochhammer[3/2, r])) (1 - (-1)^n/2)^s z^(2 (k - j - 2 r - s)) NorlundB[j, 1 - (-1)^n/2, 1 - (-1)^n/2] Pochhammer[(-1)^n/2, j] Pochhammer[(3 - (-1)^n)/4 - j + k - s, s] Pochhammer[(3 - (-1)^n)/4 - j + k - r - s, r] Pochhammer[(5 - (-1)^n)/4 - j + k - r - s, r] Pochhammer[((-1)^n - 1)/4 + j - k + r + s, r] Pochhammer[((-1)^n + 1)/4 + j - k + r + s, r] ((1 + 2 r) Subscript[A, 2 (k - j - r - s) + 1] Cos[Pi (((-1)^n - 3)/4 + j - k + r + s) + 2 z Sqrt[(2 n + 1)/4]] - (1/(8 z^2)) ((-1 + (-1)^n + 4 j - 4 k + 8 r + 4 s) (1 + (-1)^n + 4 j - 4 k + 8 r + 4 s)) Subscript[A, 2 (k - j - r - s)] Sin[Pi (((-1)^n - 1)/4 + j - k + r + s) + 2 z Sqrt[(2 n + 1)/4]]))/Floor[n/2]^k, {k, 0, Infinity}, {j, 0, k}, {r, 0, k - j}, {s, 0, k - j - r}]))/ Floor[n/2]^((1 + (-1)^n)/4) /; (n -> Infinity) && Subscript[A, 0] == 1 && Subscript[A, 1] == 0 && Subscript[A, 2] == ((-1)^(n - 1) + 2)/4 && Subscript[A, m] == ((2 m + (-1)^(n - 1) - 2)/(2 m)) Subscript[A, m - 2] - (1/2) (4 Floor[n/2] + (-1)^(n - 1) + 2) Subscript[A, m - 3] && Element[m, Integers] && m > 0

 Standard Form

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 MathML Form

 H n ( z ) ( - 1 ) n 2 2 n π z 2 2 n 2 ! n 2 - 1 + ( - 1 ) n 4 ( cos ( 1 4 ( 1 - ( - 1 ) n ) π - 2 z 1 4 ( 2 n + 1 ) ) + k = 1 n 2 - k ( j = 0 k r = 0 k - j s = 0 k - j - r ( - 1 ) j + r + s 2 2 j - 2 k + s j ! s ! ( 1 2 ) r TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "r"], Pochhammer] ( 1 - ( - 1 ) n 2 ) s z 2 ( k - j - 2 r - s ) A 2 ( k - j - r - s ) cos ( π ( 1 4 ( ( - 1 ) n - 1 ) + j - k + r + s ) + 2 1 4 ( 2 n + 1 ) z ) B TagBox["B", NorlundB] j ( 1 - ( - 1 ) n 2 ) ( 1 - ( - 1 ) n 2 ) ( ( - 1 ) n 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], "2"], ")"]], "j"], Pochhammer] ( 1 4 ( 1 - ( - 1 ) n ) - j + k - s ) s TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]]]], "-", "j", "+", "k", "-", "s"]], ")"]], "s"], Pochhammer] ( 1 4 ( 1 - ( - 1 ) n ) - j + k - r - s ) r TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]]]], "-", "j", "+", "k", "-", "r", "-", "s"]], ")"]], "r"], Pochhammer] ( 1 4 ( 3 - ( - 1 ) n ) - j + k - r - s ) r TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]]]], "-", "j", "+", "k", "-", "r", "-", "s"]], ")"]], "r"], Pochhammer] ( 1 4 ( 1 + ( - 1 ) n ) + j - k + r + s ) r TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]]]], "+", "j", "-", "k", "+", "r", "+", "s"]], ")"]], "r"], Pochhammer] ( 1 4 ( 3 + ( - 1 ) n ) + j - k + r + s ) r TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]]]], "+", "j", "-", "k", "+", "r", "+", "s"]], ")"]], "r"], Pochhammer] - 2 z 2 j = 0 k - 1 r = 0 k - j - 1 s = 0 k - j - r - 1 ( - 1 ) j + r + s 2 2 j - 2 k + s j ! s ! ( 3 2 ) r TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "r"], Pochhammer] ( 1 - ( - 1 ) n 2 ) s z 2 ( k - j - 2 r - s ) A 2 ( k - j - r - s ) - 1 B TagBox["B", NorlundB] j ( 1 - ( - 1 ) n 2 ) ( 1 - ( - 1 ) n 2 ) ( ( - 1 ) n 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], "2"], ")"]], "j"], Pochhammer] ( 1 4 ( 1 - ( - 1 ) n ) - j + k - s ) s TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]]]], "-", "j", "+", "k", "-", "s"]], ")"]], "s"], Pochhammer] ( 1 4 ( 1 - ( - 1 ) n ) - j + k - r - s ) r TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]]]], "-", "j", "+", "k", "-", "r", "-", "s"]], ")"]], "r"], Pochhammer] ( 1 4 ( 3 - ( - 1 ) n ) - j + k - r - s ) r TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]]]], "-", "j", "+", "k", "-", "r", "-", "s"]], ")"]], "r"], Pochhammer] ( 1 4 ( ( - 1 ) n + 1 ) + j - k + r + s ) r + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], "+", "1"]], ")"]]]], "+", "j", "-", "k", "+", "r", "+", "s"]], ")"]], RowBox[List["r", "+", "1"]]], Pochhammer] ( 1 4 ( 3 + ( - 1 ) n ) + j - k + r + s ) r + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]]]], "+", "j", "-", "k", "+", "r", "+", "s"]], ")"]], RowBox[List["r", "+", "1"]]], Pochhammer] sin ( π ( 1 4 ( ( - 1 ) n + 1 ) + j - k + r + s ) + 2 1 4 ( 2 n + 1 ) z ) ) + z 2 n 2 k = 0 j = 0 k r = 0 k - j s = 0 k - j - r ( - 1 ) j + r + s 2 2 j - 2 k + s j ! s ! ( 3 2 ) r TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "r"], Pochhammer] ( 1 - ( - 1 ) n 2 ) s z 2 ( k - j - 2 r - s ) n 2 - k B TagBox["B", NorlundB] j ( 1 - ( - 1 ) n 2 ) ( 1 - ( - 1 ) n 2 ) ( ( - 1 ) n 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], "2"], ")"]], "j"], Pochhammer] ( 1 4 ( 3 - ( - 1 ) n ) - j + k - s ) s TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]]]], "-", "j", "+", "k", "-", "s"]], ")"]], "s"], Pochhammer] ( 1 4 ( 3 - ( - 1 ) n ) - j + k - r - s ) r TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]]]], "-", "j", "+", "k", "-", "r", "-", "s"]], ")"]], "r"], Pochhammer] ( 1 4 ( 5 - ( - 1 ) n ) - j + k - r - s ) r TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["5", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]]]], "-", "j", "+", "k", "-", "r", "-", "s"]], ")"]], "r"], Pochhammer] ( 1 4 ( ( - 1 ) n - 1 ) + j - k + r + s ) r TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], "-", "1"]], ")"]]]], "+", "j", "-", "k", "+", "r", "+", "s"]], ")"]], "r"], Pochhammer] ( 1 4 ( ( - 1 ) n + 1 ) + j - k + r + s ) r TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], "+", "1"]], ")"]]]], "+", "j", "-", "k", "+", "r", "+", "s"]], ")"]], "r"], Pochhammer] ( ( 2 r + 1 ) A 2 ( - j + k - r - s ) + 1 cos ( π ( 1 4 ( ( - 1 ) n - 3 ) + j - k + r + s ) + 2 1 4 ( 2 n + 1 ) z ) - ( 4 j + ( - 1 ) n - 4 k + 8 r + 4 s - 1 ) ( 4 j + ( - 1 ) n - 4 k + 8 r + 4 s + 1 ) 8 z 2 A 2 ( - j + k - r - s ) sin ( π ( 1 4 ( ( - 1 ) n - 1 ) + j - k + r + s ) + 2 1 4 ( 2 n + 1 ) z ) ) ) /; ( n "\[Rule]" ) A 0 1 A 1 0 A 2 ( - 1 ) n - 1 + 2 4 A m 2 m + ( - 1 ) n - 1 - 2 2 m A m - 2 - 1 2 ( 4 n 2 + ( - 1 ) n - 1 + 2 ) A m - 3 m + Condition Proportional HermiteH n z -1 n 2 -1 2 n 1 2 -1 z 2 2 -1 n 2 -1 n 2 -1 -1 1 -1 n 4 -1 1 4 1 -1 -1 n -1 2 z 1 4 2 n 1 1 2 k 1 n 2 -1 -1 k s 0 k -1 j -1 r r 0 k -1 j j 0 k -1 j r s 2 2 j -1 2 k s j s Pochhammer 1 2 r -1 1 -1 -1 n 2 -1 s z 2 k -1 j -1 2 r -1 s Subscript A 2 k -1 j -1 r -1 s 1 4 -1 n -1 j -1 k r s 2 1 4 2 n 1 1 2 z NorlundB j 1 -1 -1 n 2 -1 1 -1 -1 n 2 -1 Pochhammer -1 n 2 -1 j Pochhammer 1 4 1 -1 -1 n -1 j k -1 s s Pochhammer 1 4 1 -1 -1 n -1 j k -1 r -1 s r Pochhammer 1 4 3 -1 -1 n -1 j k -1 r -1 s r Pochhammer 1 4 1 -1 n j -1 k r s r Pochhammer 1 4 3 -1 n j -1 k r s r -1 2 z 2 -1 s 0 k -1 j -1 r -1 r 0 k -1 j -1 j 0 k -1 -1 j r s 2 2 j -1 2 k s j s Pochhammer 3 2 r -1 1 -1 -1 n 2 -1 s z 2 k -1 j -1 2 r -1 s Subscript A 2 k -1 j -1 r -1 s -1 NorlundB j 1 -1 -1 n 2 -1 1 -1 -1 n 2 -1 Pochhammer -1 n 2 -1 j Pochhammer 1 4 1 -1 -1 n -1 j k -1 s s Pochhammer 1 4 1 -1 -1 n -1 j k -1 r -1 s r Pochhammer 1 4 3 -1 -1 n -1 j k -1 r -1 s r Pochhammer 1 4 -1 n 1 j -1 k r s r 1 Pochhammer 1 4 3 -1 n j -1 k r s r 1 1 4 -1 n 1 j -1 k r s 2 1 4 2 n 1 1 2 z z 2 n 2 -1 1 2 -1 s 0 k -1 j -1 r r 0 k -1 j j 0 k k 0 -1 j r s 2 2 j -1 2 k s j s Pochhammer 3 2 r -1 1 -1 -1 n 2 -1 s z 2 k -1 j -1 2 r -1 s n 2 -1 -1 k NorlundB j 1 -1 -1 n 2 -1 1 -1 -1 n 2 -1 Pochhammer -1 n 2 -1 j Pochhammer 1 4 3 -1 -1 n -1 j k -1 s s Pochhammer 1 4 3 -1 -1 n -1 j k -1 r -1 s r Pochhammer 1 4 5 -1 -1 n -1 j k -1 r -1 s r Pochhammer 1 4 -1 n -1 j -1 k r s r Pochhammer 1 4 -1 n 1 j -1 k r s r 2 r 1 Subscript A 2 -1 j k -1 r -1 s 1 1 4 -1 n -3 j -1 k r s 2 1 4 2 n 1 1 2 z -1 4 j -1 n -1 4 k 8 r 4 s -1 4 j -1 n -1 4 k 8 r 4 s 1 8 z 2 -1 Subscript A 2 -1 j k -1 r -1 s 1 4 -1 n -1 j -1 k r s 2 1 4 2 n 1 1 2 z Rule n Subscript A 0 1 Subscript A 1 0 Subscript A 2 -1 n -1 2 4 -1 Subscript A m 2 m -1 n -1 -2 2 m -1 Subscript A m -2 -1 1 2 4 n 2 -1 -1 n -1 2 Subscript A m -3 m SuperPlus [/itex]

 Rule Form

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

 2007-05-02