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 Cosh

 http://functions.wolfram.com/01.20.21.4841.01

 Input Form

 Integrate[z^n E^(p Sqrt[z]) Sinh[b Sqrt[z]]^m Cosh[c z]^v, z] == ((-I^m) 2^(1 - m - v) Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2]) Gamma[2 (1 + n), (-p) Sqrt[z]])/p^(2 (1 + n)) + I^m 2^(1 - m - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[(Binomial[v, s] E^(p^2/(-8 c s + 4 c v)) ((c (-2 s + v))^(2 n) Sum[(-1)^(-h + k) 4^k p^(-h - k + 2 n) (p + 2 c (2 s - v) Sqrt[z])^(h + k) (-((p + 2 c (2 s - v) Sqrt[z])^2/(c (2 s - v))))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (p (p + 2 c (2 s - v) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((p + 2 c (2 s - v) Sqrt[z])^2/(c (8 s - 4 v)))] + 2 c (2 s - v) Sqrt[-((p + 2 c (2 s - v) Sqrt[z])^2/ (c (2 s - v)))] Gamma[(1/2) (2 + h + k), -((p + 2 c (2 s - v) Sqrt[z])^2/(c (8 s - 4 v)))]), {k, 0, n}, {h, 0, k}] + E^(p^2/(4 c s - 2 c v)) (c (2 s - v))^(2 n) Sum[(-1)^(-h + k) 4^k p^(-h - k + 2 n) (p + 2 c (-2 s + v) Sqrt[z])^ (h + k) (-((p + 2 c (-2 s + v) Sqrt[z])^2/(c (-2 s + v))))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (p (p + 2 c (-2 s + v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (p + 2 c (-2 s + v) Sqrt[z])^2/(c (8 s - 4 v))] + 2 c (-2 s + v) Sqrt[(p + 2 c (-2 s + v) Sqrt[z])^2/(c (2 s - v))] Gamma[(1/2) (2 + h + k), (p + 2 c (-2 s + v) Sqrt[z])^2/(c (8 s - 4 v))]), {k, 0, n}, {h, 0, k}]))/ (2^(2 (1 + n)) (c (2 s - v))^(2 n) (c (-2 s + v))^(2 (1 + n))), {s, 0, Floor[(1/2) (-1 + v)]}] - 2^(1 - m - v) Binomial[v, v/2] (1 - Mod[v, 2]) z^(1 + n) Sum[(-1)^u Binomial[m, u] (Gamma[2 (1 + n), (-(b m + p - 2 b u)) Sqrt[z]]/ ((-(b m + p - 2 b u)) Sqrt[z])^(2 (1 + n)) + ((-1)^m Gamma[2 (1 + n), (-((-b) m + p + 2 b u)) Sqrt[z]])/ ((-((-b) m + p + 2 b u)) Sqrt[z])^(2 (1 + n))), {u, 0, Floor[(1/2) (-1 + m)]}] + (2^(-1 - m - v - 2 n)/c^2) Sum[(Binomial[v, s]/(-2 s + v)^2) Sum[(-1)^u Binomial[m, u] ((E^(I m Pi - ((-b) m + p + 2 b u)^2/(c (8 s - 4 v))) Sum[(-1)^(-h + k) 4^k ((-b) m + p + 2 b u)^(-h - k + 2 n) (p - b (m - 2 u) + 2 c (2 s - v) Sqrt[z])^(h + k) (-((p - b (m - 2 u) + 2 c (2 s - v) Sqrt[z])^2/(c (2 s - v))))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (((-b) m + p + 2 b u) (p - b (m - 2 u) + 2 c (2 s - v) Sqrt[z]) Gamma[(1/2) (1 + h + k), -(p - b (m - 2 u) + 2 c (2 s - v) Sqrt[z])^2/(c (8 s - 4 v))] + 2 c (2 s - v) Sqrt[ -(p - b (m - 2 u) + 2 c (2 s - v) Sqrt[z])^2/(c (2 s - v))] Gamma[(1/2) (2 + h + k), -(p - b (m - 2 u) + 2 c (2 s - v) Sqrt[z])^2/(c (8 s - 4 v))]), {k, 0, n}, {h, 0, k}])/ (c (2 s - v))^(2 n) + Sum[(-1)^(-h + k) 4^k (b m + p - 2 b u)^ (-h - k + 2 n) (p + b (m - 2 u) + 2 c (2 s - v) Sqrt[z])^(h + k) (-((p + b (m - 2 u) + 2 c (2 s - v) Sqrt[z])^2/(c (2 s - v))))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((b m + p - 2 b u) (p + b (m - 2 u) + 2 c (2 s - v) Sqrt[z]) Gamma[(1/2) (1 + h + k), -(p + b (m - 2 u) + 2 c (2 s - v) Sqrt[z])^2/(c (8 s - 4 v))] + 2 c (2 s - v) Sqrt[-(p + b (m - 2 u) + 2 c (2 s - v) Sqrt[z])^2/ (c (2 s - v))] Gamma[(1/2) (2 + h + k), -(p + b (m - 2 u) + 2 c (2 s - v) Sqrt[z])^2/(c (8 s - 4 v))]), {k, 0, n}, {h, 0, k}]/(E^((b m + p - 2 b u)^2/(8 c s - 4 c v)) (c (2 s - v))^(2 n)) + (E^(I m Pi + ((-b) m + p + 2 b u)^2/(8 c s - 4 c v)) Sum[(-1)^(-h + k) 4^k ((-b) m + p + 2 b u)^(-h - k + 2 n) (p - b (m - 2 u) + 2 c (-2 s + v) Sqrt[z])^(h + k) (-(p - b (m - 2 u) + 2 c (-2 s + v) Sqrt[z])^2/(c (-2 s + v)))^( (1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (((-b) m + p + 2 b u) (p - b (m - 2 u) + 2 c (-2 s + v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (p - b (m - 2 u) + 2 c (-2 s + v) Sqrt[z])^2/(c (8 s - 4 v))] + 2 c (-2 s + v) Sqrt[-(p - b (m - 2 u) + 2 c (-2 s + v) Sqrt[z])^2/ (c (-2 s + v))] Gamma[(1/2) (2 + h + k), (p - b (m - 2 u) + 2 c (-2 s + v) Sqrt[z])^2/ (c (8 s - 4 v))]), {k, 0, n}, {h, 0, k}] + E^((b m + p - 2 b u)^2/(8 c s - 4 c v)) Sum[(-1)^(-h + k) 4^k (b m + p - 2 b u)^(-h - k + 2 n) (p + b (m - 2 u) + 2 c (-2 s + v) Sqrt[z])^(h + k) (-(p + b (m - 2 u) + 2 c (-2 s + v) Sqrt[z])^2/(c (-2 s + v)))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((b m + p - 2 b u) (p + b (m - 2 u) + 2 c (-2 s + v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (p + b (m - 2 u) + 2 c (-2 s + v) Sqrt[z])^2/(c (8 s - 4 v))] + 2 c (-2 s + v) Sqrt[-(p + b (m - 2 u) + 2 c (-2 s + v) Sqrt[z])^2/ (c (-2 s + v))] Gamma[(1/2) (2 + h + k), (p + b (m - 2 u) + 2 c (-2 s + v) Sqrt[z])^2/ (c (8 s - 4 v))]), {k, 0, n}, {h, 0, k}])/ (c (-2 s + v))^(2 n)), {u, 0, Floor[(1/2) (-1 + m)]}], {s, 0, Floor[(1/2) (-1 + v)]}] /; Element[m, Integers] && m > 0 && Element[v, Integers] && v > 0 && Element[n, Integers] && n >= 0

 Standard Form

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RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["8", " ", "s"]], "-", RowBox[List["4", " ", "v"]]]], ")"]]]], ")"]]]]]], "]"]]]]]], ")"]]]]]]]]]]]], ")"]]]]]], ")"]]]]]]]], " ", ",", RowBox[List["{", RowBox[List["s", ",", "0", ",", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]]], "}"]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z n p z sinh m ( b z ) cosh v ( c z ) z - m 2 - m - v + 1 ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) p - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , - p z ) + m 2 - m - v + 1 ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] 2 - 2 ( n + 1 ) p 2 4 c v - 8 c s ( c ( 2 s - v ) ) - 2 n ( c ( v - 2 s ) ) - 2 ( n + 1 ) ( p 2 4 c s - 2 c v ( c ( 2 s - v ) ) 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k p - h - k + 2 n ( p + 2 c ( v - 2 s ) z ) h + k ( - ( p + 2 c ( v - 2 s ) z ) 2 c ( v - 2 s ) ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p ( p + 2 c ( v - 2 s ) z ) Γ ( 1 2 ( h + k + 1 ) , ( p + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) + 2 c ( p + 2 c ( v - 2 s ) z ) 2 c ( 2 s - v ) ( v - 2 s ) Γ ( 1 2 ( h + k + 2 ) , ( p + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) ) + ( c ( v - 2 s ) ) 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k p - h - k + 2 n ( p + 2 c ( 2 s - v ) z ) h + k ( - ( p + 2 c ( 2 s - v ) z ) 2 c ( 2 s - v ) ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p ( p + 2 c ( 2 s - v ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( p + 2 c ( 2 s - v ) z ) 2 c ( 8 s - 4 v ) ) + 2 c ( 2 s - v ) - ( p + 2 c ( 2 s - v ) z ) 2 c ( 2 s - v ) Γ ( 1 2 ( h + k + 2 ) , - ( p + 2 c ( 2 s - v ) z ) 2 c ( 8 s - 4 v ) ) ) ) - 2 - m - v + 1 ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - v mod 2 \$CellContext`v 2 ) z n + 1 u = 0 m - 1 2 ( - 1 ) u ( m u ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["u", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - 1 ) m Γ ( 2 ( n + 1 ) , - ( - b m + p + 2 b u ) z ) ( - ( - b m + p + 2 b u ) z ) - 2 ( n + 1 ) + ( - ( b m + p - 2 b u ) z ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , - ( b m + p - 2 b u ) z ) ) + 1 c 2 2 - m - 2 n - v - 1 s = 0 v - 1 2 1 ( v - 2 s ) 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] u = 0 m - 1 2 ( - 1 ) u ( m u ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["u", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - ( b m + p - 2 b u ) 2 8 c s - 4 c v ( c ( 2 s - v ) ) - 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b m + p - 2 b u ) - h - k + 2 n ( p + b ( m - 2 u ) + 2 c ( 2 s - v ) z ) h + k ( - ( p + b ( m - 2 u ) + 2 c ( 2 s - v ) z ) 2 c ( 2 s - v ) ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b m + p - 2 b u ) ( p + b ( m - 2 u ) + 2 c ( 2 s - v ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( p + b ( m - 2 u ) + 2 c ( 2 s - v ) z ) 2 c ( 8 s - 4 v ) ) + 2 c ( 2 s - v ) Γ ( 1 2 ( h + k + 2 ) , - ( p + b ( m - 2 u ) + 2 c ( 2 s - v ) z ) 2 c ( 8 s - 4 v ) ) - ( p + b ( m - 2 u ) + 2 c ( 2 s - v ) z ) 2 c ( 2 s - v ) ) + m π - ( - b m + p + 2 b u ) 2 c ( 8 s - 4 v ) ( c ( 2 s - v ) ) - 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b m + p + 2 b u ) - h - k + 2 n ( p - b ( m - 2 u ) + 2 c ( 2 s - v ) z ) h + k ( - ( p - b ( m - 2 u ) + 2 c ( 2 s - v ) z ) 2 c ( 2 s - v ) ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - b m + p + 2 b u ) ( p - b ( m - 2 u ) + 2 c ( 2 s - v ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( p - b ( m - 2 u ) + 2 c ( 2 s - v ) z ) 2 c ( 8 s - 4 v ) ) + 2 c ( 2 s - v ) Γ ( 1 2 ( h + k + 2 ) , - ( p - b ( m - 2 u ) + 2 c ( 2 s - v ) z ) 2 c ( 8 s - 4 v ) ) - ( p - b ( m - 2 u ) + 2 c ( 2 s - v ) z ) 2 c ( 2 s - v ) ) + ( c ( v - 2 s ) ) - 2 n ( ( b m + p - 2 b u ) 2 8 c s - 4 c v k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b m + p - 2 b u ) - h - k + 2 n ( p + b ( m - 2 u ) + 2 c ( v - 2 s ) z ) h + k ( - ( p + b ( m - 2 u ) + 2 c ( v - 2 s ) z ) 2 c ( v - 2 s ) ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b m + p - 2 b u ) ( p + b ( m - 2 u ) + 2 c ( v - 2 s ) z ) Γ ( 1 2 ( h + k + 1 ) , ( p + b ( m - 2 u ) + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) + 2 c ( v - 2 s ) Γ ( 1 2 ( h + k + 2 ) , ( p + b ( m - 2 u ) + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) - ( p + b ( m - 2 u ) + 2 c ( v - 2 s ) z ) 2 c ( v - 2 s ) ) + ( - b m + p + 2 b u ) 2 8 c s - 4 c v + m π k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b m + p + 2 b u ) - h - k + 2 n ( p - b ( m - 2 u ) + 2 c ( v - 2 s ) z ) h + k ( - ( p - b ( m - 2 u ) + 2 c ( v - 2 s ) z ) 2 c ( v - 2 s ) ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - b m + p + 2 b u ) ( p - b ( m - 2 u ) + 2 c ( v - 2 s ) z ) Γ ( 1 2 ( h + k + 1 ) , ( p - b ( m - 2 u ) + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) + 2 c ( v - 2 s ) Γ ( 1 2 ( h + k + 2 ) , ( p - b ( m - 2 u ) + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) - ( p - b ( m - 2 u ) + 2 c ( v - 2 s ) z ) 2 c ( v - 2 s ) ) ) ) /; m + v + n Condition z z n p z 1 2 b z 1 2 m c z v -1 m 2 -1 m -1 v 1 Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 p -2 n 1 Gamma 2 n 1 -1 p z 1 2 m 2 -1 m -1 v 1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 s 0 v -1 2 -1 Binomial v s 2 -2 n 1 p 2 4 c v -1 8 c s -1 c 2 s -1 v -2 n c v -1 2 s -2 n 1 p 2 4 c s -1 2 c v -1 c 2 s -1 v 2 n h 0 k k 0 n -1 k -1 h 4 k p -1 h -1 k 2 n p 2 c v -1 2 s z 1 2 h k -1 p 2 c v -1 2 s z 1 2 2 c v -1 2 s -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k p p 2 c v -1 2 s z 1 2 Gamma 1 2 h k 1 p 2 c v -1 2 s z 1 2 2 c 8 s -1 4 v -1 2 c p 2 c v -1 2 s z 1 2 2 c 2 s -1 v -1 1 2 v -1 2 s Gamma 1 2 h k 2 p 2 c v -1 2 s z 1 2 2 c 8 s -1 4 v -1 c v -1 2 s 2 n h 0 k k 0 n -1 k -1 h 4 k p -1 h -1 k 2 n p 2 c 2 s -1 v z 1 2 h k -1 p 2 c 2 s -1 v z 1 2 2 c 2 s -1 v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k p p 2 c 2 s -1 v z 1 2 Gamma 1 2 h k 1 -1 p 2 c 2 s -1 v z 1 2 2 c 8 s -1 4 v -1 2 c 2 s -1 v -1 p 2 c 2 s -1 v z 1 2 2 c 2 s -1 v -1 1 2 Gamma 1 2 h k 2 -1 p 2 c 2 s -1 v z 1 2 2 c 8 s -1 4 v -1 -1 2 -1 m -1 v 1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 z n 1 u 0 m -1 2 -1 -1 u Binomial m u -1 m Gamma 2 n 1 -1 -1 b m p 2 b u z 1 2 -1 -1 b m p 2 b u z 1 2 -2 n 1 -1 b m p -1 2 b u z 1 2 -2 n 1 Gamma 2 n 1 -1 b m p -1 2 b u z 1 2 1 c 2 -1 2 -1 m -1 2 n -1 v -1 s 0 v -1 2 -1 1 v -1 2 s 2 -1 Binomial v s u 0 m -1 2 -1 -1 u Binomial m u -1 b m p -1 2 b u 2 8 c s -1 4 c v -1 c 2 s -1 v -2 n h 0 k k 0 n -1 k -1 h 4 k b m p -1 2 b u -1 h -1 k 2 n p b m -1 2 u 2 c 2 s -1 v z 1 2 h k -1 p b m -1 2 u 2 c 2 s -1 v z 1 2 2 c 2 s -1 v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b m p -1 2 b u p b m -1 2 u 2 c 2 s -1 v z 1 2 Gamma 1 2 h k 1 -1 p b m -1 2 u 2 c 2 s -1 v z 1 2 2 c 8 s -1 4 v -1 2 c 2 s -1 v Gamma 1 2 h k 2 -1 p b m -1 2 u 2 c 2 s -1 v z 1 2 2 c 8 s -1 4 v -1 -1 p b m -1 2 u 2 c 2 s -1 v z 1 2 2 c 2 s -1 v -1 1 2 m -1 -1 b m p 2 b u 2 c 8 s -1 4 v -1 c 2 s -1 v -2 n h 0 k k 0 n -1 k -1 h 4 k -1 b m p 2 b u -1 h -1 k 2 n p -1 b m -1 2 u 2 c 2 s -1 v z 1 2 h k -1 p -1 b m -1 2 u 2 c 2 s -1 v z 1 2 2 c 2 s -1 v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k -1 b m p 2 b u p -1 b m -1 2 u 2 c 2 s -1 v z 1 2 Gamma 1 2 h k 1 -1 p -1 b m -1 2 u 2 c 2 s -1 v z 1 2 2 c 8 s -1 4 v -1 2 c 2 s -1 v Gamma 1 2 h k 2 -1 p -1 b m -1 2 u 2 c 2 s -1 v z 1 2 2 c 8 s -1 4 v -1 -1 p -1 b m -1 2 u 2 c 2 s -1 v z 1 2 2 c 2 s -1 v -1 1 2 c v -1 2 s -2 n b m p -1 2 b u 2 8 c s -1 4 c v -1 h 0 k k 0 n -1 k -1 h 4 k b m p -1 2 b u -1 h -1 k 2 n p b m -1 2 u 2 c v -1 2 s z 1 2 h k -1 p b m -1 2 u 2 c v -1 2 s z 1 2 2 c v -1 2 s -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b m p -1 2 b u p b m -1 2 u 2 c v -1 2 s z 1 2 Gamma 1 2 h k 1 p b m -1 2 u 2 c v -1 2 s z 1 2 2 c 8 s -1 4 v -1 2 c v -1 2 s Gamma 1 2 h k 2 p b m -1 2 u 2 c v -1 2 s z 1 2 2 c 8 s -1 4 v -1 -1 p b m -1 2 u 2 c v -1 2 s z 1 2 2 c v -1 2 s -1 1 2 -1 b m p 2 b u 2 8 c s -1 4 c v -1 m h 0 k k 0 n -1 k -1 h 4 k -1 b m p 2 b u -1 h -1 k 2 n p -1 b m -1 2 u 2 c v -1 2 s z 1 2 h k -1 p -1 b m -1 2 u 2 c v -1 2 s z 1 2 2 c v -1 2 s -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k -1 b m p 2 b u p -1 b m -1 2 u 2 c v -1 2 s z 1 2 Gamma 1 2 h k 1 p -1 b m -1 2 u 2 c v -1 2 s z 1 2 2 c 8 s -1 4 v -1 2 c v -1 2 s Gamma 1 2 h k 2 p -1 b m -1 2 u 2 c v -1 2 s z 1 2 2 c 8 s -1 4 v -1 -1 p -1 b m -1 2 u 2 c v -1 2 s z 1 2 2 c v -1 2 s -1 1 2 m SuperPlus v SuperPlus n [/itex]

 Rule Form

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

 2002-12-18