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 Cosh

 http://functions.wolfram.com/01.20.21.3947.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18