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 Cosh

 http://functions.wolfram.com/01.20.21.4572.01

 Input Form

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

 Standard Form

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 MathML Form

 z n sinh m ( z b + d z ) cosh v ( z c + g ) z 2 - m - v - 1 ( 1 b ( 4 - n ( 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]]]]] ( v mod 2 \$CellContext`v 2 - 1 ) ( u = 0 m - 1 2 ( - 1 ) u - b 2 ( m - 2 u ) 4 d ( m - 2 u ) - 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]]]]] ( ( - 1 ) m b 2 ( m - 2 u ) 2 d k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b ( m - 2 u ) ) - h - k + 2 n + 1 ( - ( m - 2 u ) ( b + 2 d z ) ) h + k ( ( m - 2 u ) ( b + 2 d z ) 2 d ) 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 - 2 u ) ( b + 2 d z ) Γ ( 1 2 ( h + k + 1 ) , ( m - 2 u ) ( b + 2 d z ) 2 4 d ) - 2 d ( m - 2 u ) ( b + 2 d z ) 2 d Γ ( 1 2 ( h + k + 2 ) , ( m - 2 u ) ( b + 2 d z ) 2 4 d ) ) + k = 0 n h = 0 k ( - 1 ) - h + k + 1 4 k ( b ( m - 2 u ) ) - h - k + 2 n + 1 ( ( m - 2 u ) ( b + 2 d z ) ) h + k ( - ( m - 2 u ) ( b + 2 d z ) 2 d ) 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 - 2 u ) ( b + 2 d z ) Γ ( 1 2 ( h + k + 1 ) , - ( m - 2 u ) ( b + 2 d z ) 2 4 d ) + 2 - ( m - 2 u ) ( b + 2 d z ) 2 d d Γ ( 1 2 ( h + k + 2 ) , - ( m - 2 u ) ( b + 2 d z ) 2 4 d ) ) ) ) d - 2 n - 2 ) + 2 - m z 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]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) ( v mod 2 \$CellContext`v 2 - 1 ) n + 1 + 4 c - 2 n - 2 - m ( 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]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) s = 0 v - 1 2 g ( 2 s - v ) ( 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]]]]] ( Γ ( 2 ( n + 1 ) , - c ( 2 s - v ) z ) + 2 g ( v - 2 s ) Γ ( 2 ( n + 1 ) , c ( 2 s - v ) z ) ) - 4 - n 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]]]]] s = 0 v - 1 2 - 1 4 ( c 2 ( v - 2 s ) 2 d ( m - 2 u ) + ( m - 2 u ) b 2 + 2 c ( v - 2 s ) b d ) - g ( 2 s + v ) ( - d 2 ( m - 2 u ) 2 ) - 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]]]]] ( 4 g s + b c ( v - 2 s ) d ( k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b m + 2 c s - 2 b u - c v ) - h - k + 2 n ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( 2 s - v ) ) h + k ( - ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( 2 s - v ) ) 2 d ( 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]]]]] ( ( b m + 2 c s - 2 b u - c v ) ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( 2 s - v ) ) Γ ( 1 2 ( h + k + 1 ) , - ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( 2 s - v ) ) 2 4 d ( m - 2 u ) ) + 2 d ( m - 2 u ) Γ ( 1 2 ( h + k + 2 ) , - ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( 2 s - v ) ) 2 4 d ( m - 2 u ) ) - ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( 2 s - v ) ) 2 d ( m - 2 u ) ) ) ( - d ( m - 2 u ) ) 2 n + 2 g v ( k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b m - 2 c s - 2 b u + c v ) - h - k + 2 n ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( v - 2 s ) ) h + k ( - ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( v - 2 s ) ) 2 d ( 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]]]]] ( ( b m - 2 c s - 2 b u + c v ) ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( v - 2 s ) ) Γ ( 1 2 ( h + k + 1 ) , - ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( v - 2 s ) ) 2 4 d ( m - 2 u ) ) + 2 d ( m - 2 u ) Γ ( 1 2 ( h + k + 2 ) , - ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( v - 2 s ) ) 2 4 d ( m - 2 u ) ) - ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( v - 2 s ) ) 2 d ( m - 2 u ) ) ) ( - d ( m - 2 u ) ) 2 n + ( - 1 ) m 1 2 ( ( m - 2 u ) b 2 d + c 2 ( v - 2 s ) 2 d ( m - 2 u ) + 4 g v ) ( d ( m - 2 u ) ) 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b m - 2 c s + 2 b u + c v ) - h - k + 2 n ( - b ( m - 2 u ) - 2 d z ( m - 2 u ) + c ( v - 2 s ) ) h + k ( ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( 2 s - v ) ) 2 d ( 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]]]]] ( ( b m + 2 c s - 2 b u - c v ) ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( 2 s - v ) ) Γ ( 1 2 ( h + k + 1 ) , ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( 2 s - v ) ) 2 4 d ( m - 2 u ) ) - 2 d ( m - 2 u ) ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( 2 s - v ) ) 2 d ( m - 2 u ) Γ ( 1 2 ( h + k + 2 ) , ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( 2 s - v ) ) 2 4 d ( m - 2 u ) ) ) + ( - 1 ) m 1 2 ( c 2 ( v - 2 s ) 2 d ( m - 2 u ) + 8 g s + ( m - 2 u ) b 2 + 2 c ( v - 2 s ) b d ) ( d ( m - 2 u ) ) 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b m + 2 c s + 2 b u - c v ) - h - k + 2 n ( - b ( m - 2 u ) - 2 d z ( m - 2 u ) + c ( 2 s - v ) ) h + k ( ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( v - 2 s ) ) 2 d ( 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]]]]] ( ( b m - 2 c s - 2 b u + c v ) ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( v - 2 s ) ) Γ ( 1 2 ( h + k + 1 ) , ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( v - 2 s ) ) 2 4 d ( m - 2 u ) ) - 2 d ( m - 2 u ) ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( v - 2 s ) ) 2 d ( m - 2 u ) Γ ( 1 2 ( h + k + 2 ) , ( b ( m - 2 u ) + 2 d z ( m - 2 u ) + c ( v - 2 s ) ) 2 4 d ( m - 2 u ) ) ) ) ) /; n m + v + Condition z z n z 1 2 b d z m z 1 2 c g v 2 -1 m -1 v -1 1 b -1 4 -1 n Binomial v v 2 -1 \$CellContext`v 2 -1 u 0 m -1 2 -1 -1 u -1 b 2 m -1 2 u 4 d -1 m -1 2 u -2 n 1 Binomial m u -1 m b 2 m -1 2 u 2 d -1 h 0 k k 0 n -1 k -1 h 4 k -1 b m -1 2 u -1 h -1 k 2 n 1 -1 m -1 2 u b 2 d z 1 2 h k m -1 2 u b 2 d z 1 2 2 d -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b m -1 2 u b 2 d z 1 2 Gamma 1 2 h k 1 m -1 2 u b 2 d z 1 2 2 4 d -1 -1 2 d m -1 2 u b 2 d z 1 2 2 d -1 1 2 Gamma 1 2 h k 2 m -1 2 u b 2 d z 1 2 2 4 d -1 h 0 k k 0 n -1 -1 h k 1 4 k b m -1 2 u -1 h -1 k 2 n 1 m -1 2 u b 2 d z 1 2 h k -1 m -1 2 u b 2 d z 1 2 2 d -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b m -1 2 u b 2 d z 1 2 Gamma 1 2 h k 1 -1 m -1 2 u b 2 d z 1 2 2 4 d -1 2 -1 m -1 2 u b 2 d z 1 2 2 d -1 1 2 d Gamma 1 2 h k 2 -1 m -1 2 u b 2 d z 1 2 2 4 d -1 d -2 n -2 2 -1 m z n 1 Binomial m m 2 -1 Binomial v v 2 -1 \$CellContext`m 2 -1 \$CellContext`v 2 -1 n 1 -1 4 c -2 n -2 -1 m Binomial m m 2 -1 \$CellContext`m 2 -1 s 0 v -1 2 -1 g 2 s -1 v v -1 2 s -2 n 1 Binomial v s Gamma 2 n 1 -1 c 2 s -1 v z 1 2 2 g v -1 2 s Gamma 2 n 1 c 2 s -1 v z 1 2 -1 4 -1 n u 0 m -1 2 -1 -1 u Binomial m u s 0 v -1 2 -1 -1 1 4 c 2 v -1 2 s 2 d m -1 2 u -1 m -1 2 u b 2 2 c v -1 2 s b d -1 -1 g 2 s v -1 d 2 m -1 2 u 2 -2 n -1 Binomial v s 4 g s b c v -1 2 s d -1 h 0 k k 0 n -1 k -1 h 4 k b m 2 c s -1 2 b u -1 c v -1 h -1 k 2 n b m -1 2 u 2 d z 1 2 m -1 2 u c 2 s -1 v h k -1 b m -1 2 u 2 d z 1 2 m -1 2 u c 2 s -1 v 2 d m -1 2 u -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b m 2 c s -1 2 b u -1 c v b m -1 2 u 2 d z 1 2 m -1 2 u c 2 s -1 v Gamma 1 2 h k 1 -1 b m -1 2 u 2 d z 1 2 m -1 2 u c 2 s -1 v 2 4 d m -1 2 u -1 2 d m -1 2 u Gamma 1 2 h k 2 -1 b m -1 2 u 2 d z 1 2 m -1 2 u c 2 s -1 v 2 4 d m -1 2 u -1 -1 b m -1 2 u 2 d z 1 2 m -1 2 u c 2 s -1 v 2 d m -1 2 u -1 1 2 -1 d m -1 2 u 2 n 2 g v h 0 k k 0 n -1 k -1 h 4 k b m -1 2 c s -1 2 b u c v -1 h -1 k 2 n b m -1 2 u 2 d z 1 2 m -1 2 u c v -1 2 s h k -1 b m -1 2 u 2 d z 1 2 m -1 2 u c v -1 2 s 2 d m -1 2 u -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b m -1 2 c s -1 2 b u c v b m -1 2 u 2 d z 1 2 m -1 2 u c v -1 2 s Gamma 1 2 h k 1 -1 b m -1 2 u 2 d z 1 2 m -1 2 u c v -1 2 s 2 4 d m -1 2 u -1 2 d m -1 2 u Gamma 1 2 h k 2 -1 b m -1 2 u 2 d z 1 2 m -1 2 u c v -1 2 s 2 4 d m -1 2 u -1 -1 b m -1 2 u 2 d z 1 2 m -1 2 u c v -1 2 s 2 d m -1 2 u -1 1 2 -1 d m -1 2 u 2 n -1 m 1 2 m -1 2 u b 2 d -1 c 2 v -1 2 s 2 d m -1 2 u -1 4 g v d m -1 2 u 2 n h 0 k k 0 n -1 k -1 h 4 k -1 b m -1 2 c s 2 b u c v -1 h -1 k 2 n -1 b m -1 2 u -1 2 d z 1 2 m -1 2 u c v -1 2 s h k b m -1 2 u 2 d z 1 2 m -1 2 u c 2 s -1 v 2 d m -1 2 u -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b m 2 c s -1 2 b u -1 c v b m -1 2 u 2 d z 1 2 m -1 2 u c 2 s -1 v Gamma 1 2 h k 1 b m -1 2 u 2 d z 1 2 m -1 2 u c 2 s -1 v 2 4 d m -1 2 u -1 -1 2 d m -1 2 u b m -1 2 u 2 d z 1 2 m -1 2 u c 2 s -1 v 2 d m -1 2 u -1 1 2 Gamma 1 2 h k 2 b m -1 2 u 2 d z 1 2 m -1 2 u c 2 s -1 v 2 4 d m -1 2 u -1 -1 m 1 2 c 2 v -1 2 s 2 d m -1 2 u -1 8 g s m -1 2 u b 2 2 c v -1 2 s b d -1 d m -1 2 u 2 n h 0 k k 0 n -1 k -1 h 4 k -1 b m 2 c s 2 b u -1 c v -1 h -1 k 2 n -1 b m -1 2 u -1 2 d z 1 2 m -1 2 u c 2 s -1 v h k b m -1 2 u 2 d z 1 2 m -1 2 u c v -1 2 s 2 d m -1 2 u -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b m -1 2 c s -1 2 b u c v b m -1 2 u 2 d z 1 2 m -1 2 u c v -1 2 s Gamma 1 2 h k 1 b m -1 2 u 2 d z 1 2 m -1 2 u c v -1 2 s 2 4 d m -1 2 u -1 -1 2 d m -1 2 u b m -1 2 u 2 d z 1 2 m -1 2 u c v -1 2 s 2 d m -1 2 u -1 1 2 Gamma 1 2 h k 2 b m -1 2 u 2 d z 1 2 m -1 2 u c v -1 2 s 2 4 d m -1 2 u -1 n m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18