html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Cosh

 http://functions.wolfram.com/01.20.21.4918.01

 Input Form

 Integrate[z^n Sin[a z^2 + p z + q]^m Sinh[w z^2 + s z + t]^u Cosh[c z^2 + f z + g]^v, z] == ((I^u 2^(-m - u - v) z^(1 + n))/(1 + n)) Binomial[m, m/2] Binomial[u, u/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[u, 2]) (1 - Mod[v, 2]) - I^u 2^(-1 - m - u - v) Binomial[u, u/2] Binomial[v, v/2] (1 - Mod[u, 2]) (1 - Mod[v, 2]) Sum[(-1)^k Binomial[m, k] (E^(-((I (2 k - m) p^2)/(4 a)) + (I m Pi)/2 + I (2 k - m) q) (I a (2 k - m))^(-1 - n) Sum[2^(i - n) ((-I) (2 k - m) p)^(-i + n) (I (2 k - m) p + 2 I a (2 k - m) z)^(1 + i) ((I (I (2 k - m) p + 2 I a (2 k - m) z)^2)/(a (2 k - m)))^ ((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, (I (I (2 k - m) p + 2 I a (2 k - m) z)^2)/(4 a (2 k - m))], {i, 0, n}] + E^(-((I (-2 k + m) p^2)/(4 a)) - (I m Pi)/2 + I (-2 k + m) q) (I a (-2 k + m))^(-1 - n) Sum[2^(i - n) ((-I) (-2 k + m) p)^(-i + n) (I (-2 k + m) p + 2 I a (-2 k + m) z)^(1 + i) ((I (I (-2 k + m) p + 2 I a (-2 k + m) z)^2)/(a (-2 k + m)))^ ((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, (I (I (-2 k + m) p + 2 I a (-2 k + m) z)^2)/(4 a (-2 k + m))], {i, 0, n}]), {k, 0, Floor[(1/2) (-1 + m)]}] - I^u 2^(-1 - m - u - v) Binomial[m, m/2] Binomial[u, u/2] (1 - Mod[m, 2]) (1 - Mod[u, 2]) Sum[Binomial[v, h] (E^(-((f^2 (2 h - v))/(4 c)) + g (2 h - v)) (c (2 h - v))^(-1 - n) Sum[2^(i - n) ((-f) (2 h - v))^(-i + n) (f (2 h - v) + 2 c (2 h - v) z)^(1 + i) (-((f (2 h - v) + 2 c (2 h - v) z)^2/(c (2 h - v))))^ ((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, -((f (2 h - v) + 2 c (2 h - v) z)^2/(4 c (2 h - v)))], {i, 0, n}] + E^(-((f^2 (-2 h + v))/(4 c)) + g (-2 h + v)) (c (-2 h + v))^(-1 - n) Sum[2^(i - n) ((-f) (-2 h + v))^(-i + n) (f (-2 h + v) + 2 c (-2 h + v) z)^(1 + i) (-((f (-2 h + v) + 2 c (-2 h + v) z)^2/(c (-2 h + v))))^ ((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, -((f (-2 h + v) + 2 c (-2 h + v) z)^2/(4 c (-2 h + v)))], {i, 0, n}]), {h, 0, Floor[(1/2) (-1 + v)]}] - I^u 2^(-1 - m - u - v) Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2]) Sum[(-1)^k Binomial[u, k] (E^(t (2 k - u) + (I Pi u)/2 - (s^2 (2 k - u))/(4 w)) ((2 k - u) w)^(-1 - n) Sum[2^(i - n) ((-s) (2 k - u))^(-i + n) (s (2 k - u) + 2 (2 k - u) w z)^(1 + i) (-((s (2 k - u) + 2 (2 k - u) w z)^2/((2 k - u) w)))^ ((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, -((s (2 k - u) + 2 (2 k - u) w z)^2/(4 (2 k - u) w))], {i, 0, n}] + E^((-(1/2)) I Pi u + t (-2 k + u) - (s^2 (-2 k + u))/(4 w)) ((-2 k + u) w)^(-1 - n) Sum[2^(i - n) ((-s) (-2 k + u))^(-i + n) (s (-2 k + u) + 2 (-2 k + u) w z)^(1 + i) (-((s (-2 k + u) + 2 (-2 k + u) w z)^2/((-2 k + u) w)))^ ((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, -((s (-2 k + u) + 2 (-2 k + u) w z)^2/(4 (-2 k + u) w))], {i, 0, n}]), {k, 0, Floor[(1/2) (-1 + u)]}] - I^u 2^(-1 - m - u - v) Binomial[u, u/2] (1 - Mod[u, 2]) Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, h] (E^((I m Pi)/2 + I (2 k - m) q - (I (2 k - m) p + f (2 h - v))^2/ (4 (I a (2 k - m) + c (2 h - v))) + g (2 h - v)) (I a (2 k - m) + c (2 h - v))^(-1 - n) Sum[2^(i - n) ((-I) (2 k - m) p - f (2 h - v))^(-i + n) (I (2 k - m) p + f (2 h - v) + 2 (I a (2 k - m) + c (2 h - v)) z)^ (1 + i) (-((I (2 k - m) p + f (2 h - v) + 2 (I a (2 k - m) + c (2 h - v)) z)^2/(I a (2 k - m) + c (2 h - v))))^ ((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, -((I (2 k - m) p + f (2 h - v) + 2 (I a (2 k - m) + c (2 h - v)) z)^2/(4 (I a (2 k - m) + c (2 h - v))))], {i, 0, n}] + E^((-(1/2)) I m Pi + I (-2 k + m) q - (I (-2 k + m) p + f (2 h - v))^ 2/(4 (I a (-2 k + m) + c (2 h - v))) + g (2 h - v)) (I a (-2 k + m) + c (2 h - v))^(-1 - n) Sum[2^(i - n) ((-I) (-2 k + m) p - f (2 h - v))^(-i + n) (I (-2 k + m) p + f (2 h - v) + 2 (I a (-2 k + m) + c (2 h - v)) z)^(1 + i) (-((I (-2 k + m) p + f (2 h - v) + 2 (I a (-2 k + m) + c (2 h - v)) z)^2/(I a (-2 k + m) + c (2 h - v))))^((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, -((I (-2 k + m) p + f (2 h - v) + 2 (I a (-2 k + m) + c (2 h - v)) z)^2/(4 (I a (-2 k + m) + c (2 h - v))))], {i, 0, n}] + E^((I m Pi)/2 + I (2 k - m) q + g (-2 h + v) - (I (2 k - m) p + f (-2 h + v))^2/(4 (I a (2 k - m) + c (-2 h + v)))) (I a (2 k - m) + c (-2 h + v))^(-1 - n) Sum[2^(i - n) ((-I) (2 k - m) p - f (-2 h + v))^(-i + n) (I (2 k - m) p + f (-2 h + v) + 2 (I a (2 k - m) + c (-2 h + v)) z)^(1 + i) (-((I (2 k - m) p + f (-2 h + v) + 2 (I a (2 k - m) + c (-2 h + v)) z)^2/(I a (2 k - m) + c (-2 h + v))))^((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, -((I (2 k - m) p + f (-2 h + v) + 2 (I a (2 k - m) + c (-2 h + v)) z)^2/(4 (I a (2 k - m) + c (-2 h + v))))], {i, 0, n}] + E^((-(1/2)) I m Pi + I (-2 k + m) q + g (-2 h + v) - (I (-2 k + m) p + f (-2 h + v))^2/(4 (I a (-2 k + m) + c (-2 h + v)))) (I a (-2 k + m) + c (-2 h + v))^(-1 - n) Sum[2^(i - n) ((-I) (-2 k + m) p - f (-2 h + v))^(-i + n) (I (-2 k + m) p + f (-2 h + v) + 2 (I a (-2 k + m) + c (-2 h + v)) z)^(1 + i) (-((I (-2 k + m) p + f (-2 h + v) + 2 (I a (-2 k + m) + c (-2 h + v)) z)^2/(I a (-2 k + m) + c (-2 h + v))))^((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, -((I (-2 k + m) p + f (-2 h + v) + 2 (I a (-2 k + m) + c (-2 h + v)) z)^2/(4 (I a (-2 k + m) + c (-2 h + v))))], {i, 0, n}]), {h, 0, Floor[(1/2) (-1 + v)]}], {k, 0, Floor[(1/2) (-1 + m)]}] - I^u 2^(-1 - m - u - v) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(-1)^k Binomial[m, k] Sum[(-1)^j Binomial[u, j] (E^((I m Pi)/2 + I (2 k - m) q + t (2 j - u) + (I Pi u)/2 - (I (2 k - m) p + s (2 j - u))^2/(4 (I a (2 k - m) + (2 j - u) w))) (I a (2 k - m) + (2 j - u) w)^(-1 - n) Sum[2^(i - n) ((-I) (2 k - m) p - s (2 j - u))^(-i + n) (I (2 k - m) p + s (2 j - u) + 2 (I a (2 k - m) + (2 j - u) w) z)^ (1 + i) (-((I (2 k - m) p + s (2 j - u) + 2 (I a (2 k - m) + (2 j - u) w) z)^2/(I a (2 k - m) + (2 j - u) w)))^ ((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, -((I (2 k - m) p + s (2 j - u) + 2 (I a (2 k - m) + (2 j - u) w) z)^2/(4 (I a (2 k - m) + (2 j - u) w)))], {i, 0, n}] + E^((-(1/2)) I m Pi + I (-2 k + m) q + t (2 j - u) + (I Pi u)/2 - (I (-2 k + m) p + s (2 j - u))^2/(4 (I a (-2 k + m) + (2 j - u) w))) (I a (-2 k + m) + (2 j - u) w)^(-1 - n) Sum[2^(i - n) ((-I) (-2 k + m) p - s (2 j - u))^(-i + n) (I (-2 k + m) p + s (2 j - u) + 2 (I a (-2 k + m) + (2 j - u) w) z)^(1 + i) (-((I (-2 k + m) p + s (2 j - u) + 2 (I a (-2 k + m) + (2 j - u) w) z)^2/(I a (-2 k + m) + (2 j - u) w)))^((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, -((I (-2 k + m) p + s (2 j - u) + 2 (I a (-2 k + m) + (2 j - u) w) z)^2/(4 (I a (-2 k + m) + (2 j - u) w)))], {i, 0, n}] + E^((I m Pi)/2 + I (2 k - m) q - (I Pi u)/2 + t (-2 j + u) - (I (2 k - m) p + s (-2 j + u))^2/(4 (I a (2 k - m) + (-2 j + u) w))) (I a (2 k - m) + (-2 j + u) w)^(-1 - n) Sum[2^(i - n) ((-I) (2 k - m) p - s (-2 j + u))^(-i + n) (I (2 k - m) p + s (-2 j + u) + 2 (I a (2 k - m) + (-2 j + u) w) z)^(1 + i) (-((I (2 k - m) p + s (-2 j + u) + 2 (I a (2 k - m) + (-2 j + u) w) z)^2/(I a (2 k - m) + (-2 j + u) w)))^((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, -((I (2 k - m) p + s (-2 j + u) + 2 (I a (2 k - m) + (-2 j + u) w) z)^2/(4 (I a (2 k - m) + (-2 j + u) w)))], {i, 0, n}] + E^((-(1/2)) I m Pi + I (-2 k + m) q - (I Pi u)/2 + t (-2 j + u) - (I (-2 k + m) p + s (-2 j + u))^2/(4 (I a (-2 k + m) + (-2 j + u) w))) (I a (-2 k + m) + (-2 j + u) w)^(-1 - n) Sum[2^(i - n) ((-I) (-2 k + m) p - s (-2 j + u))^(-i + n) (I (-2 k + m) p + s (-2 j + u) + 2 (I a (-2 k + m) + (-2 j + u) w) z)^(1 + i) (-((I (-2 k + m) p + s (-2 j + u) + 2 (I a (-2 k + m) + (-2 j + u) w) z)^2/(I a (-2 k + m) + (-2 j + u) w)))^((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, -((I (-2 k + m) p + s (-2 j + u) + 2 (I a (-2 k + m) + (-2 j + u) w) z)^2/(4 (I a (-2 k + m) + (-2 j + u) w)))], {i, 0, n}]), {j, 0, Floor[(1/2) (-1 + u)]}], {k, 0, Floor[(1/2) (-1 + m)]}] - I^u 2^(-1 - m - u - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[(-1)^k Binomial[u, k] Sum[Binomial[v, h] (E^(t (2 k - u) + (I Pi u)/2 + g (2 h - v) - (s (2 k - u) + f (2 h - v))^2/(4 (c (2 h - v) + (2 k - u) w))) (c (2 h - v) + (2 k - u) w)^(-1 - n) Sum[2^(i - n) ((-s) (2 k - u) - f (2 h - v))^(-i + n) (s (2 k - u) + f (2 h - v) + 2 (c (2 h - v) + (2 k - u) w) z)^ (1 + i) (-((s (2 k - u) + f (2 h - v) + 2 (c (2 h - v) + (2 k - u) w) z)^2/(c (2 h - v) + (2 k - u) w)))^ ((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, -((s (2 k - u) + f (2 h - v) + 2 (c (2 h - v) + (2 k - u) w) z)^ 2/(4 (c (2 h - v) + (2 k - u) w)))], {i, 0, n}] + E^(t (2 k - u) + (I Pi u)/2 + g (-2 h + v) - (s (2 k - u) + f (-2 h + v))^2/(4 (c (-2 h + v) + (2 k - u) w))) (c (-2 h + v) + (2 k - u) w)^(-1 - n) Sum[2^(i - n) ((-s) (2 k - u) - f (-2 h + v))^(-i + n) (s (2 k - u) + f (-2 h + v) + 2 (c (-2 h + v) + (2 k - u) w) z)^ (1 + i) (-((s (2 k - u) + f (-2 h + v) + 2 (c (-2 h + v) + (2 k - u) w) z)^2/(c (-2 h + v) + (2 k - u) w)))^ ((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, -((s (2 k - u) + f (-2 h + v) + 2 (c (-2 h + v) + (2 k - u) w) z)^2/(4 (c (-2 h + v) + (2 k - u) w)))], {i, 0, n}] + E^((-(1/2)) I Pi u + t (-2 k + u) + g (2 h - v) - (s (-2 k + u) + f (2 h - v))^2/(4 (c (2 h - v) + (-2 k + u) w))) (c (2 h - v) + (-2 k + u) w)^(-1 - n) Sum[2^(i - n) ((-s) (-2 k + u) - f (2 h - v))^(-i + n) (s (-2 k + u) + f (2 h - v) + 2 (c (2 h - v) + (-2 k + u) w) z)^ (1 + i) (-((s (-2 k + u) + f (2 h - v) + 2 (c (2 h - v) + (-2 k + u) w) z)^2/(c (2 h - v) + (-2 k + u) w)))^ ((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, -((s (-2 k + u) + f (2 h - v) + 2 (c (2 h - v) + (-2 k + u) w) z)^2/(4 (c (2 h - v) + (-2 k + u) w)))], {i, 0, n}] + E^((-(1/2)) I Pi u + t (-2 k + u) + g (-2 h + v) - (s (-2 k + u) + f (-2 h + v))^2/(4 (c (-2 h + v) + (-2 k + u) w))) (c (-2 h + v) + (-2 k + u) w)^(-1 - n) Sum[2^(i - n) ((-s) (-2 k + u) - f (-2 h + v))^(-i + n) (s (-2 k + u) + f (-2 h + v) + 2 (c (-2 h + v) + (-2 k + u) w) z)^ (1 + i) (-((s (-2 k + u) + f (-2 h + v) + 2 (c (-2 h + v) + (-2 k + u) w) z)^2/(c (-2 h + v) + (-2 k + u) w)))^ ((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, -((s (-2 k + u) + f (-2 h + v) + 2 (c (-2 h + v) + (-2 k + u) w) z)^2/(4 (c (-2 h + v) + (-2 k + u) w)))], {i, 0, n}]), {h, 0, Floor[(1/2) (-1 + v)]}], {k, 0, Floor[(1/2) (-1 + u)]}] - I^u 2^(-1 - m - u - v) Sum[Binomial[v, h] Sum[(-1)^k Binomial[m, k] Sum[(-1)^j Binomial[u, j] (E^((I m Pi)/2 + I (2 k - m) q + t (2 j - u) + (I Pi u)/2 + g (2 h - v) - (I (2 k - m) p + s (2 j - u) + f (2 h - v))^2/(4 (I a (2 k - m) + c (2 h - v) + (2 j - u) w))) (I a (2 k - m) + c (2 h - v) + (2 j - u) w)^(-1 - n) Sum[2^(i - n) ((-I) (2 k - m) p - s (2 j - u) - f (2 h - v))^( -i + n) (I (2 k - m) p + s (2 j - u) + f (2 h - v) + 2 (I a (2 k - m) + c (2 h - v) + (2 j - u) w) z)^(1 + i) (-((I (2 k - m) p + s (2 j - u) + f (2 h - v) + 2 (I a (2 k - m) + c (2 h - v) + (2 j - u) w) z)^2/ (I a (2 k - m) + c (2 h - v) + (2 j - u) w)))^((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, -((I (2 k - m) p + s (2 j - u) + f (2 h - v) + 2 (I a (2 k - m) + c (2 h - v) + (2 j - u) w) z)^2/ (4 (I a (2 k - m) + c (2 h - v) + (2 j - u) w)))], {i, 0, n}] + E^((-(1/2)) I m Pi + I (-2 k + m) q + t (2 j - u) + (I Pi u)/2 + g (2 h - v) - (I (-2 k + m) p + s (2 j - u) + f (2 h - v))^2/(4 (I a (-2 k + m) + c (2 h - v) + (2 j - u) w))) (I a (-2 k + m) + c (2 h - v) + (2 j - u) w)^ (-1 - n) Sum[2^(i - n) ((-I) (-2 k + m) p - s (2 j - u) - f (2 h - v))^(-i + n) (I (-2 k + m) p + s (2 j - u) + f (2 h - v) + 2 (I a (-2 k + m) + c (2 h - v) + (2 j - u) w) z)^(1 + i) (-((I (-2 k + m) p + s (2 j - u) + f (2 h - v) + 2 (I a (-2 k + m) + c (2 h - v) + (2 j - u) w) z)^2/ (I a (-2 k + m) + c (2 h - v) + (2 j - u) w)))^((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, -((I (-2 k + m) p + s (2 j - u) + f (2 h - v) + 2 (I a (-2 k + m) + c (2 h - v) + (2 j - u) w) z)^2/ (4 (I a (-2 k + m) + c (2 h - v) + (2 j - u) w)))], {i, 0, n}] + E^((I m Pi)/2 + I (2 k - m) q + t (2 j - u) + (I Pi u)/2 + g (-2 h + v) - (I (2 k - m) p + s (2 j - u) + f (-2 h + v))^2/(4 (I a (2 k - m) + c (-2 h + v) + (2 j - u) w))) (I a (2 k - m) + c (-2 h + v) + (2 j - u) w)^ (-1 - n) Sum[2^(i - n) ((-I) (2 k - m) p - s (2 j - u) - f (-2 h + v))^(-i + n) (I (2 k - m) p + s (2 j - u) + f (-2 h + v) + 2 (I a (2 k - m) + c (-2 h + v) + (2 j - u) w) z)^(1 + i) (-((I (2 k - m) p + s (2 j - u) + f (-2 h + v) + 2 (I a (2 k - m) + c (-2 h + v) + (2 j - u) w) z)^2/ (I a (2 k - m) + c (-2 h + v) + (2 j - u) w)))^((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, -((I (2 k - m) p + s (2 j - u) + f (-2 h + v) + 2 (I a (2 k - m) + c (-2 h + v) + (2 j - u) w) z)^2/ (4 (I a (2 k - m) + c (-2 h + v) + (2 j - u) w)))], {i, 0, n}] + E^((-(1/2)) I m Pi + I (-2 k + m) q + t (2 j - u) + (I Pi u)/2 + g (-2 h + v) - (I (-2 k + m) p + s (2 j - u) + f (-2 h + v))^2/(4 (I a (-2 k + m) + c (-2 h + v) + (2 j - u) w))) (I a (-2 k + m) + c (-2 h + v) + (2 j - u) w)^ (-1 - n) Sum[2^(i - n) ((-I) (-2 k + m) p - s (2 j - u) - f (-2 h + v))^(-i + n) (I (-2 k + m) p + s (2 j - u) + f (-2 h + v) + 2 (I a (-2 k + m) + c (-2 h + v) + (2 j - u) w) z)^(1 + i) (-((I (-2 k + m) p + s (2 j - u) + f (-2 h + v) + 2 (I a (-2 k + m) + c (-2 h + v) + (2 j - u) w) z)^2/(I a (-2 k + m) + c (-2 h + v) + (2 j - u) w)))^((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, -((I (-2 k + m) p + s (2 j - u) + f (-2 h + v) + 2 (I a (-2 k + m) + c (-2 h + v) + (2 j - u) w) z)^2/(4 (I a (-2 k + m) + c (-2 h + v) + (2 j - u) w)))], {i, 0, n}] + E^((I m Pi)/2 + I (2 k - m) q - (I Pi u)/2 + t (-2 j + u) + g (2 h - v) - (I (2 k - m) p + s (-2 j + u) + f (2 h - v))^2/(4 (I a (2 k - m) + c (2 h - v) + (-2 j + u) w))) (I a (2 k - m) + c (2 h - v) + (-2 j + u) w)^(-1 - n) Sum[2^(i - n) ((-I) (2 k - m) p - s (-2 j + u) - f (2 h - v))^( -i + n) (I (2 k - m) p + s (-2 j + u) + f (2 h - v) + 2 (I a (2 k - m) + c (2 h - v) + (-2 j + u) w) z)^(1 + i) (-((I (2 k - m) p + s (-2 j + u) + f (2 h - v) + 2 (I a (2 k - m) + c (2 h - v) + (-2 j + u) w) z)^2/ (I a (2 k - m) + c (2 h - v) + (-2 j + u) w)))^((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, -((I (2 k - m) p + s (-2 j + u) + f (2 h - v) + 2 (I a (2 k - m) + c (2 h - v) + (-2 j + u) w) z)^2/ (4 (I a (2 k - m) + c (2 h - v) + (-2 j + u) w)))], {i, 0, n}] + E^((-(1/2)) I m Pi + I (-2 k + m) q - (I Pi u)/2 + t (-2 j + u) + g (2 h - v) - (I (-2 k + m) p + s (-2 j + u) + f (2 h - v))^2/(4 (I a (-2 k + m) + c (2 h - v) + (-2 j + u) w))) (I a (-2 k + m) + c (2 h - v) + (-2 j + u) w)^(-1 - n) Sum[2^(i - n) ((-I) (-2 k + m) p - s (-2 j + u) - f (2 h - v))^(-i + n) (I (-2 k + m) p + s (-2 j + u) + f (2 h - v) + 2 (I a (-2 k + m) + c (2 h - v) + (-2 j + u) w) z)^(1 + i) (-((I (-2 k + m) p + s (-2 j + u) + f (2 h - v) + 2 (I a (-2 k + m) + c (2 h - v) + (-2 j + u) w) z)^2/ (I a (-2 k + m) + c (2 h - v) + (-2 j + u) w)))^((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, -((I (-2 k + m) p + s (-2 j + u) + f (2 h - v) + 2 (I a (-2 k + m) + c (2 h - v) + (-2 j + u) w) z)^2/ (4 (I a (-2 k + m) + c (2 h - v) + (-2 j + u) w)))], {i, 0, n}] + E^((I m Pi)/2 + I (2 k - m) q - (I Pi u)/2 + t (-2 j + u) + g (-2 h + v) - (I (2 k - m) p + s (-2 j + u) + f (-2 h + v))^2/(4 (I a (2 k - m) + c (-2 h + v) + (-2 j + u) w))) (I a (2 k - m) + c (-2 h + v) + (-2 j + u) w)^(-1 - n) Sum[2^(i - n) ((-I) (2 k - m) p - s (-2 j + u) - f (-2 h + v))^(-i + n) (I (2 k - m) p + s (-2 j + u) + f (-2 h + v) + 2 (I a (2 k - m) + c (-2 h + v) + (-2 j + u) w) z)^(1 + i) (-((I (2 k - m) p + s (-2 j + u) + f (-2 h + v) + 2 (I a (2 k - m) + c (-2 h + v) + (-2 j + u) w) z)^2/ (I a (2 k - m) + c (-2 h + v) + (-2 j + u) w)))^((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, -((I (2 k - m) p + s (-2 j + u) + f (-2 h + v) + 2 (I a (2 k - m) + c (-2 h + v) + (-2 j + u) w) z)^2/ (4 (I a (2 k - m) + c (-2 h + v) + (-2 j + u) w)))], {i, 0, n}] + E^((-(1/2)) I m Pi + I (-2 k + m) q - (I Pi u)/2 + t (-2 j + u) + g (-2 h + v) - (I (-2 k + m) p + s (-2 j + u) + f (-2 h + v))^2/(4 (I a (-2 k + m) + c (-2 h + v) + (-2 j + u) w))) (I a (-2 k + m) + c (-2 h + v) + (-2 j + u) w)^(-1 - n) Sum[2^(i - n) ((-I) (-2 k + m) p - s (-2 j + u) - f (-2 h + v))^(-i + n) (I (-2 k + m) p + s (-2 j + u) + f (-2 h + v) + 2 (I a (-2 k + m) + c (-2 h + v) + (-2 j + u) w) z)^(1 + i) (-((I (-2 k + m) p + s (-2 j + u) + f (-2 h + v) + 2 (I a (-2 k + m) + c (-2 h + v) + (-2 j + u) w) z)^2/ (I a (-2 k + m) + c (-2 h + v) + (-2 j + u) w)))^((1/2) (-1 - i)) Binomial[n, i] Gamma[(1 + i)/2, -((I (-2 k + m) p + s (-2 j + u) + f (-2 h + v) + 2 (I a (-2 k + m) + c (-2 h + v) + (-2 j + u) w) z)^2/ (4 (I a (-2 k + m) + c (-2 h + v) + (-2 j + u) w)))], {i, 0, n}]), {j, 0, Floor[(1/2) (-1 + u)]}], {k, 0, Floor[(1/2) (-1 + m)]}], {h, 0, Floor[(1/2) (-1 + v)]}] /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m > 0 && Element[u, Integers] && u > 0 && Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 z n sin m ( a z 2 + p z + q ) sinh u ( w z 2 + s z + t ) cosh v ( c z 2 + f z + g ) z ( u 2 - m - u - v 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]]]]] ( u u 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox[FractionBox["u", "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 - u mod 2 \$CellContext`u 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) n + 1 - u 2 - m - u - v - 1 ( u u 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox[FractionBox["u", "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 - u mod 2 \$CellContext`u 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - ( 2 k - m ) p 2 4 a + ( 2 k - m ) q + m π 2 ( i = 0 n 2 i - n ( - ( 2 k - m ) p ) n - i ( ( 2 k - m ) p + 2 a ( 2 k - m ) z ) i + 1 ( ( ( 2 k - m ) p + 2 a ( 2 k - m ) z ) 2 a ( 2 k - m ) ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , ( ( 2 k - m ) p + 2 a ( 2 k - m ) z ) 2 4 a ( 2 k - m ) ) ) ( a ( 2 k - m ) ) - n - 1 + - ( m - 2 k ) p 2 4 a + ( m - 2 k ) q - m π 2 ( a ( m - 2 k ) ) - n - 1 i = 0 n 2 i - n ( - ( m - 2 k ) p ) n - i ( ( m - 2 k ) p + 2 a ( m - 2 k ) z ) i + 1 ( ( ( m - 2 k ) p + 2 a ( m - 2 k ) z ) 2 a ( m - 2 k ) ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , ( ( m - 2 k ) p + 2 a ( m - 2 k ) z ) 2 4 a ( m - 2 k ) ) ) - u 2 - m - u - 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]]]]] ( u u 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox[FractionBox["u", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - u mod 2 \$CellContext`u 2 ) h = 0 v - 1 2 ( v h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( g ( 2 h - v ) - f 2 ( 2 h - v ) 4 c ( i = 0 n 2 i - n ( - f ( 2 h - v ) ) n - i ( f ( 2 h - v ) + 2 c z ( 2 h - v ) ) i + 1 ( - ( f ( 2 h - v ) + 2 c z ( 2 h - v ) ) 2 c ( 2 h - v ) ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , - ( f ( 2 h - v ) + 2 c z ( 2 h - v ) ) 2 4 c ( 2 h - v ) ) ) ( c ( 2 h - v ) ) - n - 1 + g ( v - 2 h ) - f 2 ( v - 2 h ) 4 c ( c ( v - 2 h ) ) - n - 1 i = 0 n 2 i - n ( - f ( v - 2 h ) ) n - i ( f ( v - 2 h ) + 2 c z ( v - 2 h ) ) i + 1 ( - ( f ( v - 2 h ) + 2 c z ( v - 2 h ) ) 2 c ( v - 2 h ) ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , - ( f ( v - 2 h ) + 2 c z ( v - 2 h ) ) 2 4 c ( v - 2 h ) ) ) - u 2 - m - u - 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 ) k = 0 u - 1 2 ( - 1 ) k ( u k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - ( 2 k - u ) s 2 4 w + t ( 2 k - u ) + π u 2 ( i = 0 n 2 i - n ( - s ( 2 k - u ) ) n - i ( s ( 2 k - u ) + 2 w z ( 2 k - u ) ) i + 1 ( - ( s ( 2 k - u ) + 2 w z ( 2 k - u ) ) 2 ( 2 k - u ) w ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , - ( s ( 2 k - u ) + 2 w z ( 2 k - u ) ) 2 4 ( 2 k - u ) w ) ) ( ( 2 k - u ) w ) - n - 1 + - ( u - 2 k ) s 2 4 w - π u 2 + t ( u - 2 k ) ( ( u - 2 k ) w ) - n - 1 i = 0 n 2 i - n ( - s ( u - 2 k ) ) n - i ( s ( u - 2 k ) + 2 w z ( u - 2 k ) ) i + 1 ( - ( s ( u - 2 k ) + 2 w z ( u - 2 k ) ) 2 ( u - 2 k ) w ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , - ( s ( u - 2 k ) + 2 w z ( u - 2 k ) ) 2 4 ( u - 2 k ) w ) ) - u 2 - m - u - v - 1 ( u u 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox[FractionBox["u", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - u mod 2 \$CellContext`u 2 ) k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] h = 0 v - 1 2 ( v h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - ( ( 2 k - m ) p + f ( 2 h - v ) ) 2 4 ( a ( 2 k - m ) + c ( 2 h - v ) ) + ( 2 k - m ) q + g ( 2 h - v ) + m π 2 ( i = 0 n 2 i - n ( - ( 2 k - m ) p - f ( 2 h - v ) ) n - i ( ( 2 k - m ) p + f ( 2 h - v ) + 2 ( a ( 2 k - m ) + c ( 2 h - v ) ) z ) i + 1 ( - ( ( 2 k - m ) p + f ( 2 h - v ) + 2 ( a ( 2 k - m ) + c ( 2 h - v ) ) z ) 2 / ( a ( 2 k - m ) + c ( 2 h - v ) ) ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , - ( ( 2 k - m ) p + f ( 2 h - v ) + 2 ( a ( 2 k - m ) + c ( 2 h - v ) ) z ) 2 4 ( a ( 2 k - m ) + c ( 2 h - v ) ) ) ) ( a ( 2 k - m ) + c ( 2 h - v ) ) - n - 1 + - ( ( m - 2 k ) p + f ( 2 h - v ) ) 2 4 ( a ( m - 2 k ) + c ( 2 h - v ) ) + ( m - 2 k ) q + g ( 2 h - v ) - m π 2 ( a ( m - 2 k ) + c ( 2 h - v ) ) - n - 1 i = 0 n 2 i - n ( - ( m - 2 k ) p - f ( 2 h - v ) ) n - i ( ( m - 2 k ) p + f ( 2 h - v ) + 2 ( a ( m - 2 k ) + c ( 2 h - v ) ) z ) i + 1 ( - ( ( m - 2 k ) p + f ( 2 h - v ) + 2 ( a ( m - 2 k ) + c ( 2 h - v ) ) z ) 2 a ( m - 2 k ) + c ( 2 h - v ) ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , - ( ( m - 2 k ) p + f ( 2 h - v ) + 2 ( a ( m - 2 k ) + c ( 2 h - v ) ) z ) 2 4 ( a ( m - 2 k ) + c ( 2 h - v ) ) ) + - ( ( 2 k - m ) p + f ( v - 2 h ) ) 2 4 ( a ( 2 k - m ) + c ( v - 2 h ) ) + ( 2 k - m ) q + g ( v - 2 h ) + m π 2 ( a ( 2 k - m ) + c ( v - 2 h ) ) - n - 1 i = 0 n 2 i - n ( - ( 2 k - m ) p - f ( v - 2 h ) ) n - i ( ( 2 k - m ) p + f ( v - 2 h ) + 2 ( a ( 2 k - m ) + c ( v - 2 h ) ) z ) i + 1 ( - ( ( 2 k - m ) p + f ( v - 2 h ) + 2 ( a ( 2 k - m ) + c ( v - 2 h ) ) z ) 2 a ( 2 k - m ) + c ( v - 2 h ) ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , - ( ( 2 k - m ) p + f ( v - 2 h ) + 2 ( a ( 2 k - m ) + c ( v - 2 h ) ) z ) 2 4 ( a ( 2 k - m ) + c ( v - 2 h ) ) ) + - ( ( m - 2 k ) p + f ( v - 2 h ) ) 2 4 ( a ( m - 2 k ) + c ( v - 2 h ) ) + ( m - 2 k ) q + g ( v - 2 h ) - m π 2 ( a ( m - 2 k ) + c ( v - 2 h ) ) - n - 1 i = 0 n 2 i - n ( - ( m - 2 k ) p - f ( v - 2 h ) ) n - i ( ( m - 2 k ) p + f ( v - 2 h ) + 2 ( a ( m - 2 k ) + c ( v - 2 h ) ) z ) i + 1 ( - ( ( m - 2 k ) p + f ( v - 2 h ) + 2 ( a ( m - 2 k ) + c ( v - 2 h ) ) z ) 2 a ( m - 2 k ) + c ( v - 2 h ) ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , - ( ( m - 2 k ) p + f ( v - 2 h ) + 2 ( a ( m - 2 k ) + c ( v - 2 h ) ) z ) 2 4 ( a ( m - 2 k ) + c ( v - 2 h ) ) ) ) - u 2 - m - u - 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 ) k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] j = 0 u - 1 2 ( - 1 ) j ( u j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - ( ( 2 k - m ) p + s ( 2 j - u ) ) 2 4 ( a ( 2 k - m ) + ( 2 j - u ) w ) + ( 2 k - m ) q + t ( 2 j - u ) + π u 2 + m π 2 ( i = 0 n 2 i - n ( - ( 2 k - m ) p - s ( 2 j - u ) ) n - i ( ( 2 k - m ) p + s ( 2 j - u ) + 2 ( a ( 2 k - m ) + ( 2 j - u ) w ) z ) i + 1 ( - ( ( 2 k - m ) p + s ( 2 j - u ) + 2 ( a ( 2 k - m ) + ( 2 j - u ) w ) z ) 2 / ( a ( 2 k - m ) + ( 2 j - u ) w ) ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , - ( ( 2 k - m ) p + s ( 2 j - u ) + 2 ( a ( 2 k - m ) + ( 2 j - u ) w ) z ) 2 4 ( a ( 2 k - m ) + ( 2 j - u ) w ) ) ) ( a ( 2 k - m ) + ( 2 j - u ) w ) - n - 1 + - ( ( m - 2 k ) p + s ( 2 j - u ) ) 2 4 ( a ( m - 2 k ) + ( 2 j - u ) w ) + ( m - 2 k ) q + t ( 2 j - u ) + π u 2 - m π 2 ( a ( m - 2 k ) + ( 2 j - u ) w ) - n - 1 i = 0 n 2 i - n ( - ( m - 2 k ) p - s ( 2 j - u ) ) n - i ( ( m - 2 k ) p + s ( 2 j - u ) + 2 ( a ( m - 2 k ) + ( 2 j - u ) w ) z ) i + 1 ( - ( ( m - 2 k ) p + s ( 2 j - u ) + 2 ( a ( m - 2 k ) + ( 2 j - u ) w ) z ) 2 a ( m - 2 k ) + ( 2 j - u ) w ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , - ( ( m - 2 k ) p + s ( 2 j - u ) + 2 ( a ( m - 2 k ) + ( 2 j - u ) w ) z ) 2 4 ( a ( m - 2 k ) + ( 2 j - u ) w ) ) + - ( ( 2 k - m ) p + s ( u - 2 j ) ) 2 4 ( a ( 2 k - m ) + ( u - 2 j ) w ) + ( 2 k - m ) q + t ( u - 2 j ) - π u 2 + m π 2 ( a ( 2 k - m ) + ( u - 2 j ) w ) - n - 1 i = 0 n 2 i - n ( - ( 2 k - m ) p - s ( u - 2 j ) ) n - i ( ( 2 k - m ) p + s ( u - 2 j ) + 2 ( a ( 2 k - m ) + ( u - 2 j ) w ) z ) i + 1 ( - ( ( 2 k - m ) p + s ( u - 2 j ) + 2 ( a ( 2 k - m ) + ( u - 2 j ) w ) z ) 2 a ( 2 k - m ) + ( u - 2 j ) w ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , - ( ( 2 k - m ) p + s ( u - 2 j ) + 2 ( a ( 2 k - m ) + ( u - 2 j ) w ) z ) 2 4 ( a ( 2 k - m ) + ( u - 2 j ) w ) ) + - ( ( m - 2 k ) p + s ( u - 2 j ) ) 2 4 ( a ( m - 2 k ) + ( u - 2 j ) w ) + ( m - 2 k ) q + t ( u - 2 j ) - π u 2 - m π 2 ( a ( m - 2 k ) + ( u - 2 j ) w ) - n - 1 i = 0 n 2 i - n ( - ( m - 2 k ) p - s ( u - 2 j ) ) n - i ( ( m - 2 k ) p + s ( u - 2 j ) + 2 ( a ( m - 2 k ) + ( u - 2 j ) w ) z ) i + 1 ( - ( ( m - 2 k ) p + s ( u - 2 j ) + 2 ( a ( m - 2 k ) + ( u - 2 j ) w ) z ) 2 a ( m - 2 k ) + ( u - 2 j ) w ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , - ( ( m - 2 k ) p + s ( u - 2 j ) + 2 ( a ( m - 2 k ) + ( u - 2 j ) w ) z ) 2 4 ( a ( m - 2 k ) + ( u - 2 j ) w ) ) ) - u 2 - m - u - 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 ) k = 0 u - 1 2 ( - 1 ) k ( u k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] h = 0 v - 1 2 ( v h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - ( s ( 2 k - u ) + f ( 2 h - v ) ) 2 4 ( c ( 2 h - v ) + ( 2 k - u ) w ) + t ( 2 k - u ) + π u 2 + g ( 2 h - v ) ( i = 0 n 2 i - n ( - s ( 2 k - u ) - f ( 2 h - v ) ) n - i ( s ( 2 k - u ) + f ( 2 h - v ) + 2 ( c ( 2 h - v ) + ( 2 k - u ) w ) z ) i + 1 ( - ( s ( 2 k - u ) + f ( 2 h - v ) + 2 ( c ( 2 h - v ) + ( 2 k - u ) w ) z ) 2 c ( 2 h - v ) + ( 2 k - u ) w ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , - ( s ( 2 k - u ) + f ( 2 h - v ) + 2 ( c ( 2 h - v ) + ( 2 k - u ) w ) z ) 2 4 ( c ( 2 h - v ) + ( 2 k - u ) w ) ) ) ( c ( 2 h - v ) + ( 2 k - u ) w ) - n - 1 + - ( s ( 2 k - u ) + f ( v - 2 h ) ) 2 4 ( c ( v - 2 h ) + ( 2 k - u ) w ) + t ( 2 k - u ) + π u 2 + g ( v - 2 h ) ( c ( v - 2 h ) + ( 2 k - u ) w ) - n - 1 i = 0 n 2 i - n ( - s ( 2 k - u ) - f ( v - 2 h ) ) n - i ( s ( 2 k - u ) + f ( v - 2 h ) + 2 ( c ( v - 2 h ) + ( 2 k - u ) w ) z ) i + 1 ( - ( s ( 2 k - u ) + f ( v - 2 h ) + 2 ( c ( v - 2 h ) + ( 2 k - u ) w ) z ) 2 c ( v - 2 h ) + ( 2 k - u ) w ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , - ( s ( 2 k - u ) + f ( v - 2 h ) + 2 ( c ( v - 2 h ) + ( 2 k - u ) w ) z ) 2 4 ( c ( v - 2 h ) + ( 2 k - u ) w ) ) + - ( s ( u - 2 k ) + f ( 2 h - v ) ) 2 4 ( c ( 2 h - v ) + ( u - 2 k ) w ) - π u 2 + t ( u - 2 k ) + g ( 2 h - v ) ( c ( 2 h - v ) + ( u - 2 k ) w ) - n - 1 i = 0 n 2 i - n ( - s ( u - 2 k ) - f ( 2 h - v ) ) n - i ( s ( u - 2 k ) + f ( 2 h - v ) + 2 ( c ( 2 h - v ) + ( u - 2 k ) w ) z ) i + 1 ( - ( s ( u - 2 k ) + f ( 2 h - v ) + 2 ( c ( 2 h - v ) + ( u - 2 k ) w ) z ) 2 c ( 2 h - v ) + ( u - 2 k ) w ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , - ( s ( u - 2 k ) + f ( 2 h - v ) + 2 ( c ( 2 h - v ) + ( u - 2 k ) w ) z ) 2 4 ( c ( 2 h - v ) + ( u - 2 k ) w ) ) + - ( s ( u - 2 k ) + f ( v - 2 h ) ) 2 4 ( c ( v - 2 h ) + ( u - 2 k ) w ) - π u 2 + t ( u - 2 k ) + g ( v - 2 h ) ( c ( v - 2 h ) + ( u - 2 k ) w ) - n - 1 i = 0 n 2 i - n ( - s ( u - 2 k ) - f ( v - 2 h ) ) n - i ( s ( u - 2 k ) + f ( v - 2 h ) + 2 ( c ( v - 2 h ) + ( u - 2 k ) w ) z ) i + 1 ( - ( s ( u - 2 k ) + f ( v - 2 h ) + 2 ( c ( v - 2 h ) + ( u - 2 k ) w ) z ) 2 c ( v - 2 h ) + ( u - 2 k ) w ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , - ( s ( u - 2 k ) + f ( v - 2 h ) + 2 ( c ( v - 2 h ) + ( u - 2 k ) w ) z ) 2 4 ( c ( v - 2 h ) + ( u - 2 k ) w ) ) ) - u 2 - m - u - v - 1 h = 0 v - 1 2 ( v h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] j = 0 u - 1 2 ( - 1 ) j ( u j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - ( ( 2 k - m ) p + s ( 2 j - u ) + f ( 2 h - v ) ) 2 4 ( a ( 2 k - m ) + c ( 2 h - v ) + ( 2 j - u ) w ) + ( 2 k - m ) q + t ( 2 j - u ) + π u 2 + g ( 2 h - v ) + m π 2 ( i = 0 n 2 i - n ( - ( 2 k - m ) p - s ( 2 j - u ) - f ( 2 h - v ) ) n - i ( ( 2 k - m ) p + s ( 2 j - u ) + f ( 2 h - v ) + 2 ( a ( 2 k - m ) + c ( 2 h - v ) + ( 2 j - u ) w ) z ) i + 1 ( - ( ( 2 k - m ) p + s ( 2 j - u ) + f ( 2 h - v ) + 2 ( a ( 2 k - m ) + c ( 2 h - v ) + ( 2 j - u ) w ) z ) 2 / ( a ( 2 k - m ) + c ( 2 h - v ) + ( 2 j - u ) w ) ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , - ( ( 2 k - m ) p + s ( 2 j - u ) + f ( 2 h - v ) + 2 ( a ( 2 k - m ) + c ( 2 h - v ) + ( 2 j - u ) w ) z ) 2 / ( 4 ( a ( 2 k - m ) + c ( 2 h - v ) + ( 2 j - u ) w ) ) ) ) ( a ( 2 k - m ) + c ( 2 h - v ) + ( 2 j - u ) w ) - n - 1 + - ( ( m - 2 k ) p + s ( 2 j - u ) + f ( 2 h - v ) ) 2 4 ( a ( m - 2 k ) + c ( 2 h - v ) + ( 2 j - u ) w ) + ( m - 2 k ) q + t ( 2 j - u ) + π u 2 + g ( 2 h - v ) - m π 2 ( a ( m - 2 k ) + c ( 2 h - v ) + ( 2 j - u ) w ) - n - 1 i = 0 n 2 i - n ( - ( m - 2 k ) p - s ( 2 j - u ) - f ( 2 h - v ) ) n - i ( ( m - 2 k ) p + s ( 2 j - u ) + f ( 2 h - v ) + 2 ( a ( m - 2 k ) + c ( 2 h - v ) + ( 2 j - u ) w ) z ) i + 1 ( - ( ( m - 2 k ) p + s ( 2 j - u ) + f ( 2 h - v ) + 2 ( a ( m - 2 k ) + c ( 2 h - v ) + ( 2 j - u ) w ) z ) 2 / ( a ( m - 2 k ) + c ( 2 h - v ) + ( 2 j - u ) w ) ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , - ( ( m - 2 k ) p + s ( 2 j - u ) + f ( 2 h - v ) + 2 ( a ( m - 2 k ) + c ( 2 h - v ) + ( 2 j - u ) w ) z ) 2 / ( 4 ( a ( m - 2 k ) + c ( 2 h - v ) + ( 2 j - u ) w ) ) ) + - ( ( 2 k - m ) p + s ( 2 j - u ) + f ( v - 2 h ) ) 2 4 ( a ( 2 k - m ) + c ( v - 2 h ) + ( 2 j - u ) w ) + ( 2 k - m ) q + t ( 2 j - u ) + π u 2 + g ( v - 2 h ) + m π 2 ( a ( 2 k - m ) + c ( v - 2 h ) + ( 2 j - u ) w ) - n - 1 i = 0 n 2 i - n ( - ( 2 k - m ) p - s ( 2 j - u ) - f ( v - 2 h ) ) n - i ( ( 2 k - m ) p + s ( 2 j - u ) + f ( v - 2 h ) + 2 ( a ( 2 k - m ) + c ( v - 2 h ) + ( 2 j - u ) w ) z ) i + 1 ( - ( ( 2 k - m ) p + s ( 2 j - u ) + f ( v - 2 h ) + 2 ( a ( 2 k - m ) + c ( v - 2 h ) + ( 2 j - u ) w ) z ) 2 / ( a ( 2 k - m ) + c ( v - 2 h ) + ( 2 j - u ) w ) ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , - ( ( 2 k - m ) p + s ( 2 j - u ) + f ( v - 2 h ) + 2 ( a ( 2 k - m ) + c ( v - 2 h ) + ( 2 j - u ) w ) z ) 2 / ( 4 ( a ( 2 k - m ) + c ( v - 2 h ) + ( 2 j - u ) w ) ) ) + - ( ( m - 2 k ) p + s ( 2 j - u ) + f ( v - 2 h ) ) 2 4 ( a ( m - 2 k ) + c ( v - 2 h ) + ( 2 j - u ) w ) + ( m - 2 k ) q + t ( 2 j - u ) + π u 2 + g ( v - 2 h ) - m π 2 ( a ( m - 2 k ) + c ( v - 2 h ) + ( 2 j - u ) w ) - n - 1 i = 0 n 2 i - n ( - ( m - 2 k ) p - s ( 2 j - u ) - f ( v - 2 h ) ) n - i ( ( m - 2 k ) p + s ( 2 j - u ) + f ( v - 2 h ) + 2 ( a ( m - 2 k ) + c ( v - 2 h ) + ( 2 j - u ) w ) z ) i + 1 ( - ( ( m - 2 k ) p + s ( 2 j - u ) + f ( v - 2 h ) + 2 ( a ( m - 2 k ) + c ( v - 2 h ) + ( 2 j - u ) w ) z ) 2 / ( a ( m - 2 k ) + c ( v - 2 h ) + ( 2 j - u ) w ) ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , - ( ( m - 2 k ) p + s ( 2 j - u ) + f ( v - 2 h ) + 2 ( a ( m - 2 k ) + c ( v - 2 h ) + ( 2 j - u ) w ) z ) 2 / ( 4 ( a ( m - 2 k ) + c ( v - 2 h ) + ( 2 j - u ) w ) ) ) + - ( ( 2 k - m ) p + s ( u - 2 j ) + f ( 2 h - v ) ) 2 4 ( a ( 2 k - m ) + c ( 2 h - v ) + ( u - 2 j ) w ) + ( 2 k - m ) q + t ( u - 2 j ) + g ( 2 h - v ) - π u 2 + m π 2 ( a ( 2 k - m ) + c ( 2 h - v ) + ( u - 2 j ) w ) - n - 1 i = 0 n 2 i - n ( - ( 2 k - m ) p - s ( u - 2 j ) - f ( 2 h - v ) ) n - i ( ( 2 k - m ) p + s ( u - 2 j ) + f ( 2 h - v ) + 2 ( a ( 2 k - m ) + c ( 2 h - v ) + ( u - 2 j ) w ) z ) i + 1 ( - ( ( 2 k - m ) p + s ( u - 2 j ) + f ( 2 h - v ) + 2 ( a ( 2 k - m ) + c ( 2 h - v ) + ( u - 2 j ) w ) z ) 2 / ( a ( 2 k - m ) + c ( 2 h - v ) + ( u - 2 j ) w ) ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , - ( ( 2 k - m ) p + s ( u - 2 j ) + f ( 2 h - v ) + 2 ( a ( 2 k - m ) + c ( 2 h - v ) + ( u - 2 j ) w ) z ) 2 / ( 4 ( a ( 2 k - m ) + c ( 2 h - v ) + ( u - 2 j ) w ) ) ) + - ( ( m - 2 k ) p + s ( u - 2 j ) + f ( 2 h - v ) ) 2 4 ( a ( m - 2 k ) + c ( 2 h - v ) + ( u - 2 j ) w ) + ( m - 2 k ) q + t ( u - 2 j ) + g ( 2 h - v ) - π u 2 - m π 2 ( a ( m - 2 k ) + c ( 2 h - v ) + ( u - 2 j ) w ) - n - 1 i = 0 n 2 i - n ( - ( m - 2 k ) p - s ( u - 2 j ) - f ( 2 h - v ) ) n - i ( ( m - 2 k ) p + s ( u - 2 j ) + f ( 2 h - v ) + 2 ( a ( m - 2 k ) + c ( 2 h - v ) + ( u - 2 j ) w ) z ) i + 1 ( - ( ( m - 2 k ) p + s ( u - 2 j ) + f ( 2 h - v ) + 2 ( a ( m - 2 k ) + c ( 2 h - v ) + ( u - 2 j ) w ) z ) 2 / ( a ( m - 2 k ) + c ( 2 h - v ) + ( u - 2 j ) w ) ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , - ( ( m - 2 k ) p + s ( u - 2 j ) + f ( 2 h - v ) + 2 ( a ( m - 2 k ) + c ( 2 h - v ) + ( u - 2 j ) w ) z ) 2 / ( 4 ( a ( m - 2 k ) + c ( 2 h - v ) + ( u - 2 j ) w ) ) ) + - ( ( 2 k - m ) p + s ( u - 2 j ) + f ( v - 2 h ) ) 2 4 ( a ( 2 k - m ) + c ( v - 2 h ) + ( u - 2 j ) w ) + ( 2 k - m ) q + t ( u - 2 j ) + g ( v - 2 h ) - π u 2 + m π 2 ( a ( 2 k - m ) + c ( v - 2 h ) + ( u - 2 j ) w ) - n - 1 i = 0 n 2 i - n ( - ( 2 k - m ) p - s ( u - 2 j ) - f ( v - 2 h ) ) n - i ( ( 2 k - m ) p + s ( u - 2 j ) + f ( v - 2 h ) + 2 ( a ( 2 k - m ) + c ( v - 2 h ) + ( u - 2 j ) w ) z ) i + 1 ( - ( ( 2 k - m ) p + s ( u - 2 j ) + f ( v - 2 h ) + 2 ( a ( 2 k - m ) + c ( v - 2 h ) + ( u - 2 j ) w ) z ) 2 / ( a ( 2 k - m ) + c ( v - 2 h ) + ( u - 2 j ) w ) ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , - ( ( 2 k - m ) p + s ( u - 2 j ) + f ( v - 2 h ) + 2 ( a ( 2 k - m ) + c ( v - 2 h ) + ( u - 2 j ) w ) z ) 2 / ( 4 ( a ( 2 k - m ) + c ( v - 2 h ) + ( u - 2 j ) w ) ) ) + - ( ( m - 2 k ) p + s ( u - 2 j ) + f ( v - 2 h ) ) 2 4 ( a ( m - 2 k ) + c ( v - 2 h ) + ( u - 2 j ) w ) + ( m - 2 k ) q + t ( u - 2 j ) + g ( v - 2 h ) - π u 2 - m π 2 ( a ( m - 2 k ) + c ( v - 2 h ) + ( u - 2 j ) w ) - n - 1 i = 0 n 2 i - n ( - ( m - 2 k ) p - s ( u - 2 j ) - f ( v - 2 h ) ) n - i ( ( m - 2 k ) p + s ( u - 2 j ) + f ( v - 2 h ) + 2 ( a ( m - 2 k ) + c ( v - 2 h ) + ( u - 2 j ) w ) z ) i + 1 ( - ( ( m - 2 k ) p + s ( u - 2 j ) + f ( v - 2 h ) + 2 ( a ( m - 2 k ) + c ( v - 2 h ) + ( u - 2 j ) w ) z ) 2 / ( a ( m - 2 k ) + c ( v - 2 h ) + ( u - 2 j ) w ) ) 1 2 ( - i - 1 ) ( n i ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["i", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( i + 1 2 , - ( ( m - 2 k ) p + s ( u - 2 j ) + f ( v - 2 h ) + 2 ( a ( m - 2 k ) + c ( v - 2 h ) + ( u - 2 j ) w ) z ) 2 / ( 4 ( a ( m - 2 k ) + c ( v - 2 h ) + ( u - 2 j ) w ) ) ) ) /; n m TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] m > 0 u TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] u > 0 v TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] v > 0 Condition z z n a z 2 p z q m w z 2 s z t u c z 2 f z g v u 2 -1 m -1 u -1 v z n 1 Binomial m m 2 -1 Binomial u u 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`u 2 1 -1 \$CellContext`v 2 n 1 -1 -1 u 2 -1 m -1 u -1 v -1 Binomial u u 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`u 2 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 -1 k Binomial m k -1 2 k -1 m p 2 4 a -1 2 k -1 m q m 2 -1 i 0 n 2 i -1 n -1 2 k -1 m p n -1 i 2 k -1 m p 2 a 2 k -1 m z i 1 2 k -1 m p 2 a 2 k -1 m z 2 a 2 k -1 m -1 1 2 -1 i -1 Binomial n i Gamma i 1 2 -1 2 k -1 m p 2 a 2 k -1 m z 2 4 a 2 k -1 m -1 a 2 k -1 m -1 n -1 -1 m -1 2 k p 2 4 a -1 m -1 2 k q -1 m 2 -1 a m -1 2 k -1 n -1 i 0 n 2 i -1 n -1 m -1 2 k p n -1 i m -1 2 k p 2 a m -1 2 k z i 1 m -1 2 k p 2 a m -1 2 k z 2 a m -1 2 k -1 1 2 -1 i -1 Binomial n i Gamma i 1 2 -1 m -1 2 k p 2 a m -1 2 k z 2 4 a m -1 2 k -1 -1 u 2 -1 m -1 u -1 v -1 Binomial m m 2 -1 Binomial u u 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`u 2 h 0 v -1 2 -1 Binomial v h g 2 h -1 v -1 f 2 2 h -1 v 4 c -1 i 0 n 2 i -1 n -1 f 2 h -1 v n -1 i f 2 h -1 v 2 c z 2 h -1 v i 1 -1 f 2 h -1 v 2 c z 2 h -1 v 2 c 2 h -1 v -1 1 2 -1 i -1 Binomial n i Gamma i 1 2 -1 -1 f 2 h -1 v 2 c z 2 h -1 v 2 4 c 2 h -1 v -1 c 2 h -1 v -1 n -1 g v -1 2 h -1 f 2 v -1 2 h 4 c -1 c v -1 2 h -1 n -1 i 0 n 2 i -1 n -1 f v -1 2 h n -1 i f v -1 2 h 2 c z v -1 2 h i 1 -1 f v -1 2 h 2 c z v -1 2 h 2 c v -1 2 h -1 1 2 -1 i -1 Binomial n i Gamma i 1 2 -1 -1 f v -1 2 h 2 c z v -1 2 h 2 4 c v -1 2 h -1 -1 u 2 -1 m -1 u -1 v -1 Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 k 0 u -1 2 -1 -1 k Binomial u k -1 2 k -1 u s 2 4 w -1 t 2 k -1 u u 2 -1 i 0 n 2 i -1 n -1 s 2 k -1 u n -1 i s 2 k -1 u 2 w z 2 k -1 u i 1 -1 s 2 k -1 u 2 w z 2 k -1 u 2 2 k -1 u w -1 1 2 -1 i -1 Binomial n i Gamma i 1 2 -1 -1 s 2 k -1 u 2 w z 2 k -1 u 2 4 2 k -1 u w -1 2 k -1 u w -1 n -1 -1 u -1 2 k s 2 4 w -1 -1 u 2 -1 t u -1 2 k u -1 2 k w -1 n -1 i 0 n 2 i -1 n -1 s u -1 2 k n -1 i s u -1 2 k 2 w z u -1 2 k i 1 -1 s u -1 2 k 2 w z u -1 2 k 2 u -1 2 k w -1 1 2 -1 i -1 Binomial n i Gamma i 1 2 -1 -1 s u -1 2 k 2 w z u -1 2 k 2 4 u -1 2 k w -1 -1 u 2 -1 m -1 u -1 v -1 Binomial u u 2 -1 1 -1 \$CellContext`u 2 k 0 m -1 2 -1 -1 k Binomial m k h 0 v -1 2 -1 Binomial v h -1 2 k -1 m p f 2 h -1 v 2 4 a 2 k -1 m c 2 h -1 v -1 2 k -1 m q g 2 h -1 v m 2 -1 i 0 n 2 i -1 n -1 2 k -1 m p -1 f 2 h -1 v n -1 i 2 k -1 m p f 2 h -1 v 2 a 2 k -1 m c 2 h -1 v z i 1 -1 2 k -1 m p f 2 h -1 v 2 a 2 k -1 m c 2 h -1 v z 2 a 2 k -1 m c 2 h -1 v -1 1 2 -1 i -1 Binomial n i Gamma i 1 2 -1 -1 2 k -1 m p f 2 h -1 v 2 a 2 k -1 m c 2 h -1 v z 2 4 a 2 k -1 m c 2 h -1 v -1 a 2 k -1 m c 2 h -1 v -1 n -1 -1 m -1 2 k p f 2 h -1 v 2 4 a m -1 2 k c 2 h -1 v -1 m -1 2 k q g 2 h -1 v -1 m 2 -1 a m -1 2 k c 2 h -1 v -1 n -1 i 0 n 2 i -1 n -1 m -1 2 k p -1 f 2 h -1 v n -1 i m -1 2 k p f 2 h -1 v 2 a m -1 2 k c 2 h -1 v z i 1 -1 m -1 2 k p f 2 h -1 v 2 a m -1 2 k c 2 h -1 v z 2 a m -1 2 k c 2 h -1 v -1 1 2 -1 i -1 Binomial n i Gamma i 1 2 -1 -1 m -1 2 k p f 2 h -1 v 2 a m -1 2 k c 2 h -1 v z 2 4 a m -1 2 k c 2 h -1 v -1 -1 2 k -1 m p f v -1 2 h 2 4 a 2 k -1 m c v -1 2 h -1 2 k -1 m q g v -1 2 h m 2 -1 a 2 k -1 m c v -1 2 h -1 n -1 i 0 n 2 i -1 n -1 2 k -1 m p -1 f v -1 2 h n -1 i 2 k -1 m p f v -1 2 h 2 a 2 k -1 m c v -1 2 h z i 1 -1 2 k -1 m p f v -1 2 h 2 a 2 k -1 m c v -1 2 h z 2 a 2 k -1 m c v -1 2 h -1 1 2 -1 i -1 Binomial n i Gamma i 1 2 -1 -1 2 k -1 m p f v -1 2 h 2 a 2 k -1 m c v -1 2 h z 2 4 a 2 k -1 m c v -1 2 h -1 -1 m -1 2 k p f v -1 2 h 2 4 a m -1 2 k c v -1 2 h -1 m -1 2 k q g v -1 2 h -1 m 2 -1 a m -1 2 k c v -1 2 h -1 n -1 i 0 n 2 i -1 n -1 m -1 2 k p -1 f v -1 2 h n -1 i m -1 2 k p f v -1 2 h 2 a m -1 2 k c v -1 2 h z i 1 -1 m -1 2 k p f v -1 2 h 2 a m -1 2 k c v -1 2 h z 2 a m -1 2 k c v -1 2 h -1 1 2 -1 i -1 Binomial n i Gamma i 1 2 -1 -1 m -1 2 k p f v -1 2 h 2 a m -1 2 k c v -1 2 h z 2 4 a m -1 2 k c v -1 2 h -1 -1 u 2 -1 m -1 u -1 v -1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 -1 k Binomial m k j 0 u -1 2 -1 -1 j Binomial u j -1 2 k -1 m p s 2 j -1 u 2 4 a 2 k -1 m 2 j -1 u w -1 2 k -1 m q t 2 j -1 u u 2 -1 m 2 -1 i 0 n 2 i -1 n -1 2 k -1 m p -1 s 2 j -1 u n -1 i 2 k -1 m p s 2 j -1 u 2 a 2 k -1 m 2 j -1 u w z i 1 -1 2 k -1 m p s 2 j -1 u 2 a 2 k -1 m 2 j -1 u w z 2 a 2 k -1 m 2 j -1 u w -1 1 2 -1 i -1 Binomial n i Gamma i 1 2 -1 -1 2 k -1 m p s 2 j -1 u 2 a 2 k -1 m 2 j -1 u w z 2 4 a 2 k -1 m 2 j -1 u w -1 a 2 k -1 m 2 j -1 u w -1 n -1 -1 m -1 2 k p s 2 j -1 u 2 4 a m -1 2 k 2 j -1 u w -1 m -1 2 k q t 2 j -1 u u 2 -1 -1 m 2 -1 a m -1 2 k 2 j -1 u w -1 n -1 i 0 n 2 i -1 n -1 m -1 2 k p -1 s 2 j -1 u n -1 i m -1 2 k p s 2 j -1 u 2 a m -1 2 k 2 j -1 u w z i 1 -1 m -1 2 k p s 2 j -1 u 2 a m -1 2 k 2 j -1 u w z 2 a m -1 2 k 2 j -1 u w -1 1 2 -1 i -1 Binomial n i Gamma i 1 2 -1 -1 m -1 2 k p s 2 j -1 u 2 a m -1 2 k 2 j -1 u w z 2 4 a m -1 2 k 2 j -1 u w -1 -1 2 k -1 m p s u -1 2 j 2 4 a 2 k -1 m u -1 2 j w -1 2 k -1 m q t u -1 2 j -1 u 2 -1 m 2 -1 a 2 k -1 m u -1 2 j w -1 n -1 i 0 n 2 i -1 n -1 2 k -1 m p -1 s u -1 2 j n -1 i 2 k -1 m p s u -1 2 j 2 a 2 k -1 m u -1 2 j w z i 1 -1 2 k -1 m p s u -1 2 j 2 a 2 k -1 m u -1 2 j w z 2 a 2 k -1 m u -1 2 j w -1 1 2 -1 i -1 Binomial n i Gamma i 1 2 -1 -1 2 k -1 m p s u -1 2 j 2 a 2 k -1 m u -1 2 j w z 2 4 a 2 k -1 m u -1 2 j w -1 -1 m -1 2 k p s u -1 2 j 2 4 a m -1 2 k u -1 2 j w -1 m -1 2 k q t u -1 2 j -1 u 2 -1 -1 m 2 -1 a m -1 2 k u -1 2 j w -1 n -1 i 0 n 2 i -1 n -1 m -1 2 k p -1 s u -1 2 j n -1 i m -1 2 k p s u -1 2 j 2 a m -1 2 k u -1 2 j w z i 1 -1 m -1 2 k p s u -1 2 j 2 a m -1 2 k u -1 2 j w z 2 a m -1 2 k u -1 2 j w -1 1 2 -1 i -1 Binomial n i Gamma i 1 2 -1 -1 m -1 2 k p s u -1 2 j 2 a m -1 2 k u -1 2 j w z 2 4 a m -1 2 k u -1 2 j w -1 -1 u 2 -1 m -1 u -1 v -1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 k 0 u -1 2 -1 -1 k Binomial u k h 0 v -1 2 -1 Binomial v h -1 s 2 k -1 u f 2 h -1 v 2 4 c 2 h -1 v 2 k -1 u w -1 t 2 k -1 u u 2 -1 g 2 h -1 v i 0 n 2 i -1 n -1 s 2 k -1 u -1 f 2 h -1 v n -1 i s 2 k -1 u f 2 h -1 v 2 c 2 h -1 v 2 k -1 u w z i 1 -1 s 2 k -1 u f 2 h -1 v 2 c 2 h -1 v 2 k -1 u w z 2 c 2 h -1 v 2 k -1 u w -1 1 2 -1 i -1 Binomial n i Gamma i 1 2 -1 -1 s 2 k -1 u f 2 h -1 v 2 c 2 h -1 v 2 k -1 u w z 2 4 c 2 h -1 v 2 k -1 u w -1 c 2 h -1 v 2 k -1 u w -1 n -1 -1 s 2 k -1 u f v -1 2 h 2 4 c v -1 2 h 2 k -1 u w -1 t 2 k -1 u u 2 -1 g v -1 2 h c v -1 2 h 2 k -1 u w -1 n -1 i 0 n 2 i -1 n -1 s 2 k -1 u -1 f v -1 2 h n -1 i s 2 k -1 u f v -1 2 h 2 c v -1 2 h 2 k -1 u w z i 1 -1 s 2 k -1 u f v -1 2 h 2 c v -1 2 h 2 k -1 u w z 2 c v -1 2 h 2 k -1 u w -1 1 2 -1 i -1 Binomial n i Gamma i 1 2 -1 -1 s 2 k -1 u f v -1 2 h 2 c v -1 2 h 2 k -1 u w z 2 4 c v -1 2 h 2 k -1 u w -1 -1 s u -1 2 k f 2 h -1 v 2 4 c 2 h -1 v u -1 2 k w -1 -1 u 2 -1 t u -1 2 k g 2 h -1 v c 2 h -1 v u -1 2 k w -1 n -1 i 0 n 2 i -1 n -1 s u -1 2 k -1 f 2 h -1 v n -1 i s u -1 2 k f 2 h -1 v 2 c 2 h -1 v u -1 2 k w z i 1 -1 s u -1 2 k f 2 h -1 v 2 c 2 h -1 v u -1 2 k w z 2 c 2 h -1 v u -1 2 k w -1 1 2 -1 i -1 Binomial n i Gamma i 1 2 -1 -1 s u -1 2 k f 2 h -1 v 2 c 2 h -1 v u -1 2 k w z 2 4 c 2 h -1 v u -1 2 k w -1 -1 s u -1 2 k f v -1 2 h 2 4 c v -1 2 h u -1 2 k w -1 -1 u 2 -1 t u -1 2 k g v -1 2 h c v -1 2 h u -1 2 k w -1 n -1 i 0 n 2 i -1 n -1 s u -1 2 k -1 f v -1 2 h n -1 i s u -1 2 k f v -1 2 h 2 c v -1 2 h u -1 2 k w z i 1 -1 s u -1 2 k f v -1 2 h 2 c v -1 2 h u -1 2 k w z 2 c v -1 2 h u -1 2 k w -1 1 2 -1 i -1 Binomial n i Gamma i 1 2 -1 -1 s u -1 2 k f v -1 2 h 2 c v -1 2 h u -1 2 k w z 2 4 c v -1 2 h u -1 2 k w -1 -1 u 2 -1 m -1 u -1 v -1 h 0 v -1 2 -1 Binomial v h k 0 m -1 2 -1 -1 k Binomial m k j 0 u -1 2 -1 -1 j Binomial u j -1 2 k -1 m p s 2 j -1 u f 2 h -1 v 2 4 a 2 k -1 m c 2 h -1 v 2 j -1 u w -1 2 k -1 m q t 2 j -1 u u 2 -1 g 2 h -1 v m 2 -1 i 0 n 2 i -1 n -1 2 k -1 m p -1 s 2 j -1 u -1 f 2 h -1 v n -1 i 2 k -1 m p s 2 j -1 u f 2 h -1 v 2 a 2 k -1 m c 2 h -1 v 2 j -1 u w z i 1 -1 2 k -1 m p s 2 j -1 u f 2 h -1 v 2 a 2 k -1 m c 2 h -1 v 2 j -1 u w z 2 a 2 k -1 m c 2 h -1 v 2 j -1 u w -1 1 2 -1 i -1 Binomial n i Gamma i 1 2 -1 -1 2 k -1 m p s 2 j -1 u f 2 h -1 v 2 a 2 k -1 m c 2 h -1 v 2 j -1 u w z 2 4 a 2 k -1 m c 2 h -1 v 2 j -1 u w -1 a 2 k -1 m c 2 h -1 v 2 j -1 u w -1 n -1 -1 m -1 2 k p s 2 j -1 u f 2 h -1 v 2 4 a m -1 2 k c 2 h -1 v 2 j -1 u w -1 m -1 2 k q t 2 j -1 u u 2 -1 g 2 h -1 v -1 m 2 -1 a m -1 2 k c 2 h -1 v 2 j -1 u w -1 n -1 i 0 n 2 i -1 n -1 m -1 2 k p -1 s 2 j -1 u -1 f 2 h -1 v n -1 i m -1