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/01.20.21.4550.01

 Input Form

 Integrate[z^n Sinh[d z + e]^m Cosh[c Sqrt[z]]^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) + 2 d^(-n - 1) Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[(-1)^k E^(-2 e k - e m) (m - 2 k)^(-1 - n) Binomial[m, k] ((-(-1)^n) E^(2 e m) Gamma[1 + n, d (2 k - m) z] + (-1)^m E^(4 e k) Gamma[1 + n, d (-2 k + m) z]), {k, 0, Floor[(1/2) (-1 + m)]}] + (4 c^(-2 - 2 n) Binomial[m, m/2] (-1 + Mod[m, 2]) Sum[(Binomial[v, s] (Gamma[2 (1 + n), (-c) (2 s - v) Sqrt[z]] + 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 - Sum[(-1)^u Binomial[m, u] Sum[E^((-e) m - 2 e u - (c^2 (-2 s + v)^2)/(4 d (m - 2 u))) ((-d^2) (m - 2 u)^2)^(-1 - 2 n) Binomial[v, s] (E^(2 e m) ((-d) (m - 2 u))^(2 n) Sum[(-1)^(-h + k) 4^k (c (2 s - v))^(-h - k + 2 n) (c (2 s - v) + 2 d (m - 2 u) Sqrt[z])^(h + k) (-((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] (c (2 s - v) (c (2 s - v) + 2 d (m - 2 u) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((c (2 s - v) + 2 d (m - 2 u) Sqrt[z])^2/(4 d (m - 2 u)))] + 2 d (m - 2 u) Sqrt[-((c (2 s - v) + 2 d (m - 2 u) Sqrt[z])^2/ (d (m - 2 u)))] Gamma[(1/2) (2 + h + k), -((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^(4 e u + (c^2 (-2 s + v)^2)/ (2 d (m - 2 u))) (d (m - 2 u))^(2 n) Sum[(-1)^(-h + k) 4^k (c (-2 s + v))^(-h - k + 2 n) (c (-2 s + v) - 2 d (m - 2 u) Sqrt[z])^(h + k) ((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] (c (2 s - v) (c (2 s - v) + 2 d (m - 2 u) Sqrt[z]) Gamma[ (1/2) (1 + h + k), (c (2 s - v) + 2 d (m - 2 u) Sqrt[z])^2/ (4 d (m - 2 u))] - 2 d (m - 2 u) Sqrt[(c (2 s - v) + 2 d (m - 2 u) Sqrt[z])^2/(d (m - 2 u))] Gamma[ (1/2) (2 + h + k), (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 e m) ((-d) (m - 2 u))^(2 n) Sum[(-1)^(-h + k) 4^k (c (-2 s + v))^(-h - k + 2 n) (c (-2 s + v) + 2 d (m - 2 u) Sqrt[z])^(h + k) (-((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] (c (2 s - v) (c (2 s - v) - 2 d (m - 2 u) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((c (-2 s + v) + 2 d (m - 2 u) Sqrt[z])^2/(4 d (m - 2 u)))] + 2 d (m - 2 u) Sqrt[-((c (-2 s + v) + 2 d (m - 2 u) Sqrt[z])^2/ (d (m - 2 u)))] Gamma[(1/2) (2 + h + k), -((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^(4 e u + (c^2 (-2 s + v)^2)/(2 d (m - 2 u))) (d (m - 2 u))^(2 n) Sum[(-1)^(-h + k) 4^k (c (2 s - v))^ (-h - k + 2 n) (c (2 s - v) - 2 d (m - 2 u) Sqrt[z])^(h + k) ((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] (c (2 s - v) (c (2 s - v) - 2 d (m - 2 u) Sqrt[z]) Gamma[ (1/2) (1 + h + k), (c (-2 s + v) + 2 d (m - 2 u) Sqrt[z])^2/ (4 d (m - 2 u))] - 2 d (m - 2 u) Sqrt[(c (-2 s + v) + 2 d (m - 2 u) Sqrt[z])^2/(d (m - 2 u))] Gamma[ (1/2) (2 + h + k), (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

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List[RowBox[List["d", " ", "z"]], "+", "e"]], "]"]], "m"], SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", SqrtBox["z"]]], "]"]], "v"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m", "-", "v"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["1", "+", "n"]]], RowBox[List["(", RowBox[List["2", " ", SuperscriptBox["\[ImaginaryI]", RowBox[List["-", "m"]]], SuperscriptBox["z", RowBox[List["1", "+", "n"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], ")"]]]], "+", RowBox[List["2", " ", SuperscriptBox["d", RowBox[List[RowBox[List["-", "n"]], "-", "1"]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "e", " ", "k"]], "-", RowBox[List["e", " ", "m"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", "k"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "m"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["d", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], SuperscriptBox["\[ExponentialE]", RowBox[List["4", " ", "e", " ", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["d", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", "z"]]]], "]"]]]]]], ")"]]]]]]]], "+", RowBox[List["4", " ", SuperscriptBox["\[ImaginaryI]", RowBox[List["-", "m"]]], SuperscriptBox["c", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", SqrtBox["z"]]]]], "]"]], "+", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", SqrtBox["z"]]]]], "]"]]]], ")"]]]]]]]], "-", RowBox[List[SuperscriptBox["4", RowBox[List["-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["u", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "u"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "u"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "e"]], " ", "m"]], "-", RowBox[List["2", " ", "e", " ", "u"]], "-", FractionBox[RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], "2"]]], RowBox[List["4", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["d", "2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], "2"]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "m"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], ")"]], RowBox[List["2", " ", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], ")"]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["4", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], ")"]]]]]]]], "]"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", RowBox[List["\[Sqrt]", RowBox[List["(", RowBox[List["-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], ")"]]]]]], ")"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["4", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], ")"]]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["4", " ", "e", " ", "u"]], "+", FractionBox[RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], "2"]]], RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], ")"]], RowBox[List["2", " ", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "-", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], ")"]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["4", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], ")"]]]]]], "]"]]]], "-", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", RowBox[List["\[Sqrt]", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], ")"]]]], ")"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["4", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], ")"]]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", " ", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "m"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], ")"]], RowBox[List["2", " ", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], ")"]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "-", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["4", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], ")"]]]]]]]], "]"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", RowBox[List["\[Sqrt]", RowBox[List["(", RowBox[List["-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], ")"]]]]]], ")"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["4", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], ")"]]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["4", " ", "e", " ", "u"]], "+", FractionBox[RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], "2"]]], RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], ")"]], RowBox[List["2", " ", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "-", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], ")"]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "-", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["4", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], ")"]]]]]], "]"]]]], "-", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", RowBox[List["\[Sqrt]", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], ")"]]]], ")"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], "/", RowBox[List["(", RowBox[List["4", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], ")"]]]]]], "]"]]]]]], ")"]]]]]]]]]]]], ")"]]]]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

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

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "n_"], " ", SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List[RowBox[List["d_", " ", "z_"]], "+", "e_"]], "]"]], "m_"], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c_", " ", SqrtBox["z_"]]], "]"]], "v_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "m", "-", "v"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ImaginaryI]", RowBox[List["-", "m"]]], " ", SuperscriptBox["z", RowBox[List["1", "+", "n"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], RowBox[List["1", "+", "n"]]], "+", RowBox[List["2", " ", SuperscriptBox["d", RowBox[List[RowBox[List["-", "n"]], "-", "1"]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "e", " ", "k"]], "-", RowBox[List["e", " ", "m"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "m"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["d", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["4", " ", "e", " ", "k"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["d", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]], " ", "z"]]]], "]"]]]]]], ")"]]]]]]]], "+", RowBox[List["4", " ", SuperscriptBox["\[ImaginaryI]", RowBox[List["-", "m"]]], " ", SuperscriptBox["c", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", SqrtBox["z"]]]]], "]"]], "+", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", SqrtBox["z"]]]]], "]"]]]], ")"]]]]]]]], "-", RowBox[List[SuperscriptBox["4", RowBox[List["-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["u", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "u"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "u"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "v"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "e"]], " ", "m"]], "-", RowBox[List["2", " ", "e", " ", "u"]], "-", FractionBox[RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], "2"]]], RowBox[List["4", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["d", "2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], "2"]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "m"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], ")"]], RowBox[List["2", " ", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]]]]]]], "]"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["4", " ", "e", " ", "u"]], "+", FractionBox[RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], "2"]]], RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], ")"]], RowBox[List["2", " ", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "-", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]]]]], "]"]]]], "-", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "e", " ", "m"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], ")"]], RowBox[List["2", " ", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "-", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]]]]]]], "]"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["4", " ", "e", " ", "u"]], "+", FractionBox[RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], "2"]]], RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]], ")"]], RowBox[List["2", " ", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "-", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "-", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]]]]], "]"]]]], "-", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]], " ", SqrtBox["z"]]]]], ")"]], "2"], RowBox[List["4", " ", "d", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "u"]]]], ")"]]]]]]], "]"]]]]]], ")"]]]]]]]]]]]], ")"]]]]]]]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]], "&&", RowBox[List["v", "\[Element]", "Integers"]], "&&", RowBox[List["v", ">", "0"]]]]]]]]]]

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

 2002-12-18