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; }

 Sech

 http://functions.wolfram.com/01.24.21.0272.01

 Input Form

 Integrate[z^n Sin[a z]^m Cosh[b z]^u Sech[c z], z] == 2^(1 - m - u) Binomial[m, m/2] Binomial[u, u/2] n! (1 - Mod[m, 2]) (1 - Mod[u, 2]) E^(c z) Sum[(((-1)^j z^(-j + n) c^(-1 - j))/(-j + n)!) HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, j + 1], 1}, {1 + Subscript[a, 1], \[Ellipsis], 1 + Subscript[a, j + 1]}, -E^(2 c z)], {j, 0, n}] + 2^(1 - m - u) Binomial[u, u/2] (1 - Mod[u, 2]) n! E^(c z) Sum[(-1)^k Binomial[m, k] (E^((-(1/2)) I m Pi + (I a (-2 k + m)) z) Sum[(((-1)^j z^(-j + n) (I a (-2 k + m) + c)^(-1 - j))/(-j + n)!) HypergeometricPFQ[{Subscript[b, 1], \[Ellipsis], Subscript[b, j + 1], 1}, {1 + Subscript[b, 1], \[Ellipsis], 1 + Subscript[b, j + 1]}, -E^(2 c z)], {j, 0, n}] + E^((I m Pi)/2 + ((-I) a (-2 k + m)) z) Sum[(((-1)^j z^(-j + n) ((-I) a (-2 k + m) + c)^(-1 - j))/(-j + n)!) HypergeometricPFQ[{Subscript[c, 1], \[Ellipsis], Subscript[c, j + 1], 1}, {1 + Subscript[c, 1], \[Ellipsis], 1 + Subscript[c, j + 1]}, -E^(2 c z)], {j, 0, n}]), {k, 0, Floor[(1/2) (-1 + m)]}] - 2^(1 - m - u) Binomial[m, m/2] (-1 + Mod[m, 2]) n! E^(c z) Sum[Binomial[u, s] (E^((b (-2 s + u)) z) Sum[(((-1)^j z^(-j + n) (b (-2 s + u) + c)^(-1 - j))/(-j + n)!) HypergeometricPFQ[{Subscript[d, 1], \[Ellipsis], Subscript[d, j + 1], 1}, {1 + Subscript[d, 1], \[Ellipsis], 1 + Subscript[d, j + 1]}, -E^(2 c z)], {j, 0, n}] + Sum[(((-1)^j z^(-j + n) ((-b) (-2 s + u) + c)^(-1 - j))/(-j + n)!) HypergeometricPFQ[{Subscript[e, 1], \[Ellipsis], Subscript[e, j + 1], 1}, {1 + Subscript[e, 1], \[Ellipsis], 1 + Subscript[e, j + 1]}, -E^(2 c z)], {j, 0, n}]/ E^((b (-2 s + u)) z)), {s, 0, Floor[(1/2) (-1 + u)]}] + 2^(1 - m - u) n! E^(c z) Sum[(-1)^k Binomial[m, k] Binomial[u, s] (E^((-(1/2)) I m Pi + (I a (-2 k + m) - b (-2 s + u)) z) Sum[(((-1)^j z^(-j + n) (I a (-2 k + m) - b (-2 s + u) + c)^(-1 - j))/ (-j + n)!) HypergeometricPFQ[{Subscript[f, 1], \[Ellipsis], Subscript[f, j + 1], 1}, {1 + Subscript[f, 1], \[Ellipsis], 1 + Subscript[f, j + 1]}, -E^(2 c z)], {j, 0, n}] + E^((I m Pi)/2 + ((-I) a (-2 k + m) + b (-2 s + u)) z) Sum[(((-1)^j z^(-j + n) ((-I) a (-2 k + m) + b (-2 s + u) + c)^ (-1 - j))/(-j + n)!) HypergeometricPFQ[{Subscript[g, 1], \[Ellipsis], Subscript[g, j + 1], 1}, {1 + Subscript[g, 1], \[Ellipsis], 1 + Subscript[g, j + 1]}, -E^(2 c z)], {j, 0, n}] + E^((I m Pi)/2 + ((-I) a (-2 k + m) - b (-2 s + u)) z) Sum[(((-1)^j z^(-j + n) ((-I) a (-2 k + m) - b (-2 s + u) + c)^ (-1 - j))/(-j + n)!) HypergeometricPFQ[{Subscript[h, 1], \[Ellipsis], Subscript[h, j + 1], 1}, {1 + Subscript[h, 1], \[Ellipsis], 1 + Subscript[h, j + 1]}, -E^(2 c z)], {j, 0, n}] + E^((-(1/2)) I m Pi + (I a (-2 k + m) + b (-2 s + u)) z) Sum[(((-1)^j z^(-j + n) (I a (-2 k + m) + b (-2 s + u) + c)^(-1 - j))/ (-j + n)!) HypergeometricPFQ[{Subscript[q, 1], \[Ellipsis], Subscript[q, j + 1], 1}, {1 + Subscript[q, 1], \[Ellipsis], 1 + Subscript[q, j + 1]}, -E^(2 c z)], {j, 0, n}]), {k, 0, Floor[(1/2) (-1 + m)]}, {s, 0, Floor[(1/2) (-1 + u)]}] /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n + 1] == 1/2 && Subscript[b, 1] == Subscript[b, 2] == \[Ellipsis] == Subscript[b, n + 1] == (c + I a (-2 k + m))/(2 c) && Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] == Subscript[c, n + 1] == (c - I a (-2 k + m))/(2 c) && Subscript[d, 1] == Subscript[d, 2] == \[Ellipsis] == Subscript[d, n + 1] == (c + b (-2 s + u))/(2 c) && Subscript[e, 1] == Subscript[e, 2] == \[Ellipsis] == Subscript[e, n + 1] == (c - b (-2 s + u))/(2 c) && Subscript[f, 1] == Subscript[f, 2] == \[Ellipsis] == Subscript[f, n + 1] == (I a (-2 k + m) + b (2 s - u) + c)/ (2 c) && Subscript[g, 1] == Subscript[g, 2] == \[Ellipsis] == Subscript[g, n + 1] == ((-I) a (-2 k + m) + b (-2 s + u) + c)/(2 c) && Subscript[h, 1] == Subscript[h, 2] == \[Ellipsis] == Subscript[h, n + 1] == (2 I a k - I a m + 2 b s - b u + c)/(2 c) && Subscript[q, 1] == Subscript[q, 2] == \[Ellipsis] == Subscript[q, n + 1] == (I a (-2 k + m) + b (-2 s + u) + c)/(2 c) && Element[n, Integers] && n >= 0 && Element[m, Integers] && m > 0 && Element[u, Integers] && u > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["a", " ", "z"]], "]"]], "m"], SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["b", " ", "z"]], "]"]], "u"], RowBox[List["Sech", "[", RowBox[List["c", " ", "z"]], "]"]], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m", "-", "u"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["u", ",", FractionBox["u", "2"]]], "]"]], " ", RowBox[List["n", "!"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["u", ",", "2"]], "]"]]]], ")"]], SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox["c", RowBox[List[RowBox[List["-", "1"]], "-", "j"]]], " "]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "n"]], ")"]], "!"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["j", "+", "1"]]], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", RowBox[List["j", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m", "-", "u"]]], " ", RowBox[List["Binomial", "[", RowBox[List["u", ",", FractionBox["u", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["u", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["n", "!"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], 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"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "+", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], ")"]], " ", "z"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "j"]]], " "]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "n"]], ")"]], "!"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", RowBox[List["j", "+", "1"]]], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["b", RowBox[List["j", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "2"], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], ")"]], " ", "z"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "j"]]], " "]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "n"]], ")"]], "!"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["c", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["c", RowBox[List["j", "+", "1"]]], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["c", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["c", RowBox[List["j", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]]]]]]]], ")"]]]]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m", "-", "u"]]], " ", 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["n", "!"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], RowBox[List["Sum", "[", " ", RowBox[List[RowBox[List[RowBox[List["Binomial", "[", RowBox[List["u", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]], ")"]], " ", "z"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "j"]]], " "]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "n"]], ")"]], "!"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["d", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["d", RowBox[List["j", "+", "1"]]], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["d", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["d", RowBox[List["j", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]], ")"]], " ", "z"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "j"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "n"]], ")"]], "!"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["e", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["e", RowBox[List["j", "+", "1"]]], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["e", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["e", RowBox[List["j", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]]]]]]]], ")"]]]], ",", RowBox[List["{", RowBox[List["s", ",", "0", ",", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "u"]], ")"]]]], "]"]]]], "}"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m", "-", "u"]]], " ", RowBox[List["n", "!"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], 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[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "u"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["u", ",", "s"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]]]], ")"]], " ", "z"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "j"]]], " "]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "n"]], ")"]], "!"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["f", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["f", RowBox[List["j", "+", "1"]]], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["f", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["f", RowBox[List["j", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "2"], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]]]], ")"]], " ", "z"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "j"]]], " "]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "n"]], ")"]], "!"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["g", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["g", RowBox[List["j", "+", "1"]]], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["g", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["g", RowBox[List["j", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "2"], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]]]], ")"]], " ", "z"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "j"]]], " "]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "n"]], ")"]], "!"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["h", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["h", RowBox[List["j", "+", "1"]]], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["h", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["h", RowBox[List["j", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]]]], ")"]], " ", "z"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "j"]]], " "]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "n"]], ")"]], "!"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["q", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["q", RowBox[List["j", "+", "1"]]], ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["q", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["q", RowBox[List["j", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]]]]]]]], ")"]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "\[Equal]", SubscriptBox["a", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["n", "+", "1"]]], "\[Equal]", FractionBox["1", "2"]]], "\[And]", RowBox[List[SubscriptBox["b", "1"], "\[Equal]", SubscriptBox["b", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["b", RowBox[List["n", "+", "1"]]], "\[Equal]", FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], "\[And]", RowBox[List[SubscriptBox["c", "1"], "\[Equal]", SubscriptBox["c", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["c", RowBox[List["n", "+", "1"]]], "\[Equal]", FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], "\[And]", RowBox[List[SubscriptBox["d", "1"], "\[Equal]", SubscriptBox["d", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["d", RowBox[List["n", "+", "1"]]], "\[Equal]", FractionBox[RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], "\[And]", RowBox[List[SubscriptBox["e", "1"], "\[Equal]", SubscriptBox["e", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["e", RowBox[List["n", "+", "1"]]], "\[Equal]", FractionBox[RowBox[List["c", "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], "\[And]", RowBox[List[SubscriptBox["f", "1"], "\[Equal]", SubscriptBox["f", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["f", RowBox[List["n", "+", "1"]]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "u"]], ")"]]]], "+", "c"]], RowBox[List["2", " ", "c"]]]]], "\[And]", RowBox[List[SubscriptBox["g", "1"], "\[Equal]", SubscriptBox["g", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["g", RowBox[List["n", "+", "1"]]], "\[Equal]", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]], "+", "c"]], RowBox[List["2", " ", "c"]]]]], "\[And]", RowBox[List[SubscriptBox["h", "1"], "\[Equal]", SubscriptBox["h", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["h", RowBox[List["n", "+", "1"]]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]], "-", RowBox[List["b", " ", "u"]], "+", "c"]], RowBox[List["2", " ", "c"]]]]], "\[And]", RowBox[List[SubscriptBox["q", "1"], "\[Equal]", SubscriptBox["q", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["q", RowBox[List["n", "+", "1"]]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]], "+", "c"]], RowBox[List["2", " ", "c"]]]]], "\[And]", 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["u", "\[Element]", "Integers"]], "\[And]", RowBox[List["u", ">", "0"]]]]]]]]

 MathML Form

 z n sin m ( a z ) cosh u ( b z ) sech ( c z ) z 2 - m - u + 1 c z ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["m", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( u u 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["u", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] n ! ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - u mod 2 \$CellContext`u 2 ) j = 0 n ( ( - 1 ) j z n - j c - j - 1 ) ( n - j ) ! j + 2 F j + 1 ( 1 2 , , 1 2 , 1 ; 3 2 , , 3 2 ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox["1", "2"], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox["3", "2"], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + 2 - m - u + 1 c z ( u u 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["u", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] n ! ( 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, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( m π 2 - ( a ( m - 2 k ) ) z j = 0 n ( ( - 1 ) j z n - j ( c - a ( m - 2 k ) ) - j - 1 ) ( n - j ) ! j + 2 F j + 1 ( c - a ( m - 2 k ) 2 c , , c - a ( m - 2 k ) 2 c , 1 ; c - a ( m - 2 k ) 2 c + 1 , , c - a ( m - 2 k ) 2 c + 1 ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + ( a ( m - 2 k ) ) z - m π 2 j = 0 n ( ( - 1 ) j z n - j ( c + a ( m - 2 k ) ) - j - 1 ) ( n - j ) ! j + 2 F j + 1 ( c + a ( m - 2 k ) 2 c , , c + a ( m - 2 k ) 2 c , 1 ; c + a ( m - 2 k ) 2 c + 1 , , c + a ( m - 2 k ) 2 c + 1 ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) - 2 - m - u + 1 ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["m", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( m mod 2 \$CellContext`m 2 - 1 ) n ! c z s = 0 u - 1 2 ( u s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( - b ( u - 2 s ) ) z j = 0 n ( ( - 1 ) j z n - j ( c - b ( u - 2 s ) ) - j - 1 ) ( n - j ) ! j + 2 F j + 1 ( c - b ( u - 2 s ) 2 c , , c - b ( u - 2 s ) 2 c , 1 ; c - b ( u - 2 s ) 2 c + 1 , , c - b ( u - 2 s ) 2 c + 1 ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["c", "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["c", "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["c", "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["c", "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + ( b ( u - 2 s ) ) z j = 0 n ( ( - 1 ) j z n - j ( c + b ( u - 2 s ) ) - j - 1 ) ( n - j ) ! j + 2 F j + 1 ( c + b ( u - 2 s ) 2 c , , c + b ( u - 2 s ) 2 c , 1 ; c + b ( u - 2 s ) 2 c + 1 , , c + b ( u - 2 s ) 2 c + 1 ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) + 2 - m - u + 1 c z n ! k = 0 m - 1 2 s = 0 u - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( u s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["u", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( a ( m - 2 k ) - b ( u - 2 s ) ) z - m π 2 j = 0 n ( ( - 1 ) j z n - j ( c + a ( m - 2 k ) - b ( u - 2 s ) ) - j - 1 ) ( n - j ) ! j + 2 F j + 1 ( c + a ( m - 2 k ) + b ( 2 s - u ) 2 c , , c + a ( m - 2 k ) + b ( 2 s - u ) 2 c , 1 ; c + a ( m - 2 k ) + b ( 2 s - u ) 2 c + 1 , , c + a ( m - 2 k ) + b ( 2 s - u ) 2 c + 1 ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "u"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "u"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "u"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "u"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + π m 2 + ( - a ( m - 2 k ) - b ( u - 2 s ) ) z j = 0 n ( ( - 1 ) j z n - j ( c - a ( m - 2 k ) - b ( u - 2 s ) ) - j - 1 ) ( n - j ) ! j + 2 F j + 1 ( c + 2 a k - a m + 2 b s - b u 2 c , , c + 2 a k - a m + 2 b s - b u 2 c , 1 ; c + 2 a k - a m + 2 b s - b u 2 c + 1 , , c + 2 a k - a m + 2 b s - b u 2 c + 1 ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]], "-", RowBox[List["b", " ", "u"]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]], "-", RowBox[List["b", " ", "u"]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]], "-", RowBox[List["b", " ", "u"]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]], "-", RowBox[List["b", " ", "u"]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + π m 2 + ( b ( u - 2 s ) - a ( m - 2 k ) ) z j = 0 n ( ( - 1 ) j z n - j ( c - a ( m - 2 k ) + b ( u - 2 s ) ) - j - 1 ) ( n - j ) ! j + 2 F j + 1 ( c - a ( m - 2 k ) + b ( u - 2 s ) 2 c , , c - a ( m - 2 k ) + b ( u - 2 s ) 2 c , 1 ; c - a ( m - 2 k ) + b ( u - 2 s ) 2 c + 1 , , c - a ( m - 2 k ) + b ( u - 2 s ) 2 c + 1 ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] + ( a ( m - 2 k ) + b ( u - 2 s ) ) z - m π 2 j = 0 n ( ( - 1 ) j z n - j ( c + a ( m - 2 k ) + b ( u - 2 s ) ) - j - 1 ) ( n - j ) ! j + 2 F j + 1 ( c + a ( m - 2 k ) + b ( u - 2 s ) 2 c , , c + a ( m - 2 k ) + b ( u - 2 s ) 2 c , 1 ; c + a ( m - 2 k ) + b ( u - 2 s ) 2 c + 1 , , c + a ( m - 2 k ) + b ( u - 2 s ) 2 c + 1 ; - 2 c z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["u", "-", RowBox[List["2", " ", "s"]]]], ")"]]]]]], RowBox[List["2", " ", "c"]]], "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]], HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]], HypergeometricPFQ] ) /; n m + u + Condition z z n a z m b z u c z 2 -1 m -1 u 1 c z Binomial m m 2 -1 Binomial u u 2 -1 n 1 -1 \$CellContext`m 2 1 -1 \$CellContext`u 2 j 0 n -1 j z n -1 j c -1 j -1 n -1 j -1 HypergeometricPFQ 1 2 1 2 1 3 2 3 2 -1 2 c z 2 -1 m -1 u 1 c z Binomial u u 2 -1 n 1 -1 \$CellContext`u 2 k 0 m -1 2 -1 -1 k Binomial m k m 2 -1 -1 a m -1 2 k z j 0 n -1 j z n -1 j c -1 a m -1 2 k -1 j -1 n -1 j -1 HypergeometricPFQ c -1 a m -1 2 k 2 c -1 c -1 a m -1 2 k 2 c -1 1 c -1 a m -1 2 k 2 c -1 1 c -1 a m -1 2 k 2 c -1 1 -1 2 c z a m -1 2 k z -1 m 2 -1 j 0 n -1 j z n -1 j c a m -1 2 k -1 j -1 n -1 j -1 HypergeometricPFQ c a m -1 2 k 2 c -1 c a m -1 2 k 2 c -1 1 c a m -1 2 k 2 c -1 1 c a m -1 2 k 2 c -1 1 -1 2 c z -1 2 -1 m -1 u 1 Binomial m m 2 -1 \$CellContext`m 2 -1 n c z s 0 u -1 2 -1 Binomial u s -1 b u -1 2 s z j 0 n -1 j z n -1 j c -1 b u -1 2 s -1 j -1 n -1 j -1 HypergeometricPFQ c -1 b u -1 2 s 2 c -1 c -1 b u -1 2 s 2 c -1 1 c -1 b u -1 2 s 2 c -1 1 c -1 b u -1 2 s 2 c -1 1 -1 2 c z b u -1 2 s z j 0 n -1 j z n -1 j c b u -1 2 s -1 j -1 n -1 j -1 HypergeometricPFQ c b u -1 2 s 2 c -1 c b u -1 2 s 2 c -1 1 c b u -1 2 s 2 c -1 1 c b u -1 2 s 2 c -1 1 -1 2 c z 2 -1 m -1 u 1 c z n s 0 u -1 2 -1 k 0 m -1 2 -1 -1 k Binomial m k Binomial u s a m -1 2 k -1 b u -1 2 s z -1 m 2 -1 j 0 n -1 j z n -1 j c a m -1 2 k -1 b u -1 2 s -1 j -1 n -1 j -1 HypergeometricPFQ c a m -1 2 k b 2 s -1 u 2 c -1 c a m -1 2 k b 2 s -1 u 2 c -1 1 c a m -1 2 k b 2 s -1 u 2 c -1 1 c a m -1 2 k b 2 s -1 u 2 c -1 1 -1 2 c z m 2 -1 -1 a m -1 2 k -1 b u -1 2 s z j 0 n -1 j z n -1 j c -1 a m -1 2 k -1 b u -1 2 s -1 j -1 n -1 j -1 HypergeometricPFQ c 2 a k -1 a m 2 b s -1 b u 2 c -1 c 2 a k -1 a m 2 b s -1 b u 2 c -1 1 c 2 a k -1 a m 2 b s -1 b u 2 c -1 1 c 2 a k -1 a m 2 b s -1 b u 2 c -1 1 -1 2 c z m 2 -1 b u -1 2 s -1 a m -1 2 k z j 0 n -1 j z n -1 j c -1 a m -1 2 k b u -1 2 s -1 j -1 n -1 j -1 HypergeometricPFQ c -1 a m -1 2 k b u -1 2 s 2 c -1 c -1 a m -1 2 k b u -1 2 s 2 c -1 1 c -1 a m -1 2 k b u -1 2 s 2 c -1 1 c -1 a m -1 2 k b u -1 2 s 2 c -1 1 -1 2 c z a m -1 2 k b u -1 2 s z -1 m 2 -1 j 0 n -1 j z n -1 j c a m -1 2 k b u -1 2 s -1 j -1 n -1 j -1 HypergeometricPFQ c a m -1 2 k b u -1 2 s 2 c -1 c a m -1 2 k b u -1 2 s 2 c -1 1 c a m -1 2 k b u -1 2 s 2 c -1 1 c a m -1 2 k b u -1 2 s 2 c -1 1 -1 2 c z n m SuperPlus u SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "n_"], " ", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["a_", " ", "z_"]], "]"]], "m_"], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["b_", " ", "z_"]], "]"]], "u_"], " ", RowBox[List["Sech", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m", "-", "u"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["u", ",", FractionBox["u", "2"]]], "]"]], " ", RowBox[List["n", "!"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["u", ",", "2"]], "]"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox["c", RowBox[List[RowBox[List["-", "1"]], "-", "j"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["Join", "[", RowBox[List[RowBox[List["Table", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["{", RowBox[List["K\$1", ",", "1", ",", RowBox[List["1", "+", "j"]]]], "}"]]]], "]"]], ",", RowBox[List["{", "1", "}"]]]], "]"]], ",", RowBox[List["Join", "[", RowBox[List["Table", "[", RowBox[List[FractionBox["3", "2"], ",", RowBox[List["{", RowBox[List["K\$1", ",", "1", ",", RowBox[List["1", "+", "j"]]]], "}"]]]], "]"]], "]"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "n"]], ")"]], "!"]]]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m", "-", "u"]]], " ", RowBox[List["Binomial", "[", RowBox[List["u", ",", FractionBox["u", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["u", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["n", "!"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], " ", 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"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "+", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], ")"]], " ", "z"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "j"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["Join", "[", RowBox[List[RowBox[List["Table", "[", RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["{", RowBox[List["K\$1", ",", "1", ",", RowBox[List["1", "+", "j"]]]], "}"]]]], "]"]], ",", RowBox[List["{", "1", "}"]]]], "]"]], ",", RowBox[List["Join", "[", RowBox[List["Table", "[", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["{", RowBox[List["K\$1", ",", "1", ",", RowBox[List["1", "+", "j"]]]], "}"]]]], "]"]], "]"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "n"]], ")"]], "!"]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "2"], "-", RowBox[List[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], ")"]], " ", "z"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "j"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["Join", "[", RowBox[List[RowBox[List["Table", "[", RowBox[List[FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["{", RowBox[List["K\$1", ",", "1", ",", RowBox[List["1", "+", "j"]]]], "}"]]]], "]"]], ",", RowBox[List["{", "1", "}"]]]], "]"]], ",", RowBox[List["Join", "[", RowBox[List["Table", "[", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["{", RowBox[List["K\$1", ",", "1", ",", RowBox[List["1", "+", "j"]]]], "}"]]]], "]"]], "]"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "n"]], ")"]], "!"]]]]]]]]], ")"]]]]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m", "-", "u"]]], " ", 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["n", "!"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "u"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["u", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]], ")"]], " ", "z"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "j"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["Join", "[", RowBox[List[RowBox[List["Table", "[", RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["{", RowBox[List["K\$1", ",", "1", ",", RowBox[List["1", "+", "j"]]]], "}"]]]], "]"]], ",", RowBox[List["{", "1", "}"]]]], "]"]], ",", RowBox[List["Join", "[", RowBox[List["Table", "[", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List["c", "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["{", RowBox[List["K\$1", ",", "1", ",", RowBox[List["1", "+", "j"]]]], "}"]]]], "]"]], "]"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "n"]], ")"]], "!"]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]], ")"]], " ", "z"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "j"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["Join", "[", RowBox[List[RowBox[List["Table", "[", RowBox[List[FractionBox[RowBox[List["c", "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["{", RowBox[List["K\$1", ",", "1", ",", RowBox[List["1", "+", "j"]]]], "}"]]]], "]"]], ",", RowBox[List["{", "1", "}"]]]], "]"]], ",", RowBox[List["Join", "[", RowBox[List["Table", "[", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List["c", "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["{", RowBox[List["K\$1", ",", "1", ",", RowBox[List["1", "+", "j"]]]], "}"]]]], "]"]], "]"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "n"]], ")"]], "!"]]]]]]]]], ")"]]]]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m", "-", "u"]]], " ", RowBox[List["n", "!"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], " ", 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[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "u"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["u", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]]]], ")"]], " ", "z"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "j"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["Join", "[", RowBox[List[RowBox[List["Table", "[", RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "u"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["{", RowBox[List["K\$1", ",", "1", ",", RowBox[List["1", "+", "j"]]]], "}"]]]], "]"]], ",", RowBox[List["{", "1", "}"]]]], "]"]], ",", RowBox[List["Join", "[", RowBox[List["Table", "[", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "u"]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["{", RowBox[List["K\$1", ",", "1", ",", RowBox[List["1", "+", "j"]]]], "}"]]]], "]"]], "]"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "n"]], ")"]], "!"]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "2"], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]]]], ")"]], " ", "z"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "j"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["Join", "[", RowBox[List[RowBox[List["Table", "[", RowBox[List[FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["{", RowBox[List["K\$1", ",", "1", ",", RowBox[List["1", "+", "j"]]]], "}"]]]], "]"]], ",", RowBox[List["{", "1", "}"]]]], "]"]], ",", RowBox[List["Join", "[", RowBox[List["Table", "[", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List["c", "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["{", RowBox[List["K\$1", ",", "1", ",", RowBox[List["1", "+", "j"]]]], "}"]]]], "]"]], "]"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "n"]], ")"]], "!"]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "2"], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]]]], ")"]], " ", "z"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "-", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "j"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["Join", "[", RowBox[List[RowBox[List["Table", "[", RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]], "-", RowBox[List["b", " ", "u"]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["{", RowBox[List["K\$1", ",", "1", ",", RowBox[List["1", "+", "j"]]]], "}"]]]], "]"]], ",", RowBox[List["{", "1", "}"]]]], "]"]], ",", RowBox[List["Join", "[", RowBox[List["Table", "[", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List["c", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "a", " ", "m"]], "+", RowBox[List["2", " ", "b", " ", "s"]], "-", RowBox[List["b", " ", "u"]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["{", RowBox[List["K\$1", ",", "1", ",", RowBox[List["1", "+", "j"]]]], "}"]]]], "]"]], "]"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "n"]], ")"]], "!"]]]]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", "m", " ", "\[Pi]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]]]], ")"]], " ", "z"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]], "+", "c"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "j"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["Join", "[", RowBox[List[RowBox[List["Table", "[", RowBox[List[FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]]]], RowBox[List["2", " ", "c"]]], ",", RowBox[List["{", RowBox[List["K\$1", ",", "1", ",", RowBox[List["1", "+", "j"]]]], "}"]]]], "]"]], ",", RowBox[List["{", "1", "}"]]]], "]"]], ",", RowBox[List["Join", "[", RowBox[List["Table", "[", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List["c", "+", RowBox[List["\[ImaginaryI]", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "u"]], ")"]]]]]], RowBox[List["2", " ", "c"]]]]], ",", RowBox[List["{", RowBox[List["K\$1", ",", "1", ",", RowBox[List["1", "+", "j"]]]], "}"]]]], "]"]], "]"]], ",", RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "c", " ", "z"]]]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "+", "n"]], ")"]], "!"]]]]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]], "&&", RowBox[List["u", "\[Element]", "Integers"]], "&&", RowBox[List["u", ">", "0"]]]]]]]]]]

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

 2002-12-18