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 PolyGamma

 http://functions.wolfram.com/06.15.06.0033.01

 Input Form

 PolyGamma[-n, z] == (-1)^n PolyGamma[-n, -z] + Sum[Sum[((-(z + p)^k - (-1)^(n + k) (z + p - 1)^k)/k!) Sum[((-1)^j/j!) PolyGamma[j + k - n, 1], {j, 0, n - k - 2}], {k, 0, n - 2}] + ((z + p)^(-1 + n)/(n - 1)!) (-EulerGamma + Log[-p - z] - PolyGamma[n]) + ((z + p - 1)^(-1 + n)/(n - 1)!) (EulerGamma - 2 I Pi Floor[1/2 - Arg[-1 + p - m]/(2 Pi) - (1/(2 Pi)) Arg[1 + (z + m)/(-1 + p - m)]] - Log[-1 + p - m] - Log[1 + (z + m)/(-1 + p - m)] + PolyGamma[n]), {p, 1, m - 1}] + Sum[((-(z + m)^k - (-1)^(n + k) (z + m - 1)^k)/k!) Sum[((-1)^j/j!) PolyGamma[j + k - n, 1], {j, 0, n - k - 2}], {k, 0, n - 2}] - (1/(n - 1)!) (Sum[(-2 Pi I)^(1 + k - n) (z + m)^k Binomial[n - 1, k] (n - 1 - k)! PolyLog[n - k, 1], {k, 0, n - 2}] + Sum[(-1)^(n - k) Binomial[n - 1, k] Sum[(2 Pi I)^(-n + 1 + k + j) (z + m)^(k + j) Binomial[n - 1 - k, j] (n - 1 - k - j)! PolyLog[n - k - j, E^(-2 I Pi z)], {j, 0, n - 1 - k}], {k, 0, -1 + n}]) + ((z + m - 1)^(-1 + n)/(n - 1)!) (EulerGamma - I Pi - 2 I Pi Floor[-(Arg[1 - z - m]/(2 Pi))] - Log[1 - z - m] + PolyGamma[n]) + ((z + m)^(-1 + n)/(n (-1 + n)!)) ((-I) (z + m) Pi + I n Pi + 2 I n Pi Floor[3/4 - Arg[-z - m]/(2 Pi)] + 2 I n Pi Floor[-(Arg[z + m]/(2 Pi))] + n Log[-2 I Pi]) /; (z -> -m) && Element[m, Integers] && m > 0 && ((Element[n, Integers] && n > 1) || (n == 1 && Inequality[-Pi, Less, Arg[z + m], LessEqual, Pi/2]))

 Standard Form

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"\[LessEqual]", FractionBox["\[Pi]", "2"]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 ψ TagBox["\[Psi]", PolyGamma] ( - n ) ( z ) ( - 1 ) n ψ TagBox["\[Psi]", PolyGamma] ( - n ) ( - z ) + ( m + z ) n - 1 n ( n - 1 ) ! ( - π ( m + z ) + 2 n π - arg ( m + z ) 2 π + 2 n π 3 4 - arg ( - m - z ) 2 π + n log ( - 2 π ) + n π ) + ( m + z - 1 ) n - 1 ( n - 1 ) ! ( TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] - π - 2 π - arg ( - m - z + 1 ) 2 π - log ( - m - z + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( n ) ) + p = 1 m - 1 ( ( p + z ) n - 1 ( n - 1 ) ! ( log ( - p - z ) - ψ TagBox["\[Psi]", PolyGamma] ( n ) - TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) + ( p + z - 1 ) n - 1 ( n - 1 ) ! ( - 2 π 1 2 - 1 2 π arg ( 1 + m + z - m + p - 1 ) - arg ( - m + p - 1 ) 2 π - log ( 1 + m + z - m + p - 1 ) - log ( - m + p - 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( n ) + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) + k = 0 n - 2 - ( - 1 ) k + n ( p + z - 1 ) k - ( p + z ) k k ! j = 0 - k + n - 2 ( - 1 ) j ψ TagBox["\[Psi]", PolyGamma] ( j + k - n ) ( 1 ) j ! ) + k = 0 n - 2 - ( - 1 ) k + n ( m + z - 1 ) k - ( m + z ) k k ! j = 0 - k + n - 2 ( - 1 ) j ψ TagBox["\[Psi]", PolyGamma] ( j + k - n ) ( 1 ) j ! - 1 ( n - 1 ) ! ( k = 0 n - 2 ( - 2 π ) k - n + 1 ( m + z ) k ( n - 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - k + n - 1 ) ! Li PolyLog n - k ( 1 ) + k = 0 n - 1 ( - 1 ) n - k ( n - 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] j = 0 - k + n - 1 ( 2 π ) j + k - n + 1 ( m + z ) j + k ( - k + n - 1 j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List[RowBox[List["-", "k"]], "+", "n", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - j - k + n - 1 ) ! Li PolyLog - j - k + n ( - 2 π z ) ) /; ( z "\[Rule]" - m ) m + ( ( n TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] n > 1 ) ( n 1 - π < arg ( m + z ) π 2 ) ) Condition PolyGamma -1 n z -1 n PolyGamma -1 n -1 z m z n -1 n n -1 -1 -1 m z 2 n -1 m z 2 -1 2 n 3 4 -1 -1 m -1 z 2 -1 n -2 n m z -1 n -1 n -1 -1 -1 -1 2 -1 -1 m -1 z 1 2 -1 -1 -1 m -1 z 1 PolyGamma n p 1 m -1 p z n -1 n -1 -1 -1 p -1 z -1 PolyGamma n -1 p z -1 n -1 n -1 -1 -2 1 2 -1 1 2 -1 1 m z -1 m p -1 -1 -1 -1 m p -1 2 -1 -1 1 m z -1 m p -1 -1 -1 -1 m p -1 PolyGamma n k 0 n -2 -1 -1 k n p z -1 k -1 p z k k -1 j 0 -1 k n -2 -1 j PolyGamma j k -1 n 1 j -1 k 0 n -2 -1 -1 k n m z -1 k -1 m z k k -1 j 0 -1 k n -2 -1 j PolyGamma j k -1 n 1 j -1 -1 1 n -1 -1 k 0 n -2 -2 k -1 n 1 m z k Binomial n -1 k -1 k n -1 PolyLog n -1 k 1 k 0 n -1 -1 n -1 k Binomial n -1 k j 0 -1 k n -1 2 j k -1 n 1 m z j k Binomial -1 k n -1 j -1 j -1 k n -1 PolyLog -1 j -1 k n -2 z Rule z -1 m m SuperPlus n n 1 n 1 Inequality -1 m z 2 -1 [/itex]

 Rule Form

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

 2007-05-02