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 PolyGamma

 http://functions.wolfram.com/06.15.03.0054.01

 Input Form

 PolyGamma[-2 n, m + p/q] == (1/(-1 + 2 n)!) ((m + p/q)^(-1 + 2 n) (EulerGamma - (EulerGamma (m q + p))/(2 n q) - Log[m + p/q] + PolyGamma[2 n] + Sum[(Binomial[-1 + 2 n, k] (Sum[(-m - p/q)^j Binomial[k, j] PolyGamma[1 - j + k] (Sum[(1 + i + m + p/q)^(-j + k), {i, 0, -1 + Floor[-m - p/q]}] + Zeta[j - k, 1 + m + p/q + Max[ 0, Floor[-m - p/q]]]), {j, 0, k}] - PolyGamma[1 + k] Zeta[-k] - Derivative[1][Zeta][-k]))/(m + p/q)^k, {k, 0, -1 + 2 n}]) - Sum[(1 + i + m + p/q)^(-1 + 2 n) Log[1 + i + m + p/q], {i, 0, -1 + Floor[-m - p/q]}] + ((PolyGamma[2 n] - Log[2 Pi q]) BernoulliB[2 n, p/q])/(2 n) - ((PolyGamma[2 n] - Log[2 Pi]) BernoulliB[2 n])/(q^(2 n) 2 n) + (((-1)^(n + 1) Pi)/(2 Pi q)^(2 n)) Sum[Sin[(2 Pi p j)/q] PolyGamma[2 n - 1, j/q], {j, 1, q - 1}] + (((-1)^(n + 1) 2 (2 n - 1)!)/(2 Pi q)^(2 n)) Sum[Cos[(2 Pi p j)/q] Derivative[1, 0][Zeta][2 n, j/q], {j, 1, q - 1}] + Derivative[1][Zeta][-2 n + 1]/q^(2 n) + (1/2) Sum[Log[(-((p + m q + q Max[0, Floor[-((p + m q)/q)]])/q) + k)^2]/ ((-((p + m q + q Max[0, Floor[-((p + m q)/q)]])/q) + k)^2)^(1/2 - n), {k, 0, m + Max[0, Floor[-m - p/q]]}]) /; Element[m, Integers] && Element[n, Integers] && n > 0 && Element[p, Integers] && 0 < p < q && Element[q, Integers] && q > 1

 Standard Form

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 MathML Form

 ψ TagBox["\[Psi]", PolyGamma] ( - 2 n ) ( m + p q ) 1 ( 2 n - 1 ) ! ( ζ ( 1 - 2 n ) q - 2 n + ψ TagBox["\[Psi]", PolyGamma] ( 2 n ) - log ( 2 π q ) 2 n B TagBox["B", BernoulliB] 2 n ( p q ) + 1 2 k = 0 m + max ( 0 , - m - p q ) log ( ( k - p + m q + q max ( 0 , - p + m q q ) q ) 2 ) ( ( k - p + m q + q max ( 0 , - p + m q q ) q ) 2 ) 1 2 - n - i = 0 - m - p q - 1 ( i + m + p q + 1 ) 2 n - 1 log ( i + m + p q + 1 ) + ( - 1 ) n + 1 π ( 2 π q ) 2 n j = 1 q - 1 sin ( 2 π p j q ) ψ TagBox["\[Psi]", PolyGamma] ( 2 n - 1 ) ( j q ) + ( m + p q ) 2 n - 1 ( - TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ( m q + p ) 2 n q - log ( m + p q ) + ψ TagBox["\[Psi]", PolyGamma] ( 2 n ) + k = 0 2 n - 1 ( m + p q ) - k ( 2 n - 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List[RowBox[List["2", " ", "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 ( - m - p q ) j ( k j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ψ TagBox["\[Psi]", PolyGamma] ( k - j + 1 ) ( i = 0 - m - p q - 1 ( i + m + p q + 1 ) k - j + ζ ( j - k , m + max ( 0 , - m - p q ) + p q + 1 ) TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["j", "-", "k"]], Zeta, Rule[Editable, True]], ",", TagBox[RowBox[List["m", "+", RowBox[List["max", "(", RowBox[List["0", ",", RowBox[List["\[LeftFloor]", RowBox[List[RowBox[List["-", "m"]], "-", FractionBox["p", "q"]]], "\[RightFloor]"]]]], ")"]], "+", FractionBox["p", "q"], "+", "1"]], Zeta, Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1\$, ZetaDump`e2\$], Zeta[ZetaDump`e1\$, ZetaDump`e2\$]]]] ) - ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) ζ ( - k ) TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["-", "k"]], Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e\$, Zeta[BoxForm`e\$]]]] - ζ ( - k ) ) + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) + ( - 1 ) n + 1 2 ( 2 n - 1 ) ! ( 2 π q ) 2 n j = 1 q - 1 cos ( 2 π p j q ) ζ ( 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] ( 2 n , j q ) - ψ TagBox["\[Psi]", PolyGamma] ( 2 n ) - log ( 2 π ) 2 q 2 n n B TagBox["B", BernoulliB] 2 n ) /; m TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] n TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] + p TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] 0 < p < q q TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] q > 1 Condition PolyGamma -2 n m p q -1 1 2 n -1 -1 D Zeta 1 -1 2 n 1 -1 2 n q -2 n PolyGamma 2 n -1 2 q 2 n -1 BernoulliB 2 n p q -1 1 2 k 0 m 0 -1 m -1 p q -1 k -1 p m q q 0 -1 p m q q -1 q -1 2 k -1 p m q q 0 -1 p m q q -1 q -1 2 1 2 -1 n -1 -1 i 0 -1 m -1 p q -1 -1 i m p q -1 1 2 n -1 i m p q -1 1 -1 n 1 2 q 2 n -1 j 1 q -1 2 p j q -1 PolyGamma 2 n -1 j q -1 m p q -1 2 n -1 -1 m q p 2 n q -1 -1 m p q -1 PolyGamma 2 n k 0 2 n -1 m p q -1 -1 k Binomial 2 n -1 k j 0 k -1 m -1 p q -1 j Binomial k j PolyGamma k -1 j 1 i 0 -1 m -1 p q -1 -1 i m p q -1 1 k -1 j Zeta j -1 k m 0 -1 m -1 p q -1 p q -1 1 -1 PolyGamma k 1 Zeta -1 k -1 D Zeta -1 k -1 k -1 n 1 2 2 n -1 2 q 2 n -1 j 1 q -1 2 p j q -1 1 0 Zeta 2 n j q -1 -1 PolyGamma 2 n -1 2 2 q 2 n n -1 BernoulliB 2 n m n SuperPlus p 0 p q q q 1 [/itex]

 Rule Form

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

 2007-05-02