5.15

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
5.15.E1	${\displaystyle{\displaystyle\psi'\left(z\right)=\sum_{k=0}^{\infty}\frac{1}{(k+z)^{2}}}}$ \digamma'@{z} = \sum_{k=0}^{\infty}\frac{1}{(k+z)^{2}}		diff( Psi(z), z$(1) ) = sum((1)/((k + z)^(2)), k = 0..infinity)	D[PolyGamma[z], {z, 1}] == Sum[Divide[1,(k + z)^(2)], {k, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 0]	Successful [Tested: 1]
5.15.E2	${\displaystyle{\displaystyle\psi^{(n)}\left(1\right)=(-1)^{n+1}n!\zeta\left(n+1\right)}}$ \polygamma{n}@{1} = (-1)^{n+1}n!\Riemannzeta@{n+1}		Psi(n, 1) = (- 1)^(n + 1)* factorial(n)*Zeta(n + 1)	PolyGamma[n, 1] == (- 1)^(n + 1)* (n)!*Zeta[n + 1]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
5.15.E3	${\displaystyle{\displaystyle\psi^{(n)}\left(\tfrac{1}{2}\right)=(-1)^{n+1}n!(2^{n+1}-1)\zeta\left(n+1\right)}}$ \polygamma{n}@{\tfrac{1}{2}} = (-1)^{n+1}n!(2^{n+1}-1)\Riemannzeta@{n+1}		Psi(n, (1)/(2)) = (- 1)^(n + 1)* factorial(n)*((2)^(n + 1)- 1)*Zeta(n + 1)	PolyGamma[n, Divide[1,2]] == (- 1)^(n + 1)* (n)!*((2)^(n + 1)- 1)*Zeta[n + 1]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
5.15.E4	${\displaystyle{\displaystyle\psi'\left(n-\tfrac{1}{2}\right)=\tfrac{1}{2}\pi^{2}-4\sum_{k=1}^{n-1}\frac{1}{(2k-1)^{2}}}}$ \digamma'@{n-\tfrac{1}{2}} = \tfrac{1}{2}\pi^{2}-4\sum_{k=1}^{n-1}\frac{1}{(2k-1)^{2}}		subs( temp=n -(1)/(2), diff( Psi(temp), temp$(1) ) ) = (1)/(2)*(Pi)^(2)- 4*sum((1)/((2*k - 1)^(2)), k = 1..n - 1)	(D[PolyGamma[temp], {temp, 1}]/.temp-> n -Divide[1,2]) == Divide[1,2]*(Pi)^(2)- 4*Sum[Divide[1,(2*k - 1)^(2)], {k, 1, n - 1}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 3]
5.15.E5	${\displaystyle{\displaystyle{\psi^{(n)}}\left(z+1\right)={\psi^{(n)}}\left(z\right)+(-1)^{n}n!z^{-n-1}}}$ \digamma^{(n)}@{z+1} = \digamma^{(n)}@{z}+(-1)^{n}n!z^{-n-1}		subs( temp=z + 1, diff( Psi(temp), temp$(n) ) ) = diff( Psi(z), z$(n) )+(- 1)^(n)* factorial(n)*(z)^(- n - 1)	(D[PolyGamma[temp], {temp, n}]/.temp-> z + 1) == D[PolyGamma[z], {z, n}]+(- 1)^(n)* (n)!*(z)^(- n - 1)	Failure	Failure	Successful [Tested: 21]	Successful [Tested: 21]
5.15.E6	${\displaystyle{\displaystyle{\psi^{(n)}}\left(1-z\right)+(-1)^{n-1}{\psi^{(n)}}\left(z\right)=(-1)^{n}\pi\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\cot\left(\pi z\right)}}$ \digamma^{(n)}@{1-z}+(-1)^{n-1}\digamma^{(n)}@{z} = (-1)^{n}\pi\deriv[n]{}{z}\cot@{\pi z}		subs( temp=1 - z, diff( Psi(temp), temp$(n) ) )+(- 1)^(n - 1)* diff( Psi(z), z$(n) ) = (- 1)^(n)* Pi*diff(cot(Pi*z), [z$(n)])	(D[PolyGamma[temp], {temp, n}]/.temp-> 1 - z)+(- 1)^(n - 1)* D[PolyGamma[z], {z, n}] == (- 1)^(n)* Pi*D[Cot[Pi*z], {z, n}]	Failure	Failure	Error	Failed [21 / 21] Result: Plus[Complex[-1.111486978443634, 1.4242909397222407], Times[Complex[-1.1253971041044755, 1.3474673991212198], Inactive[Sum][Times[Power[-0.5, K[1.0]], Power[Complex[2.0570132833277626, -0.06826349589921218], K[1.0]], Factorial[K[1.0]], StirlingS2[1.0, K[1.0]]] Test Values: {K[1.0], 0.0, 1.0}]]], {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[9.936030880873925, 6.770945349247037], Times[Complex[-8.466387364061939, -7.071078549251696], Inactive[Sum][Times[Power[-0.5, K[1.0]], Power[Complex[2.0570132833277626, -0.06826349589921218], K[1.0]], Factorial[K[1.0]], StirlingS2[2.0, K[1.0]]] Test Values: {K[1.0], 0.0, 2.0}]]], {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
5.15.E7	${\displaystyle{\displaystyle{\psi^{(n)}}\left(mz\right)=\frac{1}{m^{n+1}}\sum_{k=0}^{m-1}{\psi^{(n)}}\left(z+\frac{k}{m}\right)}}$ \digamma^{(n)}@{mz} = \frac{1}{m^{n+1}}\sum_{k=0}^{m-1}\digamma^{(n)}@{z+\frac{k}{m}}		diff( Psi(m*z), m*z$(n) ) = (1)/((m)^(n + 1))*sum(subs( temp=z +(k)/(m), diff( Psi(temp), temp$(n) ) ), k = 0..m - 1)	D[PolyGamma[m*z], {m*z, n}] == Divide[1,(m)^(n + 1)]*Sum[D[PolyGamma[temp], {temp, n}]/.temp-> z +Divide[k,m], {k, 0, m - 1}, GenerateConditions->None]	Error	Failure	-	Failed [63 / 63] Result: Plus[Complex[-1.1320242650810568, 1.0823171404691536], D[Complex[-0.4765906465900115, 0.839495097073875] Test Values: {Complex[0.8660254037844387, 0.49999999999999994], 1.0}]], {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[Complex[0.3478434500030721, -2.260508246850942], D[Complex[-0.4765906465900115, 0.839495097073875] Test Values: {Complex[0.8660254037844387, 0.49999999999999994], 2.0}]], {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data