Gamma Function - 5.15 Polygamma Functions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
5.15.E1 ψ ′ ⁡ ( z ) = ∑ k = 0 ∞ 1 ( k + z ) 2 diffop digamma 1 𝑧 superscript subscript 𝑘 0 1 superscript 𝑘 𝑧 2 {\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 ψ ( n ) ⁡ ( 1 ) = ( - 1 ) n + 1 ⁢ n ! ⁢ ζ ⁡ ( n + 1 ) polygamma 𝑛 1 superscript 1 𝑛 1 𝑛 Riemann-zeta 𝑛 1 {\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 ψ ( n ) ⁡ ( 1 2 ) = ( - 1 ) n + 1 ⁢ n ! ⁢ ( 2 n + 1 - 1 ) ⁢ ζ ⁡ ( n + 1 ) polygamma 𝑛 1 2 superscript 1 𝑛 1 𝑛 superscript 2 𝑛 1 1 Riemann-zeta 𝑛 1 {\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 ψ ′ ⁡ ( n - 1 2 ) = 1 2 ⁢ π 2 - 4 ⁢ ∑ k = 1 n - 1 1 ( 2 ⁢ k - 1 ) 2 diffop digamma 1 𝑛 1 2 1 2 superscript 𝜋 2 4 superscript subscript 𝑘 1 𝑛 1 1 superscript 2 𝑘 1 2 {\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 ψ ( n ) ⁡ ( z + 1 ) = ψ ( n ) ⁡ ( z ) + ( - 1 ) n ⁢ n ! ⁢ z - n - 1 digamma 𝑛 𝑧 1 digamma 𝑛 𝑧 superscript 1 𝑛 𝑛 superscript 𝑧 𝑛 1 {\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 ψ ( n ) ⁡ ( 1 - z ) + ( - 1 ) n - 1 ⁢ ψ ( n ) ⁡ ( z ) = ( - 1 ) n ⁢ π ⁢ d n d z n ⁡ cot ⁡ ( π ⁢ z ) digamma 𝑛 1 𝑧 superscript 1 𝑛 1 digamma 𝑛 𝑧 superscript 1 𝑛 𝜋 derivative 𝑧 𝑛 𝜋 𝑧 {\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 ψ ( n ) ⁡ ( m ⁢ z ) = 1 m n + 1 ⁢ ∑ k = 0 m - 1 ψ ( n ) ⁡ ( z + k m ) digamma 𝑛 𝑚 𝑧 1 superscript 𝑚 𝑛 1 superscript subscript 𝑘 0 𝑚 1 digamma 𝑛 𝑧 𝑘 𝑚 {\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
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