List of mathematical series explained

See also: Summation.

This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums.

• Here,

0 ^{0} is taken to have the value 1 \{x\} denotes the fractional part of x B _{n(x)} is a Bernoulli polynomial.B _{n} is a Bernoulli number, and here, B _{1=- (1)/(2)} . E _{n} is an Euler number.\zeta(s) is the Riemann zeta function.\Gamma(z) is the gamma function.\psi _{n(z)} is a polygamma function.\operatorname{Li} _{s(z)} is a polylogarithm.n \choose k is binomial coefficient\exp(x) denotes exponential of x

Sums of powers

See Faulhaber's formula.

\sum ^{m} _{k =0} k ^{n-1} = (B _{n(m+1)-B} _{n})/(n) The first few values are:\sum ^{m} _{k =1} k = (m(m +1 ))/(2) \sum ^{m} _{k =1} k ^{2= (m(m +1 )( 2 m +1 ))/(6)} = (m ^{3})/(+ 3) (m ^{2})/(+ 2) (m)/(6) \sum ^{m} _{k =1} k ^{3 = \left[ (m(m +1 ))/(2)} \right] ^{2= (m ^{4})/(4)} + (m ^{3})/(+ 2) (m ^{2})/(4)

See zeta constants.

\zeta( 2 n) = \sum ^{inf ty} _{k =1} (1)/(k ^{2n}) = ( -1 ) ^{n+1} (B _{2n} ( 2 \pi) ^{2n})/(2 ( 2 n)!) The first few values are:\zeta( 2 ) = \sum ^{inf ty} _{k =1} (1)/(= k ^{2}) (\pi ^{2})/(6) (the Basel problem)\zeta( 4 ) = \sum ^{inf ty} _{k =1} (1)/(= k ^{4}) (\pi ^{4})/(90) \zeta( 6 ) = \sum ^{inf ty} _{k =1} (1)/(= k ^{6}) (\pi ^{6})/(945)

Power series

Low-order polylogarithms

Finite sums:

\sum ^{n} _{k = m} z ^{k} = (z ^{m} - z ^{n+1})/(1- z) , (geometric series)\sum ^{n} _{k =0} z ^{k} = (1- z ^{n+1})/(1- z) \sum ^{n} _{k =1} z ^{k} = (1- z ^{n+1})/(1- z) -1 = (z - z ^{n+1})/(1- z) \sum ^{n} _{k =1} k z ^{k} = z (1- (n +1 )z ^{n+nz} ^{n+1})/(( 1- z) ^{2}) \sum ^{n} _{k =1} k ^{2} z ^{k} = z (1+ z - (n +1 ) ^{2z} ^{n+(2n} ^{2+2n-1)z} ^{n+1} - n ^{2z} ^{n+2})/(( 1- z) ^{3}) \sum ^{n} _{k =1} k ^{m} z ^{k} = \left(z (d)/(dz) \right) ^{m} (1- z ^{n+1})/(1- z)

Infinite sums, valid for

|z|< 1 (see polylogarithm):\operatorname{Li} _{n(z)=\sum} ^{inf ty} ^{k =1} (z ^{k})/(k ^{n}) The following is a useful property to calculate low-integer-order polylogarithms recursively in closed form:(d)/(d z) \operatorname{Li} _{n(z) = (\operatorname{Li)/(_{n-1})} (z)}{z} \operatorname{Li} _{1} (z) = \sum ^{inf ty} _{k =1} (z ^{k})/(k) =-ln ( 1- z) \operatorname{Li} _{0} (z) = \sum ^{inf ty} _{k =1} z ^{k = (z)/(1- z)} \operatorname{Li} _{-1} (z) = \sum ^{inf ty} _{k =1} k z ^{k = (z)/(( 1- z) ^{2})} \operatorname{Li} _{-2} (z) = \sum ^{inf ty} _{k =1} k ^{2} z ^{k = (z( 1+ z))/(( 1- z) ^{3})} \operatorname{Li} _{-3} (z) = \sum ^{inf ty} _{k =1} k ^{3} z ^{k} = (z( 1+4 z + z ^{2)})/(( 1- z) ^{4}) \operatorname{Li} _{-4} (z) = \sum ^{inf ty} _{k =1} k ^{4} z ^{k} = (z( 1+ z)( 1+10 z + z ^{2)})/(( 1- z) ^{5})

Exponential function

\sum ^{inf ty} _{k =0} (z ^{k})/(k!) = e ^{z} \sum ^{inf ty} _{k =0} k (z ^{k})/(k!) = z e ^{z} (cf. mean of Poisson distribution)\sum ^{inf ty} _{k =0} k ^{2} (z ^{k})/(k!) = (z + z ^{2)} e ^{z} (cf. second moment of Poisson distribution)\sum ^{inf ty} _{k =0} k ^{3} (z ^{k})/(k!) = (z + 3 z ^{2} + z ^{3)} e ^{z} \sum ^{inf ty} _{k =0} k ^{4} (z ^{k})/(k!) = (z + 7 z ^{2} + 6 z ^{3} + z ^{4)} e ^{z} \sum ^{inf ty} _{k =0} k ^{n} (z ^{k})/(k!) = z (d)/(dz) \sum ^{inf ty} _{k =0} k ^{n-1} (z ^{k})/(k!) = e ^{z} T _{n} (z)

where

T _{n} (z) is the Touchard polynomials.

Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions relationship

\sum ^{inf ty} _{k =0} (( -1 ) ^{k} z ^{2k+1})/(( 2 k +1 )!) = \sin z \sum ^{inf ty} _{k =0} (z ^{2k+1})/(( 2 k +1 )!) = \sinh z \sum ^{inf ty} _{k =0} (( -1 ) ^{k} z ^{2k})/(( 2 k)!) = \cos z \sum ^{inf ty} _{k =0} (z ^{2k})/(( 2 k)!) = \cosh z \sum ^{inf ty} _{k =1} (( -1 ) ^{k-1} ( 2 ^{2k} -1 ) 2 ^{2k} B _{2k} z ^{2k-1})/(( 2 k)!) = \tan z, |z|< (\pi)/(2) \sum ^{inf ty} _{k =1} (( 2 ^{2k} -1 ) 2 ^{2k} B _{2k} z ^{2k-1})/(( 2 k)!) = \tanh z, |z|< (\pi)/(2) \sum ^{inf ty} _{k =0} (( -1 ) ^{k2} ^{2k} B _{2k} z ^{2k-1})/(( 2 k)!) = \cot z, |z|<\pi \sum ^{inf ty} _{k =0} (2 ^{2k} B _{2k} z ^{2k-1})/(( 2 k)!) = \coth z, |z|<\pi \sum ^{inf ty} _{k =0} (( -1 ) ^{k-1} ( 2 ^{2k} -2 )B _{2k} z ^{2k-1})/(( 2 k)!) = \csc z, |z|<\pi \sum ^{inf ty} _{k =0} (- ( 2 ^{2k} -2 )B _{2k} z ^{2k-1})/(( 2 k)!) = \operatorname{csch} z, |z|<\pi \sum ^{inf ty} _{k =0} (( -1 ) ^{kE} _{2 k} z ^{2k})/(( 2 k)!) = \operatorname{sech} z, |z|< (\pi)/(2) \sum ^{inf ty} _{k =0} (E _{2k} z ^{2k})/(( 2 k)!) = \sec z, |z| < (\pi)/(2) \sum ^{inf ty} _{k =1} (( -1 ) ^{k-1} z ^{2k})/(( 2 k)!) = \operatorname{ver}z (versine)\sum ^{inf ty} _{k =1} (( -1 ) ^{k-1} z ^{2k})/(2 ( 2 k)!) = \operatorname{hav}z ^[1] (haversine)\sum ^{inf ty} _{k =0} (( 2 k)!z ^{2k+1})/(2 ^{2k} (k!) ^{2(2k+1)}) = \arcsin z, |z|\le 1 \sum ^{inf ty} _{k =0} (( -1 ) ^{k(2k)!z} ^{2k+1})/(2 ^{2k} (k!) ^{2(2k+1)}) = \operatorname{arcsinh} {z}, |z| \le 1 \sum ^{inf ty} _{k =0} (( -1 ) ^{kz} ^{2k+1})/(2 k +1) = \arctan z, |z|< 1 \sum ^{inf ty} _{k =0} (z ^{2k+1})/(2 k +1) = \operatorname{arctanh} z, |z|< 1 ln2+ \sum ^{inf ty} _{k =1} (( -1 ) ^{k-1} ( 2 k)!z ^{2k})/(2 ^{2k+1} k(k!) ^{2}) =ln \left( 1+ \sqrt{ 1+ z ^{2}\right),} |z| \le 1 \sum ^{inf ty} _{k =2} \left( k ⋅ \operatorname{arctanh}\left( (1)/(k) \right) - 1 \right) = (3-ln ( 4 \pi))/(2)

Modified-factorial denominators

\sum ^{inf ty} _{k =0} (( 4 k)!)/(2 ^{4k} \sqrt{ 2) ( 2 k)! ( 2 k +1 )!} z ^{k} = \sqrt{ (1- \sqrt{ 1- z)/() }}, |z|<1^[2] \sum ^{inf ty} _{k =0} (2 ^{2k} (k!) ^{2})/((k +1 ) ( 2 k +1 )!) z ^{2k+2} = \left(\arcsin{z}\right) ^{2,} |z|\le 1 ^[2] \sum ^{inf ty} _{n =0} (\prod ^{n -1} _{k =0} ( 4 k ^{2+\alpha} ^{2)})/(( 2 n)!) z ^{2n} + \sum ^{inf ty} _{n =0} (\alpha \prod ^{n -1} _{k =0} [( 2 k +1 ) ^{2+\alpha} ^{2]})/(( 2 n +1 )!) z ^{2n+1} = e ^{\alpha} }, |z|\le1

Binomial coefficients

( 1+ z) ^{\alpha} = \sum ^{inf ty} _{k =0} {\alpha \choose k} z ^{k,} |z|< 1 (see)

• ^[3]

\sum ^{inf ty} _{k =0} {{\alpha + k -1 } \choose k} z ^{k} = (1)/(( 1- z) ^{\alpha}) , |z|< 1

• ^[3]

\sum ^{inf ty} _{k =0} (1)/(k +1) { 2 k \choose k} z ^{k} = (1- \sqrt{ 1-4 z)/() }, |z|\leq\frac, generating function of the Catalan numbers

• ^[3]

\sum ^{inf ty} _{k =0} { 2 k \choose k} z ^{k} = (1)/(\sqrt{ 1-4 z) }, |z|<\frac, generating function of the Central binomial coefficients

• ^[3]

\sum ^{inf ty} _{k =0} { 2 k + \alpha \choose k} z ^{k} = (1)/(\sqrt{ 1-4 z) }\left(\frac\right)^\alpha, |z|<\frac

Harmonic numbers

(See harmonic numbers, themselves defined $H\_n\; =\; \backslash sum\_^\; \backslash frac$, and

H(x) generalized to the real numbers)\sum ^{inf ty} _{k =1} H _{k} z ^{k} = (-ln ( 1- z))/(1- z) , |z|< 1 \sum ^{inf ty} _{k =1} (H _{k})/(k +1) z ^{k+1} = (1)/(2) \left[ ln ( 1- z)\right] ^{2,} |z|< 1 \sum ^{inf ty} _{k =1} (( -1 ) ^{k-1} H _{2k})/(2 k +1) z ^{2k+1} = (1)/(2) \arctan{z} log {( 1+ z ^{2)},} |z|< 1 ^[2] \sum ^{inf ty} _{n =0} \sum ^{2 n} _{k =0} (( -1 ) ^{k})/(2 k +1) (z ^{4n+2})/(4 n +2) = (1)/(4) \arctan{z} log { (1+ z)/(1- z) },\qquad |z|<1 ^[2] \sum ^{inf ty} _{n =0} (x ^{2})/(n ^{2(n+x)}) = x (\pi ^{2})/(6) - H(x)

Binomial coefficients

See main article: Binomial coefficient.

\sum ^{n} _{k =0} {n \choose k} = 2 ^{n} \sum ^{n} _{k =0} {n \choose k} ^{2} = { 2 n \choose n} \sum ^{n} _{k =0} ( -1 ) ^{k} {n \choose k} = 0 , where n\geq 1 \sum ^{n} _{k =0} {k \choose m} = { n +1 \choose m +1 } \sum ^{n} _{k =0} {m + k -1 \choose k} = { n + m \choose n } (see Multiset)\sum ^{n} _{k =0} {\alpha \choose k}{\beta \choose n - k} = {\alpha + \beta \choose n}, where \alpha + \beta \geq n (see Vandermonde identity)\sum _{A} (E)} 1 = 2 ^{n} ,where E isafiniteset,andcard( E )=n \sum _{\begin{cases} (A, B) \in ( l{P}(E)) ^{2} \ A \subset B \end{cases}} 1 = 3 ^{n,whereEisafiniteset,andcard(E)=n} \sum _{A} (E)} card(A) = n 2 ^{n-1} ,where E isafiniteset,andcard( E )=n

Trigonometric functions

Sums of sines and cosines arise in Fourier series.

\sum ^{inf ty} _{k =1} (\cos(k\theta))/(=- k) (1)/(2) ln ( 2-2 \cos\theta) =-ln \left( 2 \sin (\theta)/(2) \right), 0 <\theta< 2 \pi \sum ^{inf ty} _{k =1} (\sin(k\theta))/(= k) (\pi - \theta)/(2) , 0 <\theta< 2 \pi \sum ^{inf ty} _{k =1} (( -1 ) ^{k-1})/(\cos(k\theta) = k) (1)/(2) ln ( 2+2 \cos\theta) =ln \left( 2 \cos (\theta)/(2) \right), 0 \leq\theta<\pi \sum ^{inf ty} _{k =1} (( -1 ) ^{k-1})/(\sin(k\theta) = k) (\theta)/(2) , - (\pi)/(\leq\theta\leq 2) (\pi)/(2) \sum ^{inf ty} _{k =1} (\cos( 2 k\theta))/(=- 2 k) (1)/(2) ln ( 2 \sin\theta), 0 <\theta<\pi \sum ^{inf ty} _{k =1} (\sin( 2 k\theta))/(= 2 k) (\pi -2 \theta)/(4) , 0 <\theta<\pi \sum ^{inf ty} _{k =0} (\cos[( 2 k +1 )\theta])/(= 2 k +1) (1)/(2) ln \left(\cot (\theta)/(2) \right), 0 <\theta<\pi \sum ^{inf ty} _{k =0} (\sin[( 2 k +1 )\theta])/(= 2 k +1) (\pi)/(4) , 0 <\theta<\pi ,^[4] \sum ^{inf ty} _{k =1} (\sin( 2 \pi k x))/(k) = \pi \left(\dfrac{ 1 }{ 2 } - \{x\}\right), x \in R \sum\limits ^{inf ty} _{k =1} (\sin \left( 2 \pi kx \right))/(k ^{2n-1}) = ( -1 ) ^{n} (( 2 \pi) ^{2n-1})/(2 ( 2 n -1 )!) B _{2n-1} (\{x\}), x \in R , n \in N \sum\limits ^{inf ty} _{k =1} (\cos \left( 2 \pi kx \right))/(k ^{2n}) = ( -1 ) ^{n-1} (( 2 \pi) ^{2n})/(2 ( 2 n)!) B _{2n} (\{x\}), x \in R , n \in N B _{n(x) =- (n!)/(2 ^{n-1} \pi ^{n})} \sum ^{inf ty} _{k =1} (1)/(k ^{n}) \cos\left( 2 \pi kx - (\pi n)/(2) \right), 0 <x< 1 ^[5] \sum ^{n} _{k =0} \sin(\theta + k\alpha) = (\sin ((n +1 )\alpha)/(\sin(\theta + (n\alpha)/(2) ) 2))/(\sin (\alpha)/(2)) \sum ^{n} _{k =0} \cos(\theta + k\alpha) = (\sin ((n +1 )\alpha)/(\cos(\theta + (n\alpha)/(2) ) 2))/(\sin (\alpha)/(2)) \sum ^{n -1} _{k =1} \sin (\pi k)/(= \cot n) (\pi)/(2 n) \sum ^{n -1} _{k =1} \sin (2 \pi k)/(n) =0 \sum ^{n -1} _{k =0} \csc ^{2 \left(\theta + (\pi k)/(n)} \right) = n ^{2\csc} ^{2(n\theta)} ^[6] \sum ^{n -1} _{k =1} \csc ^{2 (\pi k)/(n)} = (n ^{2-1})/(3) \sum ^{n -1} _{k =1} \csc ^{4 (\pi k)/(n)} = (n ^{4+10n} ^{2-11})/(45)

Rational functions

\sum ^{inf ty} _{n = a +1} (a)/(n ^{2} - a ^{2}) = (1)/(2) H _{2a} ^[7] \sum ^{inf ty (1)/(n ^{2+a} ^{2})} _{n =0} = (1+ a\pi\coth (a\pi))/(2 a ^{2}) \sum ^{inf ty (( -1 ) ^{n})/(n ^{2+a} ^{2})} _{n =0} = (1+ a\pi csch (a\pi))/(2 a ^{2}) \sum ^{inf ty (( 2 n +1 )( -1 ) ^{n})/(( 2 n +1 ) ^{2+a} ^{2})} _{n =0} = (\pi)/(4) sech \left( (a \pi)/(2) \right) \displaystyle \sum ^{inf ty} _{n =0} (1)/(n ^{4+4a} ^{4}) = \dfrac{ 1 }{ 8 a ^{4}+\dfrac{\pi(\sinh(2\pi} a) + \sin( 2 \pi a))}{ 8 a ^{3(\cosh(2\pi} a) - \cos( 2 \pi a))}

• An infinite series of any rational function of

n can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition,^[8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.

Exponential function

\displaystyle \dfrac{ 1 }{\sqrt{p}}\sum ^{p -1} _{n =0} \exp \left( (2 \pi i n ^{2} q)/(p) \right) = \dfrac{e ^{\pi} }\sum_^\exp \left(-\frac \right)(see the Landsberg–Schaar relation)\displaystyle \sum ^{inf ty} _{n =-inf ty} e ^{- \pi n ^{2}} = (\sqrt[ 4 ] \pi)/(\Gamma\left( (3)/(4 \right)))

Numeric series

These numeric series can be found by plugging in numbers from the series listed above.

Alternating harmonic series

\sum ^{inf ty} _{k =1} (( -1 ) ^{k+1})/(= k) (1)/(- 1) (1)/(+ 2) (1)/(- 3) (1)/(4) + … =ln 2 \sum ^{inf ty} _{k =1} (( -1 ) ^{k+1})/(= 2 k -1) (1)/(- 1) (1)/(+ 3) (1)/(- 5) (1)/(+ 7) (1)/(- … = 9) (\pi)/(4)

Sum of reciprocal of factorials

\sum ^{inf ty} _{k =0} (1)/(= k!) (1)/(+ 0 !) (1)/(+ 1 !) (1)/(+ 2 !) (1)/(+ 3 !) (1)/(4 !) + … = e \sum ^{inf ty} _{k =0} (1)/(= ( 2 k)!) (1)/(+ 0 !) (1)/(+ 2 !) (1)/(+ 4 !) (1)/(+ 6 !) (1)/(+ … = 8 !) (1)/(\left(e + 2) (1)/(e) \right) = \cosh 1 \sum ^{inf ty} _{k =0} (1)/(= ( 3 k)!) (1)/(+ 0 !) (1)/(+ 3 !) (1)/(+ 6 !) (1)/(+ 9 !) (1)/(+ … = 12 !) (1)/(\left(e + 3) (2)/(\sqrt{e) }\cos \frac\right)\sum ^{inf ty} _{k =0} (1)/(= ( 4 k)!) (1)/(+ 0 !) (1)/(+ 4 !) (1)/(+ 8 !) (1)/(+ 12 !) (1)/(+ … = 16 !) (1)/(2) \left(\cos 1+ \cosh 1 \right)

Trigonometry and π

\sum ^{inf ty} _{k =0} (( -1 ) ^{k})/(= ( 2 k +1 )!) (1)/(- 1 !) (1)/(+ 3 !) (1)/(- 5 !) (1)/(+ 7 !) (1)/(9 !) + … = \sin 1 \sum ^{inf ty} _{k =0} (( -1 ) ^{k})/(= ( 2 k)!) (1)/(- 0 !) (1)/(+ 2 !) (1)/(- 4 !) (1)/(+ 6 !) (1)/(8 !) + … = \cos 1 \sum ^{inf ty} _{k =1} (1)/(= k ^{2+1}) (1)/(+ 2) (1)/(+ 5) (1)/(+ 10) (1)/(+ … = 17) (1)/(2) (\pi \coth \pi - 1 ) \sum ^{inf ty} _{k =1} (( -1 ) ^{k})/(=- k ^{2+1}) (1)/(+ 2) (1)/(- 5) (1)/(+ 10) (1)/(+ … = 17) (1)/(2) (\pi \operatorname{csch} \pi - 1 ) 3 + (4)/(2 x 3 x 4) - (4)/(4 x 5 x 6) + (4)/(6 x 7 x 8) - (4)/(8 x 9 x 10) + … = \pi

Reciprocal of tetrahedral numbers

\sum ^{inf ty} _{k =1} (1)/(= Te _{k}) (1)/(+ 1) (1)/(+ 4) (1)/(+ 10) (1)/(+ 20) (1)/(+ … = 35) (3)/(2) Where Te ^{n} _{k =1} T _{k}

Exponential and logarithms

\sum ^{inf ty} _{k =0} (1)/(= ( 2 k +1 )( 2 k +2 )) (1)/(+ 1 x 2) (1)/(+ 3 x 4) (1)/(+ 5 x 6) (1)/(+ 7 x 8) (1)/(9 x 10) + … =ln 2 \sum ^{inf ty} _{k =1} (1)/(= 2 ^{kk}) (1)/(+ 2) (1)/(+ 8) (1)/(+ 24) (1)/(+ 64) (1)/(160) + … =ln 2 \sum ^{inf ty} _{k =1} (( -1 ) ^{k+1})/(2 ^{kk}) + \sum ^{inf ty} _{k =1} (( -1 ) ^{k+1})/(= ( 3 ^{kk}) (1)/(+ 2) (1)/() - ( 3) (1)/(+ 8) (1)/() + ( 18) (1)/(+ 24) (1)/() - ( 81) (1)/(+ 64) (1)/(324) ) + … =ln 2 \sum ^{inf ty} _{k =1} (1)/(3 ^{kk}) + \sum ^{inf ty} _{k =1} (1)/(= ( 4 ^{kk}) (1)/(+ 3) (1)/() + ( 4) (1)/(+ 18) (1)/() + ( 32) (1)/(+ 81) (1)/() + ( 192) (1)/(+ 324) (1)/(1024) ) + … =ln 2 \sum ^{inf ty} _{k =1} (1)/(=ln \left( n ^{kk}) (n)/(n -1) \right) , that is \forall n> 1

See also

• Series (mathematics)
• List of integrals
  • Taylor series
• Binomial theorem
• Gregory's series
• On-Line Encyclopedia of Integer Sequences

References

• Many books with a list of integrals also have a list of series.

Notes and References

1. Web site: Eric W. . Weisstein . Eric W. Weisstein . Haversine . . . 2015-11-06 . live . https://web.archive.org/web/20050310194740/http://mathworld.wolfram.com/Haversine.html . 2005-03-10.
2. Book: Wilf, Herbert R.. generatingfunctionology. 1994. Academic Press, Inc.
3. Web site: Theoretical computer science cheat sheet.
4. Calculate the Fourier expansion of the function f(x) = (\pi 4)/() on the interval 0 <x<\pi :(\pi 4= \sum)/(_{n=0}) ^{infty} c _{n\sin[nx]+d} _{n\cos[nx]} ⇒ \begin{cases}c _{n = \begin{cases} (1 n)/((n)} odd )\\ 0 (n even )\end{cases}\ d _{n=0} (\forall n)\end{cases}
5. Web site: Bernoulli polynomials: Series representations (subsection 06/02). 2 June 2011. Wolfram Research.
6. Web site: Hofbauer. Josef. A simple proof of 1 + 1/2² + 1/3² + &middot;&middot;&middot; = ²/6 and related identities. 2 June 2011.
7. Web site: Sondow. Jonathan. Weisstein. Eric W.. Riemann Zeta Function (eq. 52). MathWorld—A Wolfram Web Resource.
8. Book: http://people.math.sfu.ca/~cbm/aands/page_260.htm. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Milton. Abramowitz. Milton Abramowitz. Irene. Stegun. Irene Stegun. 1964. 0-486-61272-4. 260. 6.4 Polygamma functions. Courier Corporation .