Trigamma function

In mathematics, the trigamma function, denoted ψ1(z), is the second of the polygamma functions, and is defined by

\( \psi _{1}(z)={\frac {d^{2}}{dz^{2}}}\ln \Gamma (z). \)

It follows from this definition that

\( \psi _{1}(z)={\frac {d}{dz}}\psi (z) \)

where ψ(z) is the digamma function. It may also be defined as the sum of the series

\( \psi _{1}(z)=\sum _{{n=0}}^{{\infty }}{\frac {1}{(z+n)^{2}}}, \)

making it a special case of the Hurwitz zeta function

\( {\displaystyle \psi _{1}(z)=\zeta (2,z).} \)

Note that the last two formulas are valid when 1 − z is not a natural number.

Calculation

A double integral representation, as an alternative to the ones given above, may be derived from the series representation:

\( {\displaystyle \psi _{1}(z)=\int _{0}^{1}\!\!\int _{0}^{x}{\frac {x^{z-1}}{y(1-x)}}\,dx\,dy} \)

using the formula for the sum of a geometric series. Integration over y yields:

\( \psi _{1}(z)=-\int _{0}^{1}{\frac {x^{{z-1}}\ln {x}}{1-x}}\,dx \)

An asymptotic expansion as a Laurent series is

\( \psi _{1}(z)={\frac {1}{z}}+{\frac {1}{2z^{2}}}+\sum _{{k=1}}^{{\infty }}{\frac {B_{{2k}}}{z^{{2k+1}}}}=\sum _{{k=0}}^{{\infty }}{\frac {B_{k}}{z^{{k+1}}}} \)

if we have chosen B1 = 1/2, i.e. the Bernoulli numbers of the second kind.
Recurrence and reflection formulae

The trigamma function satisfies the recurrence relation

\( \psi _{1}(z+1)=\psi _{1}(z)-{\frac {1}{z^{2}}} \)

and the reflection formula

\( {\displaystyle \psi _{1}(1-z)+\psi _{1}(z)={\frac {\pi ^{2}}{\sin ^{2}\pi z}}\,} \)

which immediately gives the value for z = 1/2: \( {\displaystyle \psi _{1}({\tfrac {1}{2}})={\tfrac {\pi ^{2}}{2}}} \).
Special values

At positive half integer values we have that

\( {\displaystyle \psi _{1}\left(n+{\frac {1}{2}}\right)={\frac {\pi ^{2}}{2}}-4\sum _{k=1}^{n}{\frac {1}{(2k-1)^{2}}}.} \)

Moreover, the trigamma function has the following special values:

\( {\displaystyle {\begin{aligned}\psi _{1}\left({\tfrac {1}{4}}\right)&=\pi ^{2}+8G\quad &\psi _{1}\left({\tfrac {1}{2}}\right)&={\frac {\pi ^{2}}{2}}&\psi _{1}(1)&={\frac {\pi ^{2}}{6}}\\[6px]\psi _{1}\left({\tfrac {3}{2}}\right)&={\frac {\pi ^{2}}{2}}-4&\psi _{1}(2)&={\frac {\pi ^{2}}{6}}-1\quad \end{aligned}}} \)

where G represents Catalan's constant.

There are no roots on the real axis of ψ1, but there exist infinitely many pairs of roots zn, zn for Re z < 0. Each such pair of roots approaches Re zn = −n + 1/2 quickly and their imaginary part increases slowly logarithmic with n. For example, z1 = −0.4121345... + 0.5978119...i and z2 = −1.4455692... + 0.6992608...i are the first two roots with Im(z) > 0.
Relation to the Clausen function

The digamma function at rational arguments can be expressed in terms of trigonometric functions and logarithm by the digamma theorem. A similar result holds for the trigamma function but the circular functions are replaced by Clausen's function. Namely,[1]

\( {\displaystyle \psi _{1}\left({\frac {p}{q}}\right)={\frac {\pi ^{2}}{2\sin ^{2}(\pi p/q)}}+2q\sum _{m=1}^{(q-1)/2}\sin \left({\frac {2\pi mp}{q}}\right){\textrm {Cl}}_{2}\left({\frac {2\pi m}{q}}\right).} \)

Computation and approximation

An easy method to approximate the trigamma function is to take the derivative of the series expansion of the digamma function.

\( {\displaystyle \psi _{1}(x)\approx {\frac {1}{x}}+{\frac {1}{2x^{2}}}+{\frac {1}{6x^{3}}}-{\frac {1}{30x^{5}}}+{\frac {1}{42x^{7}}}-{\frac {1}{30x^{9}}}+{\frac {5}{66x^{11}}}-{\frac {691}{2730x^{13}}}+{\frac {7}{6x^{15}}}} \)

Appearance

The trigamma function appears in this surprising sum formula:[2]

\( {\displaystyle \sum _{n=1}^{\infty }{\frac {n^{2}-{\frac {1}{2}}}{\left(n^{2}+{\frac {1}{2}}\right)^{2}}}\left(\psi _{1}{\bigg (}n-{\frac {i}{\sqrt {2}}}{\bigg )}+\psi _{1}{\bigg (}n+{\frac {i}{\sqrt {2}}}{\bigg )}\right)=-1+{\frac {\sqrt {2}}{4}}\pi \coth {\frac {\pi }{\sqrt {2}}}-{\frac {3\pi ^{2}}{4\sinh ^{2}{\frac {\pi }{\sqrt {2}}}}}+{\frac {\pi ^{4}}{12\sinh ^{4}{\frac {\pi }{\sqrt {2}}}}}\left(5+\cosh \pi {\sqrt {2}}\right).}