Abstract analytic number theory

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Trigamma function $\psi _{1}(z)$ in the complex plane. The color of a point $z$ encodes the value of $\psi _{1}(z)$. Strong colors denote values close to zero and hue encodes the value's argument.

In mathematics, the trigamma function, denoted $\psi _{1}(z)$, 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 $\psi (z)$ 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

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

Note that the last two formulæ 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:

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

using the formula for the sum of a geometric series. Integration by parts 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 $B_{1}=1/2$, 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

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

which immediately gives the value for z=1/2.

Special values

The trigamma function has the following special values:

$\psi _{1}\left({\frac {1}{4}}\right)=\pi ^{2}+8K$

$\psi _{1}\left({\frac {1}{2}}\right)={\frac {\pi ^{2}}{2}}$

$\psi _{1}(1)={\frac {\pi ^{2}}{6}}$

$\psi _{1}\left({\frac {3}{2}}\right)={\frac {\pi ^{2}}{2}}-4$

$\psi _{1}(2)={\frac {\pi ^{2}}{6}}-1$

where K represents Catalan's constant.

There are no roots on the real axis of $\psi _{1}$, but there exist infinitely many pairs of roots $z_{n},{\overline {z_{n}}}$ for $\Re (z)<0$. Each such pair of root approach $\Re (z_{n})=-n+1/2$ quickly and their imaginary part increases slowly logarithmic with n. E.g. $z_{1}=-0.4121345\ldots +i0.5978119\ldots$ and $z_{2}=-1.4455692\ldots +i0.6992608\ldots$ are the first two roots with $\Im (z)>0$.

Appearance

The trigamma function appears in the next surprising sum formula:

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

See also

• Gamma function
• Digamma function
• Polygamma function
• Catalan's constant

Notes

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References

• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 0-486-61272-4. See section §6.4
• Eric W. Weisstein. Trigamma Function -- from MathWorld--A Wolfram Web Resource

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