Trigamma function

In mathematics, the trigamma function, denoted $ψ 1 (z)$ , is the second of the polygamma functions, and is defined by

It follows from this definition that

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

making it a special case of the Hurwitz zeta function

Note that the last two formulae are valid when $1 - z$ is not a natural number.

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

using the formula for the sum of a geometric series. Integration by parts yields:

An asymptotic expansion as a Laurent series is

if we have chosen $B 1 = 1 / 2$ , i.e. the Bernoulli numbers of the second kind.

The trigamma function satisfies the recurrence relation

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