Trigamma function

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For Barnes's gamma function of 3 variables, see triple gamma function.
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[edit]

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[edit]

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[edit]

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 + i 0.5978119\ldots and z_2 = -1.4455692\ldots + i 0.6992608\ldots are the first two roots with \Im(z) > 0.

Appearance[edit]

The trigamma function appears in the next surprising sum formula:[1]

 \sum_{n=1}^\infty\frac{n^2-\frac12}{\left(n^2+\frac12\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[edit]

Notes[edit]

  1. ^ Mező, István (2013). "Some infinite sums arising from the Weierstrass Product Theorem". Applied Mathematics and Computation 219: 9838–9846. doi:10.1016/j.amc.2013.03.122. 

References[edit]

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