trigamma function

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

: $\backslash psi\_1(z)\; =\; \backslash int\_0^1\backslash !\backslash !\backslash int\_0^x\backslash frac\{x^\{z-1\}\}\{y(1\; -\; x)\}\backslash ,dy\backslash ,dx$

using the formula for the sum of a geometric series. Integration over {{math|y}} yields:

: $\backslash psi\_1(z)\; =\; -\backslash int\_0^1\backslash frac\{x^\{z-1\}\backslash ln\{x\}\}\{1-x\}\backslash ,dx$

An asymptotic expansion as a Laurent series can be obtained via the derivative of the asymptotic expansion of the digamma function:

:$\backslash begin\{align\}$

\psi_1(z)

&\sim {\operatorname{d}\over\operatorname{d}\!z} \left(\ln z - \sum_{n=1}^\infty \frac{B_n}{nz^n}\right) \\

&= \frac{1}{z} + \sum_{n=1}^\infty \frac{B_n}{z^{n+1}} = \sum_{n=0}^{\infty}\frac{B_n}{z^{n+1}} \\

&= \frac{1}{z} + \frac{1}{2z^2} + \frac{1}{6z^3} - \frac{1}{30z^5} + \frac{1}{42z^7} - \frac{1}{30z^9} + \frac{5}{66z^{11}} - \frac{691}{2730z^{13}} + \frac{7}{6z^{15}} \cdots

\end{align}

where {{mvar|B_n}} is the {{mvar|n}}th Bernoulli number and we choose {{math|B₁ {{=}} {{sfrac|1|2}}}}.

The trigamma function satisfies the recurrence relation

: $\backslash psi\_1(z\; +\; 1)\; =\; \backslash psi\_1(z)\; -\; \backslash frac\{1\}\{z^2\}$

and the reflection formula

: $\backslash psi\_1(1\; -\; z)\; +\; \backslash psi\_1(z)\; =\; \backslash frac\{\backslash pi^2\}\{\backslash sin^2\; \backslash pi\; z\}\; \backslash ,$

which immediately gives the value for z {{=}} {{sfrac|1|2}}: $\backslash psi\_1(\backslash tfrac\{1\}\{2\})=\backslash tfrac\{\backslash pi^2\}\{2\}$.

At positive integer values we have that

:$$

\psi_1(n) = \frac{\pi^2}{6} - \sum_{k=1}^{n-1} \frac{1}{k^2}, \qquad \psi_1(1) = \frac{\pi^2}{6}, \qquad \psi_1(2) = \frac{\pi^2}{6} - 1, \qquad \psi_1(3) = \frac{\pi^2}{6} - \frac{5}{4}.

At positive half integer values we have that

:$$

\psi_1\left(n+\frac12\right)=\frac{\pi^2}{2}-4\sum_{k=1}^n\frac{1}{(2k-1)^2},

\qquad \psi_1\left(\tfrac12\right) = \frac{\pi^2}{2},

\qquad \psi_1\left(\tfrac32\right) = \frac{\pi^2}{2} - 4 .

The trigamma function has other special values such as:

: $$

\psi_1\left(\tfrac14\right) = \pi^2 + 8G

where {{mvar|G}} represents Catalan's constant.

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

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,{{Cite book|title=Structural properties of polylogarithms|editor-last=Lewin|editor-first=L. |publisher=American Mathematical Society|year=1991|isbn=978-0821816349}}

:$$

\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).

The trigamma function appears in this sum formula:

: $\backslash sum\_\{n=1\}^\backslash infty\backslash frac\{n^2-\backslash frac12\}\{\backslash left(n^2+\backslash frac12\backslash right)^2\}\backslash left(\backslash psi\_1\backslash bigg(n-\backslash frac\{i\}\{\backslash sqrt\{2\}\}\backslash bigg)+\backslash psi\_1\backslash bigg(n+\backslash frac\{i\}\{\backslash sqrt\{2\}\}\backslash bigg)\backslash right)=$

-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).

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

{{reflist|refs=

{{Cite journal

| last1 = Mező | first1 = István

| title = Some infinite sums arising from the Weierstrass Product Theorem

| journal = Applied Mathematics and Computation

| volume = 219

| issue = 18

| pages = 9838–9846

| year = 2013

| doi=10.1016/j.amc.2013.03.122

}}

}}

• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. {{ISBN|0-486-61272-4}}. See section [http://www.math.sfu.ca/~cbm/aands/page_260.htm §6.4]
• Eric W. Weisstein. [http://mathworld.wolfram.com/TrigammaFunction.html Trigamma Function -- from MathWorld--A Wolfram Web Resource]

Category:Gamma and related functions