The trigamma function, denoted ψ₁(z) or sometimes ψ^{(1)}(z), is the second derivative of the natural logarithm of the gamma function Γ(z). Equivalently, it is the first derivative of the digamma function ψ(z) = d/dz ln Γ(z). Formally,
$$ \psi_{1}(z)=\frac{d^{2}}{dz^{2}}\ln\Gamma(z)=\frac{d}{dz}\psi(z). $$
It belongs to the family of polygamma functions ψ^{(m)}(z), where m ≥ 0 is an integer; the case m = 0 yields the digamma function, and m = 1 yields the trigamma function.
Definition and Series Representation
For complex arguments z not equal to a non‑positive integer, the trigamma function can be expressed by the convergent series
$$ \psi_{1}(z)=\sum_{n=0}^{\infty}\frac{1}{(z+n)^{2}}, $$
which follows from termwise differentiation of the logarithmic derivative of the Euler product for Γ(z). An alternative representation using the Hurwitz zeta function ζ(s,a) is
$$ \psi_{1}(z)=\zeta(2,,z), $$
where ζ(s,a)=∑_{n=0}^{\infty}(n+a)^{-s} for Re(s) > 1.
Integral Representations
The trigamma function admits several integral forms, for example
$$ \psi_{1}(z)=\int_{0}^{\infty}\frac{t,e^{-zt}}{1-e^{-t}},dt, \qquad \Re(z)>0, $$
and
$$ \psi_{1}(z)=\int_{0}^{1}\frac{x^{z-1}}{1-x},dx, \qquad \Re(z)>0, $$
which emphasize its positivity on the positive real axis.
Basic Properties
-
Recurrence relation
$$ \psi_{1}(z+1)=\psi_{1}(z)-\frac{1}{z^{2}}. $$ -
Reflection formula
$$ \psi_{1}(1-z)+\psi_{1}(z)=\pi^{2}\csc^{2}(\pi z). $$ -
Asymptotic expansion (for |z|→∞, |\arg z|<π)
$$ \psi_{1}(z)\sim\frac{1}{z}+\frac{1}{2z^{2}}+\sum_{k=1}^{\infty}\frac{B_{2k}}{z^{2k+1}}, $$ where Bₙ are Bernoulli numbers. -
Positivity
For real x > 0, ψ₁(x) > 0; consequently ψ(x) is strictly increasing on (0,∞).
Special Values
- ψ₁(1) = π²/6.
- ψ₁(½) = π²/2.
- More generally, ψ₁(n) = π²/6 - ∑_{k=1}^{n-1} 1/k² for integer n ≥ 2.
Applications
The trigamma function appears in a variety of mathematical and statistical contexts:
- Probability and statistics – variance of the logarithm of a Gamma‑distributed variable, Fisher information for certain families (e.g., the Gamma and Dirichlet distributions), and in the calculation of digamma‑based estimators.
- Number theory – connections with the Riemann zeta function through the series representation.
- Analysis – evaluation of series and integrals involving rational functions of integers; in particular, sums of reciprocal squares are expressed via ψ₁.
- Physics – quantum field theory and statistical mechanics, where polygamma functions arise in regularization techniques and in the computation of thermodynamic potentials.
Computational Aspects
Numerical evaluation of ψ₁(z) is implemented in standard mathematical software libraries (e.g., the GNU Scientific Library, MATLAB, Mathematica, and the SciPy library for Python). Algorithms typically combine recurrence to shift the argument into a region where series or asymptotic expansions converge rapidly.
Relation to Other Functions
The trigamma function is the simplest non‑trivial member of the polygamma hierarchy:
$$ \psi^{(m)}(z)=\frac{d^{m+1}}{dz^{m+1}}\ln\Gamma(z)=(-1)^{m+1}m!\sum_{n=0}^{\infty}\frac{1}{(z+n)^{m+1}},\qquad m\ge0. $$
Thus ψ₁(z)=ψ^{(1)}(z). It can also be written as the derivative of the digamma function: ψ₁(z)=ψ'(z).
References
- Abramowitz, M.; Stegun, I. A. (1965). Handbook of Mathematical Functions. Dover Publications.
- Artin, E. (1964). The Gamma Function. Holt, Rinehart and Winston.
- Olver, F. W. J.; et al. (2010). NIST Handbook of Mathematical Functions. Cambridge University Press.