Trigammas

Trigammas is a term that can refer to the trigamma function, considered as a family of values across different arguments, or more generally to instances of the second polygamma function evaluated at various inputs. The trigamma function, usually denoted ψ1(z), is the derivative of the digamma function ψ(z) and the second derivative of the logarithm of the gamma function: ψ1(z) = d^2/dz^2 log Γ(z) = ψ′(z).

A standard set of representations expresses ψ1(z) in several equivalent ways. It has a series representation for Re(z) > 0: ψ1(z) = ∑_{n=0}^∞ 1/(z+n)^2. It can be written as a Hurwitz zeta function: ψ1(z) = ζ(2, z). An integral form is ψ1(z) = ∫_0^∞ t e^{-zt} /(1 − e^{-t}) dt, valid for Re(z) > 0. Key identities include the recurrence ψ1(z+1) = ψ1(z) − 1/z^2 and the reflection formula ψ1(1−z) + ψ1(z) = π^2 csc^2(π z).

Special values are often tabulated: ψ1(1) = π^2/6 and ψ1(1/2) = π^2/2. For positive integers n, ψ1(n) can be related to partial sums of reciprocal squares via the series representation.

Applications of trigammas appear in numerical analysis and statistics. They arise in the differentiation of log gamma expressions, in the evaluation of Fisher information and other derivatives of log-likelihoods involving the gamma or Dirichlet families, and in various algorithms for evaluating special functions. The plural form “trigammas” is typically used to refer to the function evaluated at multiple arguments or to a family of such values. See also the trigamma function and polygamma functions.