The polygamma function $ \psi^{(m)}(z) $ is a family of special functions defined as the $ m $-th derivative of the digamma function $ \psi(z) $, where $ m $ is a non‑negative integer and $ z $ is a complex variable not equal to a non‑positive integer. Formally
$$ \psi^{(m)}(z)=\frac{d^{,m}}{dz^{,m}}\psi(z)=\frac{d^{,m+1}}{dz^{,m+1}}\ln\Gamma(z), $$
with $ \psi(z)=\Gamma'(z)/\Gamma(z) $ and $ \Gamma(z) $ denoting the gamma function. The case $ m=0 $ corresponds to the digamma function, while $ m\ge 1 $ yields the higher‑order polygamma functions.
Series representations
For $ \Re(z)>0 $ and integer $ m\ge 0 $,
$$ \psi^{(m)}(z)=(-1)^{m+1},m!,\sum_{k=0}^{\infty}\frac{1}{(z+k)^{,m+1}} . $$
When $ m\ge 1 $ this series converges absolutely and provides a convenient computational formula.
Integral representations
For $ \Re(z)>0 $ and $ m\ge 1 $,
$$ \psi^{(m)}(z)=(-1)^{m+1}\int_{0}^{\infty}\frac{t^{,m}e^{-zt}}{1-e^{-t}},dt . $$
An alternative representation, valid for all $ m\ge 0 $, is
$$ \psi^{(m)}(z)=\int_{0}^{1}\frac{t^{z-1},\ln^{m}t}{1-t},dt . $$
Recurrence and reflection formulas
The polygamma functions satisfy the recurrence relation
$$ \psi^{(m)}(z+1)=\psi^{(m)}(z)+\frac{(-1)^{m}m!}{z^{,m+1}}, $$
which follows directly from the series definition.
A reflection formula, analogous to that for the digamma function, holds for $ m\ge 1 $:
$$ \psi^{(m)}(1-z)+(-1)^{m+1}\psi^{(m)}(z)=\pi^{,m+1}\frac{d^{,m}}{dz^{,m}}\cot(\pi z). $$
Special values
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Integer arguments: For a positive integer $ n $ and $ m\ge 1 $,
$$ \psi^{(m)}(n)=(-1)^{m+1}m!,\bigl(\zeta(m+1)-\sum_{k=1}^{n-1}\frac{1}{k^{,m+1}}\bigr), $$
where $ \zeta(s) $ denotes the Riemann zeta function.
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Half‑integer arguments: For $ n\in\mathbb{N} $,
$$ \psi^{(m)}!\left(n+\tfrac12\right)=(-1)^{m+1}m!,\bigl(2^{,m+1}-1\bigr),\zeta(m+1)-(-1)^{m+1}m!,\sum_{k=1}^{n}\frac{2^{,m+1}}{(2k-1)^{,m+1}} . $$
Relation to other functions
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The Hurwitz zeta function $ \zeta(s,q) $ provides an alternative expression:
$$ \psi^{(m)}(z)=(-1)^{m+1}m!,\zeta(m+1,z). $$
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For $ m=1 $ the function is frequently called the trigamma function, and for $ m=2 $ the tetragamma function.
Applications
Polygamma functions appear in numerous areas of mathematics and physics:
- Series and asymptotic expansions: They are used to derive expansions of the gamma function, harmonic numbers, and related combinatorial quantities.
- Number theory: Through their connection with the Hurwitz and Riemann zeta functions, they occur in evaluations of series involving reciprocal powers.
- Statistical distributions: In the theory of the beta and Dirichlet distributions, moments and Fisher information matrices involve polygamma functions.
- Quantum field theory and perturbative calculations: Certain loop integrals and renormalization-group equations yield expressions containing polygamma functions.
- Algorithmic computation: High‑precision libraries for special functions implement efficient evaluation of $ \psi^{(m)}(z) $ based on the series, recurrence, and asymptotic formulas.
Asymptotic behavior
For large $|z|$ with $|\arg z|<\pi$,
$$ \psi^{(m)}(z)\sim (-1)^{m+1} \frac{m!}{z^{,m+1}}\left(1+\frac{(m+1)}{2z}+ \frac{(m+1)(m+2)}{12z^{2}}+\cdots\right), $$
providing a useful approximation in numerical contexts.
References
- Abramowitz, M.; Stegun, I. A. (1965). Handbook of Mathematical Functions. Dover Publications.
- Artin, E. (1964). The Gamma Function. Holt, Rinehart and Winston.
- Whittaker, E. T.; Watson, G. N. (1996). A Course of Modern Analysis (4th ed.). Cambridge University Press.
These sources contain detailed proofs of the properties summarized above and further discussion of the role of the polygamma function within analytic number theory and applied mathematics.
