Polygamma Functions

📐 Definition

The polygamma functions are defined as the successive derivatives of the digamma function, or equivalently as higher logarithmic derivatives of the Gamma function.

For integers $n \ge 0$, the $n$-th polygamma function is defined by

$$\psi^{(n)}(z) = \frac{d^{\,n+1}}{dz^{\,n+1}} \log \Gamma(z)$$

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Special cases include:

• $n = 0$: digamma function $\psi(z)$,
• $n = 1$: trigamma function $\psi'(z)$,
• $n = 2$: tetragamma function $\psi''(z)$.

Domain & Codomain

Meromorphic on $\mathbb{C}$ with poles at the non-positive integers $\{0, -1, -2, \dots\}$

Values in $\mathbb{C}$

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⚙️ Key Properties

Series Representation

For $z \notin \{0,-1,-2,\dots\}$ and integers $n \ge 1$, the polygamma functions admit the convergent series representation

$$\psi^{(n)}(z) = (-1)^{n+1} n! \sum_{k=0}^{\infty} \frac{1}{(k+z)^{n+1}}$$

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This representation highlights their close connection to generalized harmonic sums.

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Relation to the Hurwitz Zeta Function

The polygamma functions are directly related to the Hurwitz zeta function:

$$\psi^{(n)}(z) = (-1)^{n+1} n!\, \zeta(n+1, z)$$

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This identity places polygamma functions at the intersection of special functions and analytic number theory.

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Recurrence Relation

The recurrence relation generalizing that of the digamma function is

$$\psi^{(n)}(z+1) = \psi^{(n)}(z) + (-1)^n \frac{n!}{z^{n+1}}$$

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This identity is useful for numerical evaluation and analytic continuation.

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Poles and Principal Parts

Each $\psi^{(n)}(z)$ has poles of order $n+1$ at

$$z = 0, -1, -2, \dots$$

Near a pole $z = -m$, the leading singular behavior is

$$\psi^{(n)}(z) \sim (-1)^{n+1} \frac{n!}{(z+m)^{n+1}}$$

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Asymptotic Behavior

As $|z| \to \infty$ with $|\arg z| < \pi$, the polygamma functions satisfy

$$\psi^{(n)}(z) \sim (-1)^{n-1} \frac{(n-1)!}{z^n} \left( 1 + \frac{n}{2z} + \mathcal{O}(z^{-2}) \right)$$

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Higher-order terms involve Bernoulli numbers.

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🎯 Special Cases

Trigamma Function ($n = 1$)

The trigamma function is the first derivative of the digamma function:

$$\psi'(z) = \frac{d^2}{dz^2} \log \Gamma(z)$$

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It has the series representation

$$\psi'(z) = \sum_{k=0}^{\infty} \frac{1}{(k+z)^2}$$

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For real $x > 0$, the trigamma function is strictly positive and monotonically decreasing, reflecting the convexity of $\log \Gamma(x)$.

Special Values

The trigamma function $\psi'(z)$ has several important closed-form values.

$z$	$\psi'(z)$ (exact)	Approximate value
$1$	$\tfrac{\pi^2}{6}$	$1.6449340668$
$2$	$\tfrac{\pi^2}{6} - 1$	$0.6449340668$
$3$	$\tfrac{\pi^2}{6} - \tfrac{5}{4}$	$0.3949340668$
$\tfrac12$	$\tfrac{\pi^2}{2}$	$4.9348022005$
$n \in \mathbb{N}$	$\zeta(2) - \sum_{k=1}^{n-1} \tfrac{1}{k^2}$	—

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Tetragamma Function ($n = 2$)

The tetragamma function is the second derivative of the digamma function:

$$\psi''(z) = \frac{d^3}{dz^3} \log \Gamma(z)$$

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It admits the representation

$$\psi''(z) = -2 \sum_{k=0}^{\infty} \frac{1}{(k+z)^3}$$

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For real $x > 0$, $\psi''(x)$ is strictly negative.

Special Values

The tetragamma function $\psi''(z)$ admits the following special values:

$z$	$\psi''(z)$ (exact)	Approximate value
$1$	$-2\zeta(3)$	$-2.4041138063$
$2$	$-2\zeta(3) + 2$	$-0.4041138063$
$\tfrac12$	$-14\zeta(3)$	$-16.8287966442$
$n \in \mathbb{N}$	$-2\!\left(\zeta(3) - \sum_{k=1}^{n-1} \tfrac{1}{k^3}\right)$	—

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Values at Integers

For integers $m \ge 1$ and $n \in \mathbb{N}$,

$$\psi^{(m)}(n) = (-1)^{m+1} m! \left( \zeta(m+1) - \sum_{k=1}^{n-1} \frac{1}{k^{m+1}} \right)$$

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These formulas are frequently used in exact and high-precision computation.

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🔗 Related Functions

Gamma Function Polygamma functions are logarithmic derivatives of Γ(z)

Digamma Function The n=0 case: ψ(z) = ψ^(0)(z)

Riemann Zeta Function Related via ψ^(n)(z) = (-1)^(n+1) n! ζ(n+1, z)

Stirling Series Provides asymptotic expansions for large arguments

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Usage in Oakfield

Applications in Oakfield

Oakfield uses polygamma functions as part of its special-function and warp implementation:

• Trigamma and tetragamma are implemented (real and complex) in the special-function core and used by the analytic_warp operator when evaluating gradients for DIGAMMA/TRIGAMMA profiles
• Safe evaluation paths include reflection/recurrence and guarded asymptotics; callers can route failures through fallback handlers and receive diagnostic reports
• Tetragamma is not currently exposed as a standalone Lua function, but it is used internally for warp gradients

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Historical Foundations

1. 📜 Early Development

   Euler introduced logarithmic derivatives of the Gamma function in his work on harmonic series. Gauss and later analysts systematized their higher derivatives within complex analysis.

2. 🔬 Modern Usage

   Polygamma functions are now standard tools in statistics, probability theory, and computational mathematics, particularly in variance, curvature, and optimization problems.

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📚 References

• NIST Digital Library of Mathematical Functions (DLMF), §5.15
• Abramowitz & Stegun, Handbook of Mathematical Functions
• Olver et al., NIST Handbook of Mathematical Functions