Polygamma Approximation

Approximate ψ^(m)(x) for positive x and integer m=1..6 via finite absolutely convergent reciprocal-power series, with integral-test absolute tail upper bound; finite truncation not hidden exact value.

Description

Higher polygamma derivatives have absolutely convergent reciprocal-power series for positive arguments.

Higher polygamma derivatives have absolutely convergent reciprocal-power series for positive arguments. Truncation can be bounded separately from the signed approximation.^{[1, 2]}

When to use Polygamma Approximation

Use this tool to approximate ψ^(m)(x) for positive x and integer m=1..6 via finite absolutely convergent reciprocal-power series, with integral-test absolute tail upper bound; finite truncation not hidden exact value.

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.

How to use Polygamma Approximation and understand its results

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
Enter derivativeOrder, x, seriesTerms using the field table and calculation rules, then run the tool.
Read truncatedPolygammaValue, absoluteTailUpperBound. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
derivativeOrder	Derivative order	number	Required
x	X	number	Required
seriesTerms	Series terms	number	Required

Use integer derivative order 1–6, 1–100,000 terms and 0<x≤one million. Return the signed partial sum and integral-test tail bound m!(x+N−1)^(−m)/m.

ψ^(m)(x) ≈ (−1)^(m+1)m! Σ[k=0..N−1](x+k)^(−m−1)

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
truncatedPolygammaValue	Truncated polygamma value	number
absoluteTailUpperBound	Absolute tail upper bound	number

json

{
  "input": {
    "derivativeOrder": 1,
    "x": 1,
    "seriesTerms": 3
  },
  "output": {
    "truncatedPolygammaValue": 1.3611111111111112,
    "absoluteTailUpperBound": 0.3333333333333333
  }
}

Limitations and assumptions

The bound covers omitted series terms, not floating-point roundoff. The digamma case m=0 uses a different tool.

Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use Digamma Approximation: Evaluate real positive digamma ψ(x) using upward recurrence and Bernoulli-corrected large-argument expansion, not finite-difference differentiation or negative-pole continuation. For a complementary task, use Gamma Function Approximation: Evaluate positive real Γ(x) and log Γ(x) by Lanczos/reflection approximation over bounded finite domain; negative poles and complex continuation intentionally excluded.

Further background: Gamma function: definitions.

References

Polygamma Approximation — tool.io
Gamma function: definitions — NIST Digital Library of Mathematical Functions

Similar or alternative tools

Digamma Approximation
Evaluate real positive digamma ψ(x) using upward recurrence and Bernoulli-corrected large-argument expansion, not finite-difference differentiation or negative-pole continuation.
Gamma Function Approximation
Evaluate positive real Γ(x) and log Γ(x) by Lanczos/reflection approximation over bounded finite domain; negative poles and complex continuation intentionally excluded.