Hypergeometric 2F1 Approximation

Compute finite Gauss 2F1(a,b;c;x) power series with explicit included term count, last term and observed numerator termination; denominator poles rejected, last term is not a tail bound or convergence certificate.

Description

Gauss 2F1 is defined by a power series with rising-factorial weights.

Gauss 2F1 is defined by a power series with rising-factorial weights. A small last term alone is not an upper bound for the whole omitted tail.^{[1, 2]}

When to use Hypergeometric 2F1 Approximation

Use this tool to compute finite Gauss 2F1(a,b;c;x) power series with explicit included term count, last term and observed numerator termination; denominator poles rejected, last term is not a tail bound or convergence certificate.

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

How to use Hypergeometric 2F1 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 a, c, x, seriesTerms, b using the field table and calculation rules, then run the tool.
Read finiteHypergeometricSum, lastIncludedTermMagnitude, polynomialTerminationObserved. 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
a	A	number	Required
c	C	number	Required
x	X	number	Required
seriesTerms	Series terms	number	Required
b	B	number	Required

Use |a|,|b|,|c|≤100, |x|≤0.95 and integer total terms 1–1000. Reject nonpositive integer c. Sum up to the supplied count including t0; mark termination only when a zero numerator factor is encountered during iteration. Return sum, last-term magnitude and that flag.

t0=1; t_k=t_(k−1)(a+k−1)(b+k−1)x/[(c+k−1)k]

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

Output field	Meaning	Value type
finiteHypergeometricSum	Finite hypergeometric sum	number
lastIncludedTermMagnitude	Last included term magnitude	number
polynomialTerminationObserved	Polynomial termination observed	boolean

json

{
  "input": {
    "a": 1,
    "c": 2,
    "x": 0.5,
    "seriesTerms": 4,
    "b": 1
  },
  "output": {
    "finiteHypergeometricSum": 1.3645833333333333,
    "lastIncludedTermMagnitude": 0.03125,
    "polynomialTerminationObserved": false
  }
}

Limitations and assumptions

There is no analytic continuation or certified total truncation bound. Insufficient requested terms can leave an eventual polynomial termination unobserved.

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. A successfully returned false flag is an evaluated outcome; its meaning is explained above.

Alternative or Complementary methods

Use Confluent Hypergeometric 1F1 Approximation: Compute finite confluent 1F1(a;c;x) power series with explicit included term count, last term and observed numerator termination; denominator poles rejected, last term is not a tail bound or convergence certificate. For a complementary task, use Struve Function Approximation: Evaluate Struve H_ν real power series for integer order 0..5 and x∈[0,20]; fixed family/order convention, bounded convergence, no ambiguous unnamed special function.

Further background: Hypergeometric function: definitions.

References

Hypergeometric 2F1 Approximation — tool.io
Hypergeometric function: definitions — NIST Digital Library of Mathematical Functions

Similar or alternative tools

Confluent Hypergeometric 1F1 Approximation
Compute finite confluent 1F1(a;c;x) power series with explicit included term count, last term and observed numerator termination; denominator poles rejected, last term is not a tail bound or convergence certificate.
Struve Function Approximation
Evaluate Struve H_ν real power series for integer order 0..5 and x∈[0,20]; fixed family/order convention, bounded convergence, no ambiguous unnamed special function.