Laguerre Polynomial Approximation

Evaluate laguerre family polynomial by three-term recurrence for degree 0..128 and |x|≤100; family parameters/domain explicit, not arbitrary caller polynomial evaluation.

Description

Generalized Laguerre polynomials depend on an alpha parameter as well as degree.

Generalized Laguerre polynomials depend on an alpha parameter as well as degree. Their normalization and recurrence differ from Legendre or Hermite polynomials.^{[1, 2]}

Evaluate laguerre family polynomial by three-term recurrence for degree 0..128 and |x|≤100; family parameters/domain explicit, not arbitrary caller polynomial evaluation.

When to use Laguerre Polynomial Approximation

Use this tool to evaluate laguerre family polynomial by three-term recurrence for degree 0..128 and |x|≤100; family parameters/domain explicit, not arbitrary caller polynomial evaluation.

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

How to use Laguerre Polynomial 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 degree, x, alpha using the field table and calculation rules, then run the tool.
Read polynomialValue. 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
degree	Degree	number	Required
x	X	number	Required
alpha	Alpha	number	Required

Use integer degree 0–128, |x|≤100 and −1<alpha≤100. Advance the three-term recurrence and return its floating-point value.

L0=1, L1=1+alpha−x; L_(k+1)=[(2k+1+alpha−x)L_k−(k+alpha)L_(k−1)]/(k+1)

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

Output field	Meaning	Value type
polynomialValue	Polynomial value	number

json

{
  "input": {
    "degree": 2,
    "x": 2,
    "alpha": 0
  },
  "output": {
    "polynomialValue": -1
  }
}

Limitations and assumptions

No arbitrary caller polynomial or negative-parameter continuation is evaluated. Overflow is rejected.

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 Jacobi Polynomial Approximation: Evaluate jacobi family polynomial by three-term recurrence for degree 0..128 and |x|≤100; family parameters/domain explicit, not arbitrary caller polynomial evaluation. For a complementary task, use Hermite Polynomial Approximation: Evaluate hermite family polynomial by three-term recurrence for degree 0..128 and |x|≤100; physicists H_n convention.

Further background: Classical orthogonal polynomials: recurrence relations.

References

Laguerre Polynomial Approximation — tool.io
Classical orthogonal polynomials: recurrence relations — NIST Digital Library of Mathematical Functions

Similar or alternative tools

Jacobi Polynomial Approximation
Evaluate jacobi family polynomial by three-term recurrence for degree 0..128 and |x|≤100; family parameters/domain explicit, not arbitrary caller polynomial evaluation.
Hermite Polynomial Approximation
Evaluate hermite family polynomial by three-term recurrence for degree 0..128 and |x|≤100; physicists H_n convention.