Laguerre polynomials are defined as solutions of Laguerre's differential equation:

Solutions corresponding to the non-negative integer *n* can be expressed using Rodrigues' formula

or can be constructed using the three term recurrence relation:

The recurrence relation given above is the most efficient way to calculate the Laguerre polynomial. The **LaguerreCalculate** subroutine uses this relation to calculate *L _{n }(x)* for any given

The **LaguerreSum** subroutine calculates the sum of Laguerre polynomials *c _{0 }L_{0 }(x) + c_{1 }L_{1 }(x) + ... + c_{n }L_{n }(x)* using Clenshaw's recurrence formula.

The **LaguerreCoefficients** subroutine can represent *L _{n }(x)* as a sum of powers of

*This article is licensed for personal use only.*

ALGLIB Project offers you two editions of ALGLIB:

**ALGLIB Free Edition**:

delivered for free

offers full set of numerical functionality

extensive algorithmic optimizations

no low level optimizations

non-commercial license

**ALGLIB Commercial Edition**:

flexible pricing

offers full set of numerical functionality

extensive algorithmic optimizations

high performance (SMP, SIMD)

commercial license with support plan

Links to download sections for Free and Commercial editions can be found below:

C++ library.

Delivered with sources.

Monolithic design.

Extreme portability.

Delivered with sources.

Monolithic design.

Extreme portability.

C# library with native kernels.

Delivered with sources.

VB.NET and IronPython wrappers.

Extreme portability.

Delivered with sources.

VB.NET and IronPython wrappers.

Extreme portability.

Delphi wrapper around C core.

Delivered as precompiled binary.

Compatible with FreePascal.

Delivered as precompiled binary.

Compatible with FreePascal.

CPython wrapper around C core.

Delivered as precompiled binary.

Delivered as precompiled binary.