Mathematical Methods (10/24.539)

VIII. Special Functions

Summary of Several Special Functions

As indicated, there are a number of special functions that occur frequently in many different fields of application. As a sample, a few of the more important functions and some of their properties are tabulated below. Note that orthogonality is a common characteristic for these special functions.

Legendre Polynomials

Differential Equation

(n is a non-negative integer)

Rodrique’s Formula

Generating Function

Recurrence Relation

Orthogonality

Associated Legendre Functions (for m = 0, these reduce to Legendre Polynomials)

Differential Equation

(n and m are non negative integers)

Rodrique’s Formula

Orthogonality

Hermite Polynomials

Differential Equation

(n is a non-negative integer)

Rodrique’s Formula

Generating Function

Recurrence Relation

Orthogonality

Laguerre Polynomials

Differential Equation

(n is a non-negative integer)

Rodrique’s Formula

Generating Function

Recurrence Relation

Orthogonality

Bessel Functions

Ordinary Bessel Equation

General Solution (ordinary)

Modified Bessel Equation

General Solution (modified)

Hankel Functions

Note: Several recurrence, derivative, and integral relationships for the Bessel functions are given in a subsequent subsection. Additional relationships and some specific examples are also given in later subsections. The orthogonality properties of the ordinary Bessel functions, which are somewhat complicated because of their relationship to the specified boundary conditions for a given problem, are also treated later (in Section IX The Sturm-Liouville Problem).

10/24.539 Lecture Notes by Dr. J. R. White, UMass-Lowell (updated October 1998).

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