# What is the generating function of Legendre polynomial?

## What is the generating function of Legendre polynomial?

The Legendre polynomials can be alternatively given by the generating function ( 1 − 2 x z + z 2 ) − 1 / 2 = ∑ n = 0 ∞ P n ( x ) z n , but there are other generating functions.

## What is the Legendre differential equation?

Legendre’s differential equation has the form (1 − x2)y − 2xy + l(l + 1)y = 0, (2) where the parameter l, which is a real number, (we take l = 0,1,2,ททท), is called the degree.

**Are Legendre polynomials even?**

One of the varieties of special functions which are encountered in the solution of physical problems is the class of functions called Legendre polynomials. They are solutions to a very important differential equation, the Legendre equation: The polynomials are either even or odd functions of x for even or odd orders n.

**Why are Legendre polynomials useful?**

For example, Legendre and Associate Legendre polynomials are widely used in the determination of wave functions of electrons in the orbits of an atom [3], [4] and in the determination of potential functions in the spherically symmetric geometry [5], etc.

### Are Legendre polynomials linearly independent?

Any polynomial of degree m can be represented as a linear combination of Legendre polynomials of degree at most m. show that the legendre polynomials of degree ≤ n, are linearly independent, and thus form a basis for all polynomials of degree ≤ n.

### What is Legendre differential equation?

Since the Legendre differential equation is a second-order ordinary differential equation, it has two linearly independent solutions. A solution which is regular at finite points is called a Legendre function of the first kind, while a solution which is singular at is called a Legendre function of the second kind.

**What are the coefficients of the Legendre polynomials?**

The Legendre polynomials can also be defined as the coefficients in a formal expansion in powers of t {\\displaystyle t} of the generating function. 1 1 − 2 x t + t 2 = ∑ n = 0 ∞ P n ( x ) t n .

**How are Legendre polynomials generated in orthonormalization?**

The Legendre polynomials can also be generated using Gram-Schmidt orthonormalization in the open interval with the weighting function 1. Normalizing so that gives the expected Legendre polynomials. The “shifted” Legendre polynomials are a set of functions analogous to the Legendre polynomials, but defined on the interval (0, 1).

## Where do Legendre polynomials occur in Laplace’s equation?

Legendre polynomials occur in the solution of Laplace’s equation of the static potential, ∇2 Φ(x) = 0, in a charge-free region of space, using the method of separation of variables, where the boundary conditions have axial symmetry (no dependence on an azimuthal angle ).

## How are Legendre polynomials used in Newtonian potential expansion?

Applications of Legendre polynomials Expanding a 1/r potential The Legendre polynomials were first introduced in 1782 by Adrien-Marie Legendre as the coefficients in the expansion of the Newtonian potential where r and r′ are the lengths of the vectors x and x′ respectively and γ is the angle between those two vectors.