What is the generating function for Legendre polynomial?

What is the generating function for 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 Legendre function used for?

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.

What is meant by generating function?

In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence.

What is Fourier Legendre series?

Therefore, the Legendre polynomial series is a type of Fourier Series written in the system of orthogonal polynomials. The partial sums of a Legendre series bring the functions f(x) closer in the sense of a root-mean-square deviation and the condition limn→ ∞ cn = 0 is satisfied.

What is Legendre function of first kind?

real, the Legendre function of the first kind simplifies to a polynomial, called the Legendre polynomial. The associated Legendre function of first kind is given by the Wolfram Language command LegendreP[n, m, z], and the unassociated function by LegendreP[n, z].

What is generating function with example?

The generating function for 1,2,3,4,5,… is 1(1−x)2. Take a second derivative: 2(1−x)3=2+6x+12×2+20×3+⋯. So 1(1−x)3=1+3x+6×2+10×3+⋯ is a generating function for the triangular numbers, 1,3,6,10… (although here we have a0=1 while T0=0 usually).

How do you find a generating function?

To find the generating function for a sequence means to find a closed form formula for f(x), one that has no ellipses. (for all x less than 1 in absolute value). Problem: Suppose f(x) is the generating function for a and g(x) is the generating function for b.

What do you mean by Legendre polynomial?

In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applications.

What is the singular point of Legendre differential equation?

Hence the Legendre Equation has regular singular point at X=I1. 1. Z=0 is a singular point of the Legendre Equations but since 2 P(z) and Z² A(z) are well-behaved this is a regular singular point.

What is the formula for for generating function?

The generating function associated to the class of binary sequences (where the size of a sequence is its length) is A(x) = ∑n≥0 2nxn since there are an = 2n binary sequences of size n. (k n ) xn = (1 + x)k. Here the second equality uses the binomial theorem.

What is the generating function of generating Series 12345?

What is the generating function for generating series 1, 2, 3, 4, 5,…? Explanation: Basic generating function is \frac{1}{1-x}. If we differentiate term by term in the power series, we get (1 + x + x2 + x3 +⋯)′ = 1 + 2x + 3×2 + 4×3 +⋯ which is the generating series for 1, 2, 3, 4,…. 4.

How do you generate the next Legendre polynomial?

The recurrence relations obtained are often the best way to generate the next Legendre polynomial if you have two, i.e., you can take P 0(x) and P

Why are recurrence relations important in calculus?

These relations are useful and their derivation using the re- currence relation is a useful exercise in manipulating series, but none of the material in this section is essential. The recurrence relations obtained are often the best way to generate the next Legendre polynomial if you have two, i.e., you can take P

How do you find the Legendre series of a function?

This is called a Legendre series. To calculate the C n’s, we write Z1 1 f(x)P n(x)dx = X1 n=0 Z1 1 P m(x)P n(x)dx C n= Z1 1 P2 m (x) C m. (8.12) 91 Let’s calculate the integral using the generating function: 1 1 22xt+t = ” X1 n=0

What is the relation between P2 and Legendre polynomial P2(x)?

The relation isn’t correct. For n = 1 the formula gives P2(x) = 1 2(x2 − 2x − 1), but the Legendre polynomial P2 is P2(x) = 1 2(3×2 − 1). Thanks for contributing an answer to Mathematics Stack Exchange!