Orthogonal properties of associated laguerre polynomial

Thread starter Arafat Sagar
Start date Jul 21, 2012
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Laguerre Orthogonal Polynomial Properties

In summary, orthogonal properties refer to the property that two functions are perpendicular to each other when their inner product is equal to zero. Associated Laguerre polynomials are derived from the Laguerre polynomials through a process called Rodrigues' formula and have various applications in mathematics, physics, and engineering. They are also used to express the probability density functions and calculate moments for probability distributions. Associated Laguerre polynomials and Hermite polynomials are both orthogonal families of polynomials but differ in their weight functions and domains.

Jul 21, 2012

Arafat Sagar

i need the derivation of orthogonal properties of associated laguerre polynomial (with intermediate steps). someone please tell me where can i get it (for easy understanding).

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Jul 22, 2012

Marioeden

I'm pretty sure you just appeal to usual Sturm Liouville Orthogonality, just work out the weight function which I think is e^(-x) but haven't looked at in a while.

Related to Orthogonal properties of associated laguerre polynomial

1. What are orthogonal properties of associated Laguerre polynomials?

Orthogonal properties refer to the property that two functions are perpendicular to each other when their inner product is equal to zero. In the case of associated Laguerre polynomials, they are orthogonal with respect to the weight function e^(-x) on the interval [0, ∞).

2. How are associated Laguerre polynomials derived?

Associated Laguerre polynomials are derived from the Laguerre polynomials through a process called Rodrigues' formula. This involves expressing the polynomials as a series of derivatives and integrating them to obtain the final form.

3. What are the applications of associated Laguerre polynomials?

Associated Laguerre polynomials have various applications in mathematics, physics, and engineering. They are commonly used in solving differential equations, studying quantum mechanics, and in signal processing and control systems.

4. What is the relationship between associated Laguerre polynomials and Hermite polynomials?

Associated Laguerre polynomials and Hermite polynomials are both orthogonal families of polynomials. However, they differ in their weight functions and domains. Associated Laguerre polynomials are orthogonal on the interval [0, ∞) with respect to the weight function e^(-x), while Hermite polynomials are orthogonal on the entire real line with respect to the weight function e^(-x^2).

5. How are associated Laguerre polynomials used in probability distributions?

Associated Laguerre polynomials are used to express the probability density functions of various probability distributions, such as the Chi-squared distribution and the Gamma distribution. They also play a role in calculating moments and generating functions for these distributions.