Student t orthogonal polynomials

In summary, the conversation discusses the use of student-t orthogonal polynomials and the process of deriving them. The speaker asks for help in obtaining the student-t polynomials to check their derivation. The response provides the form of the orthogonal polynomials and an integral equation that can be used to verify the derivation. The speaker expresses their gratitude for the information.
  • #1
64jnk
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0
I've just read a paper that references the use of student-t orthogonal polynomials. I understand how the Gauss-Hermite polynomials are derived, however applying the same process to the weight function (1 + t^2/v)^-(v+1)/2 I can't quite get an answer that looks anything like a polynomial.

Would anyone be able to provide me with the student-t polynomials, which I can check my derivation against?

Thank you.
 
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  • #2
As far as I can remember you should end up with the prthogonal polynomials taking the form

/phi_{m}(t) = A_{m}/[1+/frac{t^{2}}{v}/]/frac{d^{m}}{dt^{m}}/[/frac{1}{1+/frac(t^{2}}{v}}^{/frac{v-1}{2}-m}/]

Then,

/int_{- /infty}^{+ /infty} /frac{1}{1+/frac{t^{2}}{v}}^{frac{v+1}{2}}/phi_{m}(t}/phi_{n}(t) dt=0

(hope all teX commands are in the right place!)
 
  • #3
That's done the trick. Thank you.
 

FAQ: Student t orthogonal polynomials

1. What are Student t orthogonal polynomials?

Student t orthogonal polynomials, also known as Chebyshev polynomials of the first kind, are a set of polynomials that are orthogonal with respect to a weight function that follows the Student t distribution. They are commonly used in statistical analysis and numerical computation.

2. How are Student t orthogonal polynomials calculated?

The coefficients for the Student t orthogonal polynomials can be calculated using the recurrence relation: Tn+1(x) = 2xTn(x) - Tn-1(x), where Tn(x) represents the nth order polynomial and x is the variable. Alternatively, they can also be calculated using the Gram-Schmidt process.

3. What is the significance of Student t orthogonal polynomials?

Student t orthogonal polynomials are important in statistical analysis as they provide a basis for approximating other functions and can be used to solve various types of differential equations. They also have applications in fields such as physics, engineering, and finance.

4. Can Student t orthogonal polynomials be used for multidimensional problems?

Yes, Student t orthogonal polynomials can be extended to solve multidimensional problems by using tensor products of the one-dimensional polynomials. This allows for the efficient computation of integrals and solutions to differential equations in higher dimensions.

5. What are the limitations of using Student t orthogonal polynomials?

While Student t orthogonal polynomials have many useful applications, they are not suitable for all types of problems. They may not provide accurate results for functions that do not follow the Student t distribution or for highly oscillatory functions. In addition, the accuracy of the polynomials may decrease as the order increases.

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