Function Repository Resource:

# OrthogonalPolynomialVandermondeMatrix

Generate the orthogonal polynomial Vandermonde matrix corresponding to a given vector

Contributed by: Jan Mangaldan
 ResourceFunction["OrthogonalPolynomialVandermondeMatrix"][poly,{a1,a2,…}] generates the orthogonal polynomial Vandermonde matrix V(a1,a2,…) with respect to the basis represented by poly.

## Details and Options

An orthogonal polynomial Vandermonde matrix is a square matrix with elements of the form pi-1(aj), where pi(x) is an orthogonal polynomial represented by poly.
The argument poly can be any of the following:
 "ChebyshevFirst" Chebyshev polynomial of the first kind ChebyshevT[i,x] "ChebyshevSecond" Chebyshev polynomial of the second kind ChebyshevU[i,x] "Hermite" Hermite polynomial HermiteH[i,x] "Laguerre" Laguerre polynomial LaguerreL[i,x] "Legendre" Legendre polynomial LegendreP[i,x] {"Gegenbauer",m} Gegenbauer polynomial GegenbauerC[i,m,x] {"Laguerre",a} associated Laguerre polynomial LaguerreL[i,a,x] {"Jacobi",a,b} Jacobi polynomial JacobiP[i,a,b,x]
Strings in specifications like "Legendre" and {"Jacobi",a,b} can be replaced with the corresponding symbols, as in LegendreP and {JacobiP,a,b}.
If the argument poly is set to "Monomial", the classical Vandermonde matrix is generated.
ResourceFunction["OrthogonalPolynomialVandermondeMatrix"][poly,m,{a1,a2,}] generates a matrix with m rows.
ResourceFunction["OrthogonalPolynomialVandermondeMatrix"][poly,{a1,a2,},"Transpose"True] generates a transposed version of the matrix.

## Examples

### Basic Examples (1)

A Chebyshev–Vandermonde matrix:

 In[1]:=
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### Scope (3)

Jacobi–Vandermonde matrix with symbolic parameters and vector:

 In[2]:=
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An equivalent specification:

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A numerical Hermite–Vandermonde matrix:

 In[4]:=
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A rectangular Legendre–Vandermonde matrix:

 In[5]:=
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### Options (1)

#### Transpose (1)

With "Transpose"True, OrthogonalPolynomialVandermondeMatrix generates a transposed matrix:

 In[6]:=
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 In[7]:=
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## Version History

• 1.0.1 – 27 April 2021