Function Repository Resource:

OrthogonalPolynomialDividedDifference

Evaluate the divided difference of a finite orthogonal polynomial series

Contributed by: Jan Mangaldan
 ResourceFunction["OrthogonalPolynomialDividedDifference"][cof,poly,x,y] evaluates the divided difference , where ci is the (i+1)th element of the list cof.

Details

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}.

Examples

Basic Examples (2)

Divided difference of a Laguerre series:

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Compare with an explicit evaluation using the resource function OrthogonalPolynomialSum:

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Scope (2)

Divided difference of a Jacobi series with symbolic coefficients, parameters and arguments:

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An equivalent specification:

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Applications (2)

Directly evaluating the divided difference using the resource function OrthogonalPolynomialSum gives a result that is not very accurate:

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OrthogonalPolynomialDividedDifference gives a more accurate result:

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Evaluate the q-derivative of a Chebyshev series:

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In the limit q1, the q-derivative reduces to the derivative:

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Properties and Relations (2)

OrthogonalPolynomialDividedDifference is symmetric in the arguments x and y:

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If x=y, the result of OrthogonalPolynomialDividedDifference is equal to the derivative of the orthogonal polynomial series, evaluated at x:

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Version History

• 1.0.0 – 06 April 2021