Function Repository Resource:

# StieltjesJacobiE

Evaluate the Stieltjes-Jacobi polynomial

Contributed by: Jan Mangaldan
 ResourceFunction["StieltjesJacobiE"][n,a,b,x] gives the Stieltjes-Jacobi polynomial .

## Details and Options

Mathematical function, suitable for both symbolic and numerical manipulation.
Explicit polynomials are given when possible.
The Stieltjes-Jacobi polynomial is the polynomial whose roots are the additional abscissas in Jacobi-Kronrod quadrature, apart from the abscissas that are the roots of the Jacobi polynomial .
is a polynomial of degree n+1.
For certain special arguments, ResourceFunction["StieltjesJacobiE"] automatically evaluates to exact values.
ResourceFunction["StieltjesJacobiE"] can be evaluated to arbitrary numerical precision.

## Examples

### Basic Examples (2)

Compute the 2nd Stieltjes-Jacobi polynomial:

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Plot over a subset of the reals:

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### Scope (4)

Evaluate numerically:

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Evaluate to high precision:

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The precision of the output tracks the precision of the input:

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### Applications (2)

Compute the abscissas and weights of a (2n+1)-point Gauss-Kronrod quadrature:

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Use the Gauss-Kronrod abscissas and weights to approximate the area under a curve:

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Compare to the output of NIntegrate:

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Compute the abscissas and weights of a (2n+1)-point Lobatto-Kronrod quadrature:

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Use the Lobatto-Kronrod abscissas and weights to approximate the area under a curve:

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Compare to the output of NIntegrate:

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### Properties and Relations (3)

The second (nonpolynomial) solution of the Jacobi differential equation, :

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StieltjesJacobiE is the polynomial part of the asymptotic expansion of at Infinity:

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StieltjesJacobiE is a polynomial of degree n+1:

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The roots of JacobiP and StieltjesJacobiE interlace each other:

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### Neat Examples (1)

A curve with multiple loops:

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## Version History

• 1.0.0 – 04 January 2021