On a Derivation of the Classical Differential Equation for Chebyshev Polynomials of the First Kind
DOI:
https://doi.org/10.22105/kmisj.v3i1.120Keywords:
Chebyshev polynomials, Orthogonal polynomials, Pearson equation, Second-order differen tial equations, Sturm–Liouville theory, Spectral methods, Orthogonality, Approximation theoryAbstract
In this study, the derivation of the second-order differential equation for first-order Chebyshev polynomials is revisited using orthogonality theory and the Pearson equation approach. By employing the Pearson-type relation associated with the Chebyshev weight function, the relationship between the weight function and the corresponding differential operator is systematically analyzed. This approach leads directly to the classical second-order differential equation satisfied by Chebyshev polynomials and reveals the Sturm–Liouville structure of the associated operator. The interplay between orthogonal polynomial systems and second-order differential equations is emphasized, highlighting the analytical structure underlying these polynomials. In particular, the Pearson equation framework provides an alternative and systematic derivation of the differential equation governing the Chebyshev polynomial system.
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