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Express each Legendre polynomial [tex]L_i[/tex] as a linear combination of the Hermite polynomials [tex]H_j[/tex]. Provide the mathematical expressions for the Legendre polynomials in terms of the Hermite polynomials.

Answer :

Final answer:

Expressing Legendre polynomials as a linear combination of Hermite polynomials is a complex mathematical task that involves calculating coefficients through a projection method.

Explanation:

The concept of expressing Legendre polynomials in terms of Hermite polynomials involves writing a given Legendre polynomial Li as a linear combination of Hermite polynomials Hj. Legendre polynomials are solutions to Legendre's differential equation, while Hermite polynomials are solutions to Hermite's differential equation.

However, transforming one polynomial basis to another is non-trivial, as they satisfy different orthogonality relations, and there's no standard formula for such conversion. To express a Legendre polynomial as a series of Hermite polynomials, one would need to use a projection method.

It is important to mention that any Legendre polynomial Li can theoretically be expressed as a linear combination of Hermite polynomials Hj, but calculating the exact coefficients is a mathematically complex process and is usually not covered in standard curricula.

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