Symbolic Legendre Polynomials

Is there an implemented version of the legendre Polynomials with symbolic variables?

I have checked

But they either don’t implement the Legendre Polynomials or a symbolic equivalent.

What would be the most convenient way to implement them?

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Their combination does:

julia> using ClassicalOrthogonalPolynomials, Symbolics

julia> @variables x;

julia> p4 = legendrep(4, x)
0.375 + 1.75x*(1.6666666666666667x*(1.5(x^2) - 0.5) - 0.6666666666666666x) - 1.125(x^2)

julia> expand(p4)
0.375 + 4.375(x^4) - 3.75(x^2)

Gotta love composability!


Following up on @stevengj’s enthusiasm for composability, if you needed exact coefficients, you might mix in SpecialPolynomials:

julia> using Symbolics, SpecialPolynomials

julia> @variables x
1-element Vector{Num}:

julia> basis(Legendre{Rational{Int}}, 4)(x) |> expand
(3//8) + (35//8)*(x^4) - (15//4)*(x^2)

Excelent! As a rule of thumb, which functions should I expect to work with Symbolics.jl?

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Really nice! But this option seems to overflow for degrees n > 25

basis(Legendre{Rational{Int}}, 25)(x) |> expand
ERROR: OverflowError: -148257227381443 * 74290 overflowed for type Int64

Replacing Int for Int128 does not seem to work, because I’ll need to work with polynomials of degree 50 or more

Use BigInt if you need exact rational coefficients for high degrees?