Chebyshev polynomial generalization for non-integer degrees

Question

I am trying to generalize the Chebyshev polynomials (especially of first kind) for non-integer degree. The properties I would like to keep is $$2 T_m(x) T_n(x) = T_{m+n}(x) + T_{|m-n|}(x)$$ and $$T_m(\cos(x))=\cos(mx)$$ So to be more specific generalization of the $T_r(x)$ for non integer $r$ values. I have found a document about the half degree polynomials: Pseudo-Chebishev Polynomial

Answer 1

Well, if you want $T_m(\cos(x)) = \cos(mx)$, you might as well simply define $$ T_m(z) = \cos(m \arccos(z))$$ either as a multivalued function, or use a particular branch of arccos.

EDIT: With $\arccos(z)=t$, note that you do have $$ 2 T_m(z) T_n(z) = 2\cos(m t) \cos(nt) = \cos((m-n)t) + \cos((m+n)t) = T_{m-n}(z) + T_{m+n}(z) $$ And of course $T_{m-n} = T_{n-m}$. But it's not $T_{|m-n|}$ if $m-n$ is complex.