# Chebyshev polynomial generalization for non-integer degrees

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

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.
• OK. This is some kind of obvious but not sure if this keeps the $2 T_m(x) T_n(x) = T_{m+n}(x) + T_{|m-n|}(x)$. And if so is there any series representing this. – Gevorg Hmayakyan Jan 6 '20 at 19:54