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
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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.
answered Jan 6 '20 at 19:52
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$\begingroup$ 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. $\endgroup$ – Gevorg Hmayakyan Jan 6 '20 at 19:54
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$\begingroup$ Probably some of those relations are strongly depend on degree being integer. $\endgroup$ – Gevorg Hmayakyan Jan 6 '20 at 19:57
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$\begingroup$ Thanks. And is there any paper about this ? $\endgroup$ – Gevorg Hmayakyan Jan 6 '20 at 20:11
