I'm interested in the properties of particular families of recursively-generated, complex polynomials. In particular, let $p_0(x) = e^{i \phi_0}$ and $p_1(x) = e^{i [\phi_1 - \phi_0]} x$. Then, for $n \geq 2$,

$$p_{n+1}(x) = e^{i [\phi_{n + 1} - \phi_n]} \left[ 2 x \cos \phi_n p_n(x) - p_{n - 1}(x) \right]$$

I'm trying to demonstrate that for appropriate choice of $\phi_k$, this Chebyshev-like recursion can yield polynomials of a particular form (i.e. $p_n(x)$ can be an arbitrary degree-$n$ complex polynomial with some parity and boundary conditions at $-1$ and $1$). I'm also interested more generally in more complicated recursive families of polynomials, so are there any well-developed resources on understanding the properties of families of this form (those which are parameterized by some sequence of angles $\phi_k$).

  • $\begingroup$ It is not clear what the question is. $\endgroup$ Sep 20, 2022 at 6:13


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