Universal validity of the Kadar–Yu polynomial formulae

Let $C^{(l+1,1)}(\alpha)$, $C^{(l+1,1)^T}(\alpha)$, $P_{l+4+k}^{(l+1,1)}$, and $P_{l+4+k}^{(l+1,1)^T}$ denote the polynomials and Chebyshev series defined in the preceding formulae. Kadar–Yu formula conjecture. The formulae for $C^{(l'+1,1)}(\alpha)$, $C^{(l+1,1)^T}(\alpha)$, $P_{l”+4+k}^{(l”+1,1)}$, and $P_{l+4+k}^{(l+1,1)^T}$ hold for all $l\geq 1$, $l'\geq 4$, and $l”\geq 2$. These identities extend patterns verified by direct computation and would provide the corresponding Gram-determinant and Chebyshev-series formulas beyond the explicitly checked cases.