Asymptotic normality of the coefficients of polynomials associated with the Gegenbauer ones (Q5946617)

It is shown that the coefficients in certain scaled versions of Gegenbauer polynomials obey a local limit theorem, with explicitly given asymptotic expressions for mean and variance. This is deduced by making use of the three-term recurrence obeyed by the Gegenbauer polynomials and a classical asymptotic expansion for Gegenbauer polynomials.