Asymptotics of the Basis Functions of Generalized Taylor Series for the Class $H_{\rho,2}$

We study the basis functions $\varphi_{n,k}$ and $\psi_{n,p}$ of generalized Taylor series for the class $H_{\rho,2}$ and obtain asymptotic expansions of the functions $\varphi^{(l)}_{n,0}$ and $\psi^{(l)}_{n,2\cdot4^n-1}$. We prove the existence of an asymptotics for the functions $\varphi_{n,k}$ and $\psi_{n,p}$ for $k\ne 0$ and $p\ne {2\cdot4^n}-1$. The first term of the asymptotic expansions of these functions is obtained.