Continuing his previous investigations [ibid. 79, No. 1, 65-92 (1988; Zbl 0696.05004), \((*)\) Adv. Math. 58, 89-108 (1985; Zbl 0583.05010) and ibid. 85, 215-242 (1991)], the author studies sequences of polynomials which are well suited for Hermite interpolation from the point of view of convergence, computing the coefficient, evaluations and manipulations of interpolants. The case considered in this paper is that of sequence \(\{u_ k\}\) of monic polynomials \(u_ k\) of degree \(k\), satisfying the recurrence relation \([u_{n+1}(z)-u_{n+1}(x)]/(z-x)=\sum^ n_{k=0} u_ k(z)\cdot u_{n-k}(x)\). Such sequences are called polynomial sequences of interpolatory type. A typical example is that of interpolation polynomials based on the roots of Chebyshev polynomials. The relevance of Chebyshev polynomials for the considered problems is emphasized in Section 3: the Chebyshev polynomials of the first kind form a basic sequence of interpolatory type and those of the second kind form an associated sequence, in a sense defined in the paper via generating functions. Characterizations of interpolatory sequences in terms of generating functions and of some formal power series are also given. The sequences of polynomials of interpolatory type are related to the Newton Umbral Calculus [see \textit{S. Roman}, J. Math. Anal. Appl. 89, 290-314 (1982; Zbl 0526.05007)], which is also suitable to the study of polynomial families in several variables, as pointed out the author in \((*)\).