MODIFIED APOSTOL-TYPE POLYNOMIALS ARISING FROM UMBRAL CALCULUS TOUFIK MANSOUR, HACER OZDEN, AND YILMAZ SIMSEK Abstract. In this paper, by using the orthogonality type as defined in the um- bral calculus, we derive an explicit formula for several well-known polynomials as a linear combination of the modified Apostol-type polynomials. 1. Introduction Let Π be the algebra of polynomials in a single variable x over the complex field C and let Π ∗ be the vector space of all linear functionals on Π. Let H denote the algebra of formal power series in a single variable t over C. The formal power series f (t)= ∑ k≥0 a k t k k! in the variable t defines a linear functional on Π by setting 〈f (t)|x n 〉 = a n , for all n ≥ 0 (see [23, 24, 25]). Thus, 〈t k |x n 〉 = n!δ n,k , for all n, k ≥ 0, (see [23, 24, 25]), (1.1) where δ n,k is the Kronecker’s symbol. Let f L (t)= ∑ k≥0 〈L|x k 〉 t k k! . By (1.1), we have 〈f L (t)|x n 〉 = 〈L|x n 〉. So, the map L → f L (t) is a vector space isomorphism from Π ∗ onto H. Therefore, H is thought of as set of both formal power series and linear functionals. We call H the umbral algebra. The umbral calculus is the study of umbral algebra (for recent examples, see [8, 9, 14]). Let f (t) ∈H be any non-zero power series, the order O(f (t)) of f (t) is the small- est integer k for which the coefficient of t k in f (t) does not vanish. If O(f (t)) = 1 (respectively, O(f (t)) = 0), then f (t) is called a delta (respectively, an invertible ) se- ries. Suppose that f (t),g(t) ∈H such that O(f (t)) = 1 and O(g(t)) = 0. Then there exists a unique sequence S n (x) of polynomials such that 〈g(t)f (t) k |S n (x)〉 = n!δ n,k , where n, k ≥ 0. The sequence S n (x) is called the Sheffer sequence for (g(t),f (t)) which is denoted by S n (x) ∼ (g(t),f (t)) (see [23, 24, 25]). For f (t) ∈H and p(x) ∈ Π, we have 〈e yt |p(x)〉 = p(y), 〈f (t)g(t)|p(x)〉 = 〈g(t)|f (t)p(x)〉, and f (t)=  k≥0 〈f (t)|x k 〉 t k k! , p(x)=  k≥0 〈t k |p(x)〉 x k k! , (1.2) (see [23, 24, 25]). By (1.2), we obtain 〈t k |p(x)〉 = p (k) (0), 〈1|p (k) (x)〉 = p (k) (0), (1.3) where p (k) (0) denotes the k-th derivative of p(x) with respect to x at x = 0. Thus, by (1.3), we have t k p(x)= p (k) (x)= d k dx k p(x), for all k ≥ 0, (see [23, 24, 25]). Let 2000 Mathematics Subject Classification. 05A40. Key words and phrases. Apostol-type polynomials, Bernoulli polynomial, Bessel polynomial, Euler polynomial,, Umbral Calculus. 1