Umbral calculus and Frobenius-Euler polynomials

In this paper, we study some properties of umbral calculus related to Appell sequence. From those properties, we derive new and interesting identities of Frobenius-Euler polynomials.


Introduction
Let C be the complex number field. For λ ∈ C with λ = 1, the Frobenius-Euler polynomials are defined by the generating function to be 1 − λ e t − λ e xt = e H(x|λ)t = ∞ n=0 H n (x|λ) t n n! , (see [7−11]) , with the usual convention about replacing H n (x|λ) by H n (x|λ). In the special case, x = 0, H n (0|λ) = H n (λ) are called the n-th Frobenius-Euler numbers. By (1), we get H n (x|λ) = n l=0 n l H n−l (λ)x l = (H(λ) + x) n , (see [1,2,3,4]), (2) with the usual convention about replacing H n (λ) by H n (λ). Thus, from (1) and (2), we note that (H(λ) + 1) n − λH n (λ) = (1 − λ)δ 0,n , where δ n,k is the kronecker symbol (see [6,7]). For r ∈ Z + , the Frobenius-Euler polynomials of order r are defined by the generating function to be In the special case, n (λ) are called the n-th Frobenius-Euler numbers of order r (see [6,7]). From (3), we can derive the following equation: and By (4) and (5), we see that H (r) n (x|λ) is a monic polynomial of degree n with coefficients in Q(λ). Let P be the algebra of polynomials in the single variable x over C and let P * be the vector space of all linear functionals on P. As is known, L|p(x) denotes the action of the linear functional L on a polynomial p(x) and we remind that the addition and scalar multiplication on P * are respectively defined by where c is a complex constant (see [5,8]). Let F denote the algebra of formal power series: [5,8]).
The formal power series define a linear functional on P by setting f (t)|x n = a n , for all n ≥ 0.
This kind of algebra is called an umbral algebra.
The order O(f (t)) of a nonzero power series f (t) is the smallest integer k for which the coefficient of t k does not vanish. A series f (t) for which O(f (t)) = 1 is said to be an invertible series (see [5,8]). For f (t), g(t) ∈ F and p(x) ∈ P, we have , (see [5]).
One should keep in mind that each f (t) ∈ F plays three roles in the umbral calculus : a formal power series, a linear functional and a linear operator. To illustrate this, let p(x) ∈ P and f (t) = e yt ∈ F. As a linear functional, e yt satisfies e yt |p(x) = p(y). As a linear operator, e yt satisies e yt p(x) = p(x + y) (see [5]). Let s n (x) denote a polynomial in x with degree n. Let us assume that f (t) is a delta series and g(t) is an invertible series. 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 for all n, k ≥ 0 (see [5,8]). This sequence s n (x) is called the Sheffer sequence for . Then we see that and wheref (t) is the compositional inverse of f (t) (see [8]). In this paper, we study some properties of umbral calculus related to Appell sequence. For those properties, we derive new and interesting of Frobenius-Euler polynomials.
Theorem 2. For any r, s ≥ 0, we have Let us take s = r − 1(r ≥ 1) in Theorem 2. Then we obtain the following corollary.
Let us take s = r(r ≥ 1) in Theorem 2. Then we obtain the following corollary.
Corollary 4. For n ≥ 0, r ≥ 1, we have n−l (x|λ). Now, we define the analogue of Stirling numbers of the second kind as follows: Note that S 1 (n, k) = S(n, k) is the stirling number of the second kind. From the definition of△ λ , we havẽ By (27) and (28), we get Let us take s = 2r. Then we have By (30) and (31), we get n−l (x|λ).