Umbral Calculus and the Frobenius-Euler Polynomials

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 (see [1–5]) with the usual convention about replacing Hⁿ(x∣λ) by H_n(x∣λ).

In the special case, x = 0,  H_n(0∣λ) = H_n(λ) are called the nth Frobenius-Euler numbers. By (1), we get (see [6–9]) with the usual convention about replacing Hⁿ(λ) by H_n(λ).

Thus, from (1) and (2), we note that where δ_n,k is the kronecker symbol (see [1, 10, 11]).

For r ∈ Z₊, the Frobenius-Euler polynomials of order r are defined by the generating function to be In the special case, $$ are called the nth Frobenius-Euler numbers of order r (see [1, 10]).

From (4), we can derive the following equation: By (5), we see that $$ 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 [3, 12]).

Let F denote the algebra of formal power series: (see [3, 12]). The formal power series define a linear functional on ℙ by setting Indeed, by (7) and (8), we get (see [3, 12]). 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 [2, 12]). For f(t), g(t) ∈ F, and p(x) ∈ ℙ, we have (see [12]). 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 satisfies e^ytp(x) = p(x + y) (see [12]). 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 [3, 12]). This 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)). If s_n(x)~(1, f(t)), then s_n(x) is called the associated sequence for f(t). If s_n(x)~(g(t), t), then s_n(x) is called the Appell sequence.

Let s_n(x)~(g(t), f(t)). Then we see that where $$ is the compositional inverse of f(t) (see [3]). In this paper, we study some properties of umbral calculus related to the Appell sequence. For those properties, we derive new and interesting identities of the Frobenius-Euler polynomials.

By (4) and (12), we see that Thus, by (13), we get Let Then it is an (n + 1)-dimensional vector space over Q(λ).

So we see that $$ is a basis for ℙ_n(λ). For p(x) ∈ ℙ_n(λ), let Then, by (13), (14), and (16), we get From (17), we have Therefore, by (16) and (18), we obtain the following theorem.

From Theorem 1, we note that Let us consider the operator $$ with $$ and let $$. Then we have Thus, by (22), we get From (4), we can derive By (23) and (24), we get From (25), we have For s ∈ Z₊, from (25), we have On the other hand, by (12), (13), and (25), Thus, by (28), we get Therefore, by (27) and (29), we obtain the following theorem.

Now, we define the analogue of Stirling numbers of the second kind as follows: Note that S₁(n, k) = S(n, k) is the Stirling number of the second kind.

From the definition of $$, we have By (33) and (34), we get Let us take s = 2r. Then we have By (36), we get Let us take x = 0 in (37). Then we obtain the following theorem.

Let us consider s = 2r − 1 in the identity of Theorem 2. Then we have Let us take x = 0 in (39). Then we obtain the following theorem.