It is well known that the Bernoulli polynomials are defined by the generating function
When x = 0, Bn = Bn (0) are called the Bernoulli numbers. From (1), we note that
where δn k is the Kronecker’s symbol.
In , L. Carlitz considered the degenerate Bernoulli polynomials which are given by the generating function
Thus, by (5), we get
By (4), we get
From (7), we have
Now, we consider the degenerate Bernoulli polynomials which are different from the degenerate Bernoulli polynomials of L. Carlitz as follows:
When x = 0, bn, λ = bn, λ (0) are called the degenerate Bernoulli numbers.
Note. The degenerate Bernoulli polynomials are also called Daehee polynomials with λ-parameter (see ).
From (10), we note that
The classical polylogarithm function Lik (x) is defined by
It is known that the poly-Bernoulli polynomials are defined by the generating function
When k = 1, we have
By (14), we easily get
Let x = 0. Then are called the poly-Bernoulli numbers.
In this paper, we introduce the new fully degenerate poly-Bernoulli numbers and polynomials and inverstigate some properties of these polynomials and numbers. From our investigation, we derive some identities for the fully degenerate poly-Bernoulli numbers and polynomials.
2 Fully degenerate poly-Bernoulli polynomials
For k ∈ ℤ, we define the fully degenerate poly-Bernoulli polynomials which are given by the generating function
Thus, we get
By (15), we get
Thus, from (18), we have
Theorem 2.1: For k ∈ ℤ, ≥ 0, we have(20)andwhere .From (15), we can derive the following equation:(21)Thus, by (21), we get(22)where S2 (n, l) and S1 (n, l) are the Stirling numbers of the second kind and of the first kind, respectively. Therefore, by (22), we obtain the following theorem.
Theorem 2.2: For k ∈ ℤ, n ≥ 1, we haveFrom (12), we can easily derive the following equation:(23)Thus, by (23), the generating function of the fully degenerate poly-Bernoulli numbers is also written in terms of the following iterated integral:(24)For k = 2, we have(25)Therefore, by (25), we obtain the following theorem.
Theorem 2.6: For k ∈ ℤ and n ≥ 0, we haveNote thatFrom Theorem 2.1, we haveThe generalized falling factorial (x | λ)n is given by(33)As is well known, the Bernoulli numbers of the second kind are defined by the generating funcdtion(34)We observe that(35)On the other hand,(36)From (35) and (36), we have(37)By Theorem 2.1, we get
3 Further remarks
Let ℂ be complex number field and let be the set of all formal power series in the variable t over ℂ with
Let ℙ be the algebra of polynomials in a single variable x over ℂ and let ℙ* be the vector space of all linear functionals on ℙ. The action of linear functional L ∈ ℙ* on a polynomial p (x) is denoted by 〈L| p (x)〉, and linearly extended as
where c, c′ ∈ ℂ.
For , we define a linear functional on ℙ by setting
Thus, by (39), we get
For , by (40), we get 〈fL (i)| xn〉 = 〈L| xn〉. In addition, the mapping L ↦ fL (t) is a vector space isomorphism from ℙ* onto . Henceforth, denotes both the algebra of the formal power series in t and the vector space of all linear functionals on ℙ and so an element f (i) of can be regarded as both a formal power series and a linear functional. We refer to umbral algebra. The umbral calculus is the study of umbral algebra (see [5, 15, 20]). The order o (f (i)) of the non-zero power series f (i) is the smallest integer k for which the coefficient of tk does not vanish.
If o (f (t)) = 1(respectively, o (f (t)) = 0), then f (t) is called a delta (respectively, an invertible) series (see ). For o (f (t)) = 1 and o (g (t)) = 0, there exists a unique sequence sn (x) of polynomials such that 〈g(t) f (t)k| sn (x〉 = n!δn, k, (n, k ≥ 0).
The sequence sn (x) is called the Sheffer sequence for (g (t), f (t)), and we write sn (x) ~ (g (t), f (t)) (see ).
Let f (t) ∈ and p (x) ∈ ℙ. Then, by (40), we get
From (41), we have
By (42), we easily get
From (43), we have
Let f (t) be the linear functional such that
for all polynomials p (x). Then it can be determined by (41) to be
Thus, for p (x) ∈ ℙ, we have
It is known that
Thus, by (48),
On the other hand,
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Published Online: 2016-07-26
Published in Print: 2016-01-01
© 2016 Kim et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)