Abstract
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 properties, we derive some identities for the fully degenerate poly-Bernoulli numbers and polynomials.
1 Introduction
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
and
where δn k is the Kronecker’s symbol.
In [3], L. Carlitz considered the degenerate Bernoulli polynomials which are given by the generating function
When x = 0, βn, λ = βn, λ (0) are called the degenerate Bernoulli numbers. From (1) and (4), we note that
Thus, by (5), we get
By (4), we get
and
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 [13]).
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
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
When
Thus, we get
By (15), we get
Thus, from (18), we have
and
Therefore, by (17) and (19), we obtain the following theorem.
For k ∈ ℤ, ≥ 0, we have
and
where
From (15), we can derive the following equation:
Thus, by (21), we get
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.
Fork ∈ ℤ, n ≥ 1, we have
From (12), we can easily derive the following equation:
Thus, by (23), the generating function of the fully degenerate poly-Bernoulli numbers is also written in terms of the following iterated integral:
For k = 2, we have
Therefore, by (25), we obtain the following theorem.
For n ≥ 0, we have
Note that
From (15), we have
Therefore, by (26), we obtain the following theorem.
Forn ≥ 0, we have
Note that
From (23), we have
On the other hand,
where
Thus, by (29), we have
Therefore, by (30), we obtain the following theorem
For n ≥ 1, we have
Note that
Now, we observe that
By (31), we get
Therefore, by (32), we obtain the following theorem.
Fork ∈ ℤ and n ≥ 0, we have
Note that
From Theorem 2.1, we have
The generalized falling factorial (x | λ)n is given by
As is well known, the Bernoulli numbers of the second kind are defined by the generating funcdtion
We observe that
On the other hand,
By Theorem 2.1, we get
3 Further remarks
Let ℂ be complex number field and let
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
Thus, by (39), we get
For
If o (f (t)) = 1(respectively, o (f (t)) = 0), then f (t) is called a delta (respectively, an invertible) series (see [20]). 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 [20]).
Let f (t) ∈
From (41), we have
where
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
where
That is,
Thus, by (48),
On the other hand,
Therefore, by (49) and (50), we obtain the following theorem.
For n ∈ ℕ, we have
By (46), we get
From (51), we have
Thus, by (52), we get
Therefore, by (53), we obtain the following theorem.,
For n ≥ 0 we have
Note that
Let
For p(x) ∈ ℙn with
From (48), we note that
Therefore, by (56), we obtain the following theorem.
For p (x) ∈ ℙn, we have
where
For example, let us take
where
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