1 Introduction

Carlitz is the first one who initiated the study of degenerate versions of some special numbers and polynomials, namely the degenerate Bernoulli and Euler polynomials and numbers (see [2]). In recent years, studying degenerate versions of some special numbers and polynomials regained interests of some mathematicians with their interests not only in combinatorial and arithmetic properties but also in applications to differential equations, identities of symmetry, and probability theory (see [9, 10, 12–14, 17, 19, 21] and the references therein). It is noteworthy that studying degenerate versions is not only limited to polynomials but also can be extended to transcendental functions like gamma functions (see [13]).

The Rota’s theory of umbral calculus is based on linear functionals and differential operators (see [3–7, 20, 23–27]). The Sheffer sequences occupy the central position in the theory and are characterized by the generating functions where the usual exponential function enters. The motivation for the paper [10] starts from the question that what if the usual exponential function is replaced with the degenerate exponential functions (see (2)). As it turns out, it corresponds to replacing the linear functional with the family of λ-linear functionals (see (12)) and the differential operator with the family of λ-differential operators (see (14)). Indeed, these replacements lead us to defining λ-Sheffer polynomials and degenerate Sheffer polynomials (see (16)).

As is well known, poly-Bernoulli polynomials are defined in terms of polylogarithm functions. Recently, as degenerate versions of such functions and polynomials, degenerate polylogarithm functions were introduced and degenerate poly-Bernoulli polynomials were defined by means of the degenerate polylogarithm functions, and some properties of the degenerate poly-Bernoulli polynomials were investigated (see [17]).

The aim of this paper is to further study the degenerate poly-Bernoulli polynomials, which is a λ-Sheffer sequence and hence a degenerate Sheffer sequence, by using the above-mentioned λ-linear functionals and λ-differential operators. In more detail, these polynomials are investigated by three different tools, namely a formula about representing a λ-Sheffer sequence by another (see (19)), a formula coming from the generating functions of λ-Sheffer sequences (see Theorem 1), and a formula arising from the definitions for λ-Sheffer sequences (see Theorems 6, 7). Then, among other things, we represent the degenerate poly-Bernoulli polynomials by higher-order degenerate Bernoulli polynomials and by higher-order degenerate derangement polynomials. The rest of this section is devoted to recalling the necessary facts that are needed throughout the paper, which includes the ‘λ-umbral calculus’.

For \(k\in \mathbb{Z}\) and \(0 \ne \lambda \in \mathbb{R}\), the degenerate polylogarithm functions are defined by

$$ \operatorname{Li}_{k,\lambda }(x)=\sum_{n=1}^{\infty } \frac{(-\lambda )^{n-1}(1)_{n,1/\lambda }}{(n-1)!n^{k}}x^{n}\quad ( \text{see [9, 17]}), $$
(1)

where \((x)_{0,\lambda }=1\), \((x)_{n,\lambda }=x(x-\lambda )\cdots (x-(n-1) \lambda )\), (\(n\ge 1\)).

For any \(\lambda \in \mathbb{R}\), the degenerate exponential functions are given by

$$ e_{\lambda }^{x}(t)=\sum_{n=0}^{\infty } \frac{(x)_{n,\lambda }}{n!}t^{n},\qquad e_{\lambda }(t)=e_{\lambda }^{1}(t)= \sum_{n=0}^{\infty } \frac{(1)_{n,\lambda }}{n!}t^{n} \quad (\text{see [9, 13]}). $$
(2)

Note here that, for \(\lambda \ne 0\), \(e_{\lambda }^{x}(t)=(1+\lambda t)^{\frac{x}{\lambda }}\).

The compositional inverse \(\log _{\lambda }(t)\) of \(e_{\lambda }(t)\) is given by

$$ \log _{\lambda }(t)=\sum_{n=1}^{\infty } \frac{\lambda ^{n-1}}{n!}(1)_{n,1/ \lambda }(t-1)^{n},\quad (\text{see [9, 13]}). $$
(3)

From (1) and (3), we have

$$ \operatorname{Li}_{1,\lambda }(x)=-\log _{\lambda }(1-x) \quad \text{and}\quad \lim_{\lambda \rightarrow 0}\operatorname{Li}_{k,\lambda }(x)= \operatorname{Li}_{k}(x), $$
(4)

where \(\operatorname{Li}_{k}(x)\) are the polylogarithm functions defined by

$$\begin{aligned}& \operatorname{Li}_{k}(x)=\sum_{n=1}^{\infty } \frac{x^{n}}{n^{k}} \quad \bigl( \vert x \vert < 1\bigr), (\text{see [1, 8, 9, 22]}). \end{aligned}$$

In [9], the degenerate poly-Bernoulli polynomials are defined by Kim–Kim as

$$ \frac{\operatorname{Li}_{k,\lambda }(1-e_{\lambda }(-t))}{e_{\lambda }(t)-1}e_{ \lambda }^{x}(t)=\sum _{n=0}^{\infty }B_{n,\lambda }^{(k)}(x) \frac{t^{n}}{n!}. $$
(5)

When \(x=0\), \(B_{n,\lambda }^{(k)}= B_{n,\lambda }^{(k)}(0)\) are called the degenerate poly-Bernoulli numbers.

It is well known that Carlitz’s degenerate Bernoulli polynomials of order r are defined by

$$ \biggl(\frac{t}{e_{\lambda }(t)-1} \biggr)^{r}e_{\lambda }^{x}(t)= \sum_{n=0}^{ \infty }\beta _{n,\lambda }^{(r)}(x) \frac{t^{n}}{n!}\quad (\text{see [2]}). $$
(6)

For \(r=1\), \(\beta _{n,\lambda }^{(1)}(x)=\beta _{n,\lambda }(x)\) are called the degenerate Bernoulli polynomials.

From (5), we note that

$$ \sum_{n=0}^{\infty }B_{n,\lambda }^{(1)}(x) \frac{t^{n}}{n!}= \frac{\operatorname{Li}_{1,\lambda }(1-e_{\lambda }(-t))}{e_{\lambda }(t)-1} = \frac{t}{e_{\lambda }(t)-1}e_{\lambda }^{x}(t)= \sum_{n=0}^{\infty } \beta _{n,\lambda }(x) \frac{t^{n}}{n!}. $$
(7)

By (7), we get \(B_{n,\lambda }^{(1)}(x)=\beta _{n,\lambda }(x)\), (\(n\ge 0\)).

The degenerate Stirling numbers of the second kind appear as the coefficients in the expansion

$$ (x)_{n,\lambda }=\sum_{l=0}^{n}S_{2,\lambda }(n,l) (x)_{l} \quad (n\ge 0), (\text{see [9, 11]}). $$
(8)

As the inversion formula of (8), the degenerate Stirling numbers of first kind appear as the coefficients in the expansion

$$ (x)_{n}=\sum_{l=0}^{n}S_{1,\lambda }(n,l) (x)_{l,\lambda } \quad (n\ge 0), (\text{see [9]}). $$
(9)

Thus, by (8) and (9), we have

$$ \frac{1}{k!} \bigl(e_{\lambda }(t)-1 \bigr)^{k}=\sum _{n=k}^{\infty }S_{2, \lambda }(n,k) \frac{t^{n}}{n!} $$
(10)

and

$$\frac{1}{k!} \bigl(\log _{\lambda }(1+t) \bigr)^{k}= \sum_{n=k}^{\infty }S_{1, \lambda }(n,k) \frac{t^{n}}{n!}\quad (k\ge 0), (\text{see [9]}). $$

In view of (6), the degenerate derangement polynomials of order \(r(\in \mathbb{N})\) are defined by

$$ \frac{1}{(1-t)^{r}}e_{\lambda }^{-1}(t) e_{\lambda }^{x}(t)= \sum_{n=0}^{ \infty }d_{n,\lambda }^{(r)}(x) \frac{t^{n}}{n!} \quad (\text{see [12, 19]}). $$
(11)

When \(r=1\), \(d_{n,\lambda }(x)=d_{n,\lambda }^{(1)}(x)\) are called the degenerate derangement polynomials.

Note that \(\lim_{\lambda \rightarrow 0}d_{n,\lambda }(x)=d_{n}(x) \), where \(d_{n}(x)\) are the derangement polynomials and \(d_{n}=d_{n}(0)\) are the derangement numbers (see [12, 15, 16, 18]).

We remark that the umbral calculus has long been studied by many people (see [3–7, 20, 23–27]). For the rest of this section, we recall the necessary facts on the λ-linear functionals, λ-differential operators, λ-Sheffer sequences, and so on. The details on these can be found in the recent paper [10].

Let \(\mathbb{C}\) be the field of complex numbers,

$$\mathcal{F}= \Biggl\{ f(t)=\sum_{k=0}^{\infty }a_{k} \frac{t^{k}}{k!} \Big|a_{k}\in \mathbb{C} \Biggr\} , $$

and let

$$\mathbb{P}=\mathbb{C}[x]= \Biggl\{ \sum_{i=0}^{\infty }a_{i}x^{i} \Big|a_{i}\in \mathbb{C} \textrm{ with }a_{i}=0\text{ for all but finite number of }i \Biggr\} . $$

For \(f(t)\in \mathcal{F}\) with \(f(t)=\sum_{k=0}^{\infty }a_{k}\frac{t^{k}}{k!} \) and \(\lambda \in \mathbb{R}\), the λ-linear functional \(\langle f(t)|\cdot \rangle _{\lambda }\) on \(\mathbb{P}\) is defined by

$$ \bigl\langle f(t)|(x)_{n,\lambda }\bigr\rangle _{\lambda }=a_{n} \quad (n\ge 0), (\text{see [10]}). $$
(12)

By (12), we get

$$ \bigl\langle t^{k}|(x)_{n,\lambda }\bigr\rangle _{\lambda }=n!\delta _{n,k}\quad (n,k \ge 0), (\text{see [10]}), $$
(13)

where \(\delta _{n,k}\) is the Kronecker symbol.

The λ-differential operators on \(\mathbb{P}\) are defined by

$$ \bigl(t^{k}\bigr)_{\lambda }(x)_{n,\lambda }=\textstyle\begin{cases} (n)_{k}(x)_{n-k,\lambda }, & \text{if }0 \le k\le n, \\ 0, & \text{if }k>n. \end{cases} $$
(14)

For \(f(t)=\sum_{k=0}^{\infty }a_{k}\frac{t^{k}}{k!}\in \mathcal{F}\), and by (14), we get

$$\begin{aligned} & \bigl(f(t) \bigr)_{\lambda }(x)_{n,\lambda }= \sum _{k=0}^{n} \binom{n}{k}a_{k}(x)_{n-k,\lambda } \quad (n\ge 0), \\ & \bigl(e_{\lambda }^{y}(t) \bigr)_{\lambda }(x)_{n,\lambda }= (x+y)_{n, \lambda } \quad (n\ge 0), (\text{see [10]}). \end{aligned}$$
(15)

Let \(f(t)\) be a delta series, and let \(g(t)\) be an invertible series. Then there exists a unique sequence \(S_{n,\lambda }(x) (\deg S_{n,\lambda }(x)=n)\) of polynomials satisfying the orthogonality conditions

$$ \bigl\langle g(t) \bigl(f(t) \bigr)^{k} | S_{n,\lambda }(x) \bigr\rangle _{\lambda }=n!\delta _{n,k} \quad (n,k\ge 0), ( \text{see [10]}). $$
(16)

Here, \(S_{n,\lambda }(x)\) is called the λ-Sheffer sequence for \((g(t), f(t) )\), which is denoted by \(S_{n,\lambda }(x) \sim (g(t),f(t) )_{\lambda }\). The sequence \(S_{n,\lambda }(x)\) is the λ-Sheffer sequence for \((g(t),f(t))\) if and only if

$$ \frac{1}{g(\overline{f}(t))}e_{\lambda }^{y} \bigl(\overline{f}(t) \bigr)= \sum_{n=0}^{\infty }S_{n,\lambda }(y) \frac{t^{n}}{n!}\quad ( \text{see [10]}) $$
(17)

for all \(y\in \mathbb{C}\), where \(\overline{f}(t)\) is the compositional inverse of \(f(t)\) such that \(f(\overline{f}(t))=\overline{f}(f(t))=t\).

Let \(S_{n,\lambda }(x)\sim (g(t),f(t) )_{\lambda }\). Then, from Theorem 16 of [10], we recall that

$$ \bigl(f(t)\bigr)_{\lambda }S_{n,\lambda }(x)=nS_{n-1,\lambda }(x), \quad (n\ge 1). $$
(18)

For \(S_{n,\lambda }(x)\sim (g(t),f(t))_{\lambda }\), \(r_{n,\lambda }(x)\sim (h(t),l(t))_{\lambda }\), we have

$$ S_{n,\lambda }(x)=\sum_{k=0}^{n}C_{n,k}r_{k,\lambda }(x), \quad (n\ge 0), (\text{see [10]}), $$
(19)

where

$$ C_{n,k}=\frac{1}{k!} \biggl\langle \frac{h(\overline{f}(t))}{g(\overline{f}(t))} \bigl(l\bigl(\overline{f}(t)\bigr) \bigr)^{k} \Big|(x)_{n,\lambda } \biggr\rangle _{\lambda }. $$
(20)

Finally, we note that λ-umbral calculus has some merit over umbral calculus when dealing with λ-Sheffer sequences. As one example, we illustrate this with the problem of representing the degenerate Bernoulli polynomial \(\beta _{n,\lambda }(x)\) in terms of the degenerate falling factorials \((x)_{k,\lambda }\). As before, let \(f(t)\) and \(g(t)\) be respectively a delta series and an invertible series. First, we recall that \(S_{n}(x)\) is Sheffer for \((g(t), f(t) )\) denoted by \(S_{n}(x) \sim (g(t),f(t))\) if and only if

$$ \frac{1}{g(\overline{f}(t))}e^{x\overline{f}(t)}=\sum_{n=0}^{\infty }S_{n}(x) \frac{t^{n}}{n!} \quad (\text{see [24]}). $$

Next, we recall that, for \(S_{n}(x)\sim (g(t),f(t))\), \(r_{n}(x)\sim (h(t),l(t))\), we have

$$ S_{n}(x)=\sum_{k=0}^{n}C_{n,k}r_{k}(x), \quad (n\ge 0), ( \text{see [24]}), $$
(21)

where

$$ C_{n,k}=\frac{1}{k!} \biggl\langle \frac{h(\overline{f}(t))}{g(\overline{f}(t))} \bigl(l\bigl(\overline{f}(t)\bigr) \bigr)^{k} \Big|x^{n} \biggr\rangle . $$
(22)

Now, we observe that \(\beta _{n,\lambda }(x)\) is λ-Sheffer for \((\frac{e_{\lambda }(t)-1}{t}, t )\), \(\beta _{n,\lambda }(x) \sim (\frac{e_{\lambda }(t)-1}{t}, t )_{\lambda }\) and Sheffer for \((\frac{\lambda (e^{t} -1)}{e^{\lambda t}-1}, \frac{1}{\lambda } (e^{\lambda t} -1) )\), \(\beta _{n,\lambda }(x) \sim ( \frac{\lambda (e^{t} -1)}{e^{\lambda t}-1}, \frac{1}{\lambda } (e^{ \lambda t} -1) )\). Also, \((x)_{n,\lambda }\) is λ-Sheffer for \((1,t)\), \((x)_{n,\lambda } \sim (1,t)_{\lambda }\) and Sheffer for \((1,\frac{1}{\lambda }(e^{\lambda t}-1) )\), \((x)_{n,\lambda } \sim (1,\frac{1}{\lambda }(e^{\lambda t}-1) )\). Let \(\beta _{n,\lambda }(x)=\sum_{k=0}^{n}C_{n,k}(x)_{k,\lambda }\). Then it is obviously easier to compute the coefficients \(C_{n,k}\) by viewing \(\beta _{n,\lambda }(x)\) and \((x)_{n,\lambda }\) as λ-Sheffer sequences and using (20) than by viewing them as Sheffer sequences and using (22).

In this paper, we study the properties of degenerate poly-Bernoulli polynomial arising from degenerate polylogarithmic function and give some identities of those polynomials associated with special polynomials which are derived from the properties of λ-Sheffer sequences.

2 Representations of degenerate poly-Bernoulli polynomials

For \(S_{n,\lambda }(x)\sim (g(t),f(t))_{\lambda }\), (\(n\ge 0\)), we have

$$\begin{aligned} \biggl\langle \frac{1}{g(\overline{f}(t))}e_{\lambda }^{x} \bigl( \overline{f}(t) \bigr) \Big|(x)_{n,\lambda } \biggr\rangle _{\lambda } &= \sum_{k=0}^{\infty }S_{k,\lambda }(x) \frac{1}{k!} \bigl\langle t^{k}|(x)_{n, \lambda } \bigr\rangle _{\lambda } \\ &= S_{n,\lambda }(x), \quad (n\ge 0). \end{aligned}$$
(23)

In (23), we note that

$$\begin{aligned} \biggl\langle \frac{1}{g(\overline{f}(t))}e_{\lambda }^{x} \bigl( \overline{f}(t) \bigr) \Big|(x)_{n,\lambda } \biggr\rangle _{\lambda } &= \sum_{j=0}^{\infty } \frac{1}{j!} \biggl\langle \frac{1}{g(\overline{f}(t))} \bigl(\overline{f}(t) \bigr)^{j} \Big|(x)_{n, \lambda } \biggr\rangle _{\lambda }(x)_{j,\lambda } \\ &= \sum_{j=0}^{n} \frac{1}{j!} \biggl\langle \frac{1}{g(\overline{f}(t))} \bigl(\overline{f}(t) \bigr)^{j} \Big|(x)_{n, \lambda } \biggr\rangle _{\lambda }(x)_{j,\lambda }. \end{aligned}$$
(24)

Therefore, by (23) and (24), we obtain the following theorem.

Theorem 1

For \(S_{n,\lambda }(x)\sim (g(t),f(t))_{\lambda }\), we have

$$S_{n,\lambda }(x)=\sum_{j=0}^{n} \frac{1}{j!} \biggl\langle \frac{1}{g(\overline{f}(t))} \bigl(\overline{f}(t) \bigr)^{j} \Big|(x)_{n, \lambda } \biggr\rangle _{\lambda }(x)_{j,\lambda }. $$

From (5) and (17), we have

$$ B_{n,\lambda }^{(k)}(x)\sim \biggl( \frac{e_{\lambda }(t)-1}{\operatorname{Li}_{k,\lambda }(1-e_{\lambda }(-t))},t \biggr)_{\lambda }. $$
(25)

By Theorem 1 applied to (25), we get the following corollary.

Corollary 2

For \(n\ge 0\), we have

$$B_{n,\lambda }^{(k)}(x)=\sum_{j=0}^{n} \binom{n}{j}B_{n-j,\lambda }^{(k)}(x)_{j, \lambda }. $$

Here we remark that Corollary 2 can also be obtained by combining (15) and the first line of (27).

From (25), we note that

$$ \biggl( \frac{e_{\lambda }(t)-1}{\operatorname{Li}_{k,\lambda }(1-e_{\lambda }(-t))} \biggr)_{\lambda }B_{n,\lambda }^{(k)}(x)=(x)_{n,\lambda } \sim (1,t)_{ \lambda }. $$
(26)

By (26), and noting that \((x)_{n,\lambda }=\sum_{l=0}^{n}\sum_{j=0}^{l}S_{2,\lambda }(n,l)S_{1, \lambda }(l,j)(x)_{j,\lambda }\), we get

$$\begin{aligned} B_{n,\lambda }^{(k)}(x) &= \biggl( \frac{\operatorname{Li}_{k,\lambda }(1-e_{\lambda }(-t))}{e_{\lambda }(t)-1} \biggr)_{\lambda }(x)_{n,\lambda } \\ &= \sum_{l=0}^{n}\sum _{j=0}^{l}S_{2,\lambda }(n,l)S_{1,\lambda }(l,j) \biggl( \frac{\operatorname{Li}_{k,\lambda }(1-e_{\lambda }(-t))}{e_{\lambda }(t)-1} \biggr)_{\lambda }(x)_{j,\lambda } \\ &= \sum_{l=0}^{n}\sum _{j=0}^{l}S_{2,\lambda }(n,l)S_{1,\lambda }(l,j) \sum_{m=0}^{j}B_{m,\lambda }^{(k)} \frac{1}{m!}\bigl(t^{m}\bigr)_{\lambda }(x)_{j, \lambda } \\ &= \sum_{l=0}^{n}\sum _{j=0}^{l}\sum_{m=0}^{j} \binom{j}{m}S_{2, \lambda }(n,l)S_{1,\lambda }(l,j)B_{m,\lambda }^{(k)}(x)_{j-m,\lambda } \\ &= \sum_{l=0}^{n}\sum _{m=0}^{l}\sum_{j=m}^{l} \binom{j}{m}S_{2, \lambda }(n,l)S_{1,\lambda }(l,j)B_{m,\lambda }^{(k)}(x)_{j-m,\lambda } \\ &= \sum_{l=0}^{n}\sum _{m=0}^{l}\sum_{j=0}^{l-m} \binom{j+m}{m}S_{2, \lambda }(n,l)S_{1,\lambda }(l,j+m)B_{m,\lambda }^{(k)}(x)_{j,\lambda }. \end{aligned}$$
(27)

Therefore, by (27), we obtain the following theorem.

Theorem 3

For \(n\ge 0\), we have

$$B_{n,\lambda }^{(k)}(x)= \sum_{l=0}^{n} \sum_{m=0}^{l}\sum _{j=0}^{l-m} \binom{j+m}{m}S_{2,\lambda }(n,l)S_{1,\lambda }(l,j+m)B_{m,\lambda }^{(k)}(x)_{j, \lambda }. $$

Now, we observe that

$$\begin{aligned} B_{n,\lambda }^{(k)}(y) &= \biggl\langle \frac{\operatorname{Li}_{k,\lambda }(1-e_{\lambda }(-t))}{e_{\lambda }(t)-1}e_{ \lambda }^{y}(t) \Big|(x)_{n,\lambda } \biggr\rangle _{\lambda } \\ &= \biggl\langle \frac{\operatorname{Li}_{k,\lambda }(1-e_{\lambda }(-t))}{t} \Big|\biggl(\frac{t}{e_{\lambda }(t)-1}e_{\lambda }^{y}(t) \biggr)_{ \lambda }(x)_{n,\lambda } \biggr\rangle _{\lambda } \\ &= \sum_{l=0}^{n}\binom{n}{l} \beta _{l,\lambda }(y) \biggl\langle \frac{1}{t} \operatorname{Li}_{k,\lambda } \bigl(1-e_{\lambda }(-t) \bigr) \Big|(x)_{n-l,\lambda } \biggr\rangle _{\lambda } \\ &= \sum_{l=0}^{n}\binom{n}{l} \beta _{l,\lambda }(y) \Biggl\langle \frac{1}{t}\sum _{m=1}^{\infty } \frac{(-\lambda )^{m-1}(1)_{m,1/\lambda }}{m^{k-1}} \frac{(-1)^{m}}{m!} \bigl(e_{\lambda }(-t)-1 \bigr)^{m} \Big|(x)_{n-l,\lambda } \Biggr\rangle _{\lambda } \\ &= \sum_{l=0}^{n}\binom{n}{l} \beta _{l,\lambda }(y) \Biggl\langle \frac{1}{t}\sum _{j=1}^{\infty } (\sum_{m=1}^{j} \frac{(-\lambda )^{m-1}(1)_{m,1/\lambda }}{m^{k-1}} (-1)^{j-m}S_{2, \lambda }(j,m) \frac{t^{j}}{j!} \Big|(x)_{n-l,\lambda } \Biggr\rangle _{ \lambda } \\ &= \sum_{l=0}^{n}\binom{n}{l} \beta _{l,\lambda }(y)\sum_{j=0}^{ \infty } \Biggl(\sum_{m=1}^{j+1} \frac{(-\lambda )^{m-1}(1)_{m,1/\lambda }}{m^{k-1}}(-1)^{j+1-m} \frac{S_{2,\lambda }(j+1,m)}{(j+1)!} \Biggr)\bigl\langle t^{j}|(x)_{n-l, \lambda }\bigr\rangle _{\lambda } \\ &= \sum_{l=0}^{n}\binom{n}{l} \beta _{l,\lambda }(y)\sum_{m=1}^{n-l+1} \frac{(-\lambda )^{m-1}(1)_{m,1/\lambda }}{m^{k-1}} \frac{(-1)^{n-l+1-m}}{(n-l+1)!}S_{2,\lambda }(n-l+1,m) (n-l)! \\ &= \sum_{l=0}^{n}\sum _{m=1}^{n-l+1} \frac{(-1)^{n-l}\binom{n}{l}\lambda ^{m-1}}{(n-l+1)m^{k-1}}(1)_{m,1/ \lambda }S_{2,\lambda }(n-l+1,m) \beta _{l,\lambda }(y). \end{aligned}$$
(28)

Therefore, by (28), we obtain the following theorem.

Theorem 4

For \(n\ge 0\), we have

$$B_{n,\lambda }^{(k)}(x)=\sum_{l=0}^{n} \sum_{m=1}^{n-l+1} \frac{(-1)^{n-l}\binom{n}{l}\lambda ^{m-1}}{(n-l+1)m^{k-1}}(1)_{m,1/ \lambda }S_{2,\lambda }(n-l+1,m) \beta _{l,\lambda }(x). $$

By (6) and (17), we note that

$$ \beta _{n,\lambda }^{(s)}(x)\sim \biggl( \frac{(e_{\lambda }(t)-1)^{s}}{t^{s}},t \biggr)_{\lambda }. $$
(29)

From (19), (25), and (29), we have

$$ B_{n,\lambda }^{(k)}(x)=\sum_{m=0}^{n}C_{n,m} \beta _{m,\lambda }^{(s)}(x), $$
(30)

where

$$\begin{aligned} C_{n,m} &= \frac{1}{m!} \biggl\langle \frac{\operatorname{Li}_{k,\lambda }(1-e_{\lambda }(-t))}{e_{\lambda }(t)-1} \frac{(e_{\lambda }(t)-1)^{s}}{t^{s}}t^{m} \Big|(x)_{n,\lambda } \biggr\rangle _{\lambda } \\ &= \binom{n}{m} \biggl\langle \frac{\operatorname{Li}_{k,\lambda }(1-e_{\lambda }(-t))}{e_{\lambda }(t)-1} \frac{(e_{\lambda }(t)-1)^{s}}{t^{s}} \Big|(x)_{n-m,\lambda } \biggr\rangle _{\lambda } \\ &= \binom{n}{m}\sum_{l=0}^{n-m} \frac{S_{2,\lambda }(l+s,s)}{\binom{l+s}{s}l!} \biggl\langle \frac{\operatorname{Li}_{k,\lambda }(1-e_{\lambda }(-t))}{e_{\lambda }(t)-1}t^{l} \Big|(x)_{n-m,\lambda } \biggr\rangle _{\lambda } \\ &= \binom{n}{m}\sum_{l=0}^{n-m} \frac{S_{2,\lambda }(l+s,s)}{\binom{l+s}{s}}\binom{n-m}{l} \biggl\langle \frac{\operatorname{Li}_{k,\lambda }(1-e_{\lambda }(-t))}{e_{\lambda }(t)-1} \Big|(x)_{n-m-l,\lambda } \biggr\rangle _{\lambda } \\ &= \binom{n}{m}\sum_{l=0}^{n-m} \frac{S_{2,\lambda }(l+s,s)}{\binom{l+s}{s}}\binom{n-m}{l}B_{n-m-l, \lambda }^{(k)}. \end{aligned}$$
(31)

Therefore, by (30) and (31), we obtain the following theorem.

Theorem 5

For \(n\ge 0\), we have

$$B_{n,\lambda }^{(k)}(x)=\sum_{m=0}^{n} \binom{n}{m} \Biggl(\sum_{l=0}^{n-m} \frac{\binom{n-m}{l}}{\binom{l+s}{s}}S_{2,\lambda }(l+s,s)B_{n-m-l, \lambda }^{(k)} \Biggr)\beta _{m,\lambda }^{(s)}(x). $$

For \(n\ge 0\), we let

$$\mathbb{P}_{n}= \bigl\{ p(x)\in \mathbb{C}[x] | \deg p(x)\le n \bigr\} . $$

Then \(\mathbb{P}_{n}\) is an \((n+1)\)-dimensional vector space over \(\mathbb{C}\).

From (11), we note that \(d_{n,\lambda }(x)\sim ((1-t)e_{\lambda }(t),t)_{\lambda }\), (\(n\ge 0\)). For \(p(x)\in \mathbb{P}_{n}\), we let

$$ p(x)=\sum_{l=0}^{n}C_{l}d_{l,\lambda }(x). $$
(32)

By (16), we have

$$\begin{aligned} \bigl\langle (1-t)e_{\lambda }(t)t^{m} | p(x)\bigr\rangle _{\lambda } &= \sum_{l=0}^{n}C_{l} \bigl\langle (1-t)e_{\lambda }(t)t^{m} | d_{l, \lambda }(x) \bigr\rangle _{\lambda } \\ &= \sum_{l=0}^{n}C_{l}m! \delta _{m,l} = C_{m}m!, \end{aligned}$$
(33)

where \(0\le m\le n\).

Therefore, by (32) and (33), we obtain the following theorem.

Theorem 6

For \(p(x)\in \mathbb{P}_{n}\), we have

$$p(x)=\sum_{l=0}^{n}C_{l}d_{l,\lambda }(x), $$

where

$$C_{l}=\frac{1}{l!} \bigl\langle (1-t)e_{\lambda }t^{l}|p(x) \bigr\rangle _{\lambda }. $$

Let \(p(x)=B_{n,\lambda }^{(k)}(x)\in \mathbb{P}_{n}\). Then we have

$$ B_{n,\lambda }^{(k)}(x)=\sum_{l=0}^{n}C_{l}d_{l,\lambda }(x), $$
(34)

where

$$\begin{aligned} C_{l} &= \frac{1}{l!} \bigl\langle (1-t)e_{\lambda }(t)t^{l}|B_{n, \lambda }^{(k)}(x) \bigr\rangle _{\lambda } \\ &= \binom{n}{l} \bigl\langle (1-t)e_{\lambda }(t)|B_{n-l,\lambda }^{(k)}(x) \bigr\rangle _{\lambda } \\ &= \binom{n}{l} \bigl\langle (1-t)|B_{n-l,\lambda }^{(k)}(x+1) \bigr\rangle _{\lambda } \\ &= \binom{n}{l} \bigl\langle 1 | B_{n-l,\lambda }^{(k)}(x+1) \bigr\rangle _{\lambda }-\binom{n}{l}(n-l) \bigl\langle 1 | B_{n-l-1, \lambda }^{(k)}(x+1) \bigr\rangle _{\lambda } \\ &= \binom{n}{l}B_{n-l,\lambda }^{(k)}-n \binom{n-1}{l}B_{n-l-1,\lambda }^{(k)}(1). \end{aligned}$$
(35)

Thus, by (34) and (35), we get

$$ B_{n,\lambda }^{(k)}(x)=\sum_{l=0}^{n} \biggl(\binom{n}{l}B_{n-l, \lambda }^{(k)}-n \binom{n-1}{l}B_{n-l-1,\lambda }^{(k)} \biggr)d_{l, \lambda }(x). $$
(36)

From (11), we note that \(d_{n,r}^{(r)}(x)\sim ((1-t)^{r}e_{\lambda }(t),t )_{\lambda }\).

Let us assume that

$$ p(x)=\sum_{m=0}^{n}C_{m}^{(r)}d_{m,\lambda }^{(r)}(x) \in \mathbb{P}_{n}. $$
(37)

Then, by (16), we get

$$\begin{aligned} \bigl\langle (1-t)^{r}e_{\lambda }(t)t^{m}|p(x) \bigr\rangle _{ \lambda } &= \sum_{l=0}^{n}C_{l}^{(r)} \bigl\langle (1-t)^{r}e_{ \lambda }(t)t^{m} | d_{l,\lambda }^{(r)}(x) \bigr\rangle _{\lambda } \\ &= m!C_{m}^{(r)} \quad (0\le m\le n). \end{aligned}$$
(38)

Therefore, by (37) and (38), we obtain the following theorem.

Theorem 7

For \(n\ge 0\), we have

$$p(x)=\sum_{m=0}^{n}C_{m}^{(r)}d_{m,\lambda }^{(r)}(x), $$

where

$$C_{m}^{(r)}=\frac{1}{m!} \bigl\langle (1-t)^{r}e_{\lambda }(t)t^{m} | p(x) \bigr\rangle _{\lambda }. $$

We let

$$ d_{n,\lambda }(x)=\sum_{m=0}^{n}C_{m}^{(r)}d_{m,\lambda }^{(r)}(x), $$
(39)

where

$$\begin{aligned} C_{m}^{(r)} &= \frac{1}{m!} \bigl\langle (1-t)^{r}e_{\lambda }(t)t^{m} | d_{n,\lambda }(x) \bigr\rangle _{\lambda } \\ &= \binom{n}{m} \bigl\langle (1-t)^{r} | d_{n-m,\lambda }(x+1) \bigr\rangle _{\lambda } \\ &= \binom{n}{m}\sum_{j=0}^{r} \binom{r}{j}(-1)^{j}(n-m)_{j} \bigl\langle 1|d_{n-m-j,\lambda }(x+1)\bigr\rangle _{\lambda } \\ &= \binom{n}{m}\sum_{j=0}^{r} \binom{r}{j}\binom{n-m}{j}(-1)^{j}d_{n-m-j, \lambda }(1)j!. \end{aligned}$$
(40)

By (39) and (40), we get

$$ d_{n,\lambda }(x)=\sum_{m=0}^{n} \binom{n}{m} \Biggl(\sum_{j=0}^{r} \binom{r}{j}\binom{n-m}{j}j!(-1)^{j}d_{n-m-j,\lambda }(1) \Biggr)d_{m, \lambda }^{(r)}(x), \quad (n\ge 0). $$
(41)

Let us take \(p(x)=B_{n,\lambda }^{(k)}(x)\in \mathbb{P}_{n}\), (\(n\ge 0\)). Then, by Theorem 7, we get

$$ B_{n,\lambda }^{(k)}(x)=\sum_{m=0}^{n}C_{m}^{(r)}d_{m,\lambda }^{(r)}(x), \quad (n\ge 0), $$
(42)

where

$$\begin{aligned} C_{m}^{(r)} &= \frac{1}{m!} \bigl\langle (1-t)^{r}e_{\lambda }(t)t^{m} |B_{n,\lambda }^{(k)}(x) \bigr\rangle _{\lambda } \\ &= \binom{n}{m} \bigl\langle (1-t)^{r}|B_{n-m,\lambda }^{(k)}(x+1) \bigr\rangle _{\lambda } \\ &= \binom{n}{m}\sum_{j=0}^{r} \binom{r}{j}(-1)^{j}\binom{n-m}{j}j! \bigl\langle 1|B_{n-m-j}^{(k)}(x+1) \bigr\rangle _{\lambda } \\ &= \binom{n}{m}\sum_{j=0}^{r} \binom{r}{j}(-1)^{j}\binom{n-m}{j}j!B_{n-m-j, \lambda }^{(k)}(1). \end{aligned}$$
(43)

Therefore, by (42) and (43), we obtain the following theorem.

Theorem 8

For \(n\ge 0\), \(k\in \mathbb{Z}\), and \(r\in \mathbb{N}\), we have

$$B_{n,\lambda }^{(k)}(x)=\sum_{m=0}^{n} \binom{n}{m} \Biggl(\sum_{j=0}^{r} \binom{r}{j}(-1)^{j}\binom{n-m}{j}j!B_{n-m-j,\lambda }^{(k)}(1) \Biggr)d_{m, \lambda }^{(r)}(x). $$

3 Conclusion

The umbral calculus had been laid as a rigorous foundation by Rota and is based on linear functionals, differential operators, and Sheffer sequences. Here, for an invertible series \(g(t)\) and a delta series \(f(t)\), \(S_{n}(x)\) is the Sheffer sequence for \((g(t), f(t) )\) if and only if

$$ \frac{1}{g(\overline{f}(t))}e^{x\overline{f}(t)}=\sum _{n=0}^{\infty }S_{n}(x) \frac{t^{n}}{n!}. $$
(44)

Recently, the ‘λ-umbral calculus’ was developed by the motivation that what if the usual exponential function appearing in (44) is replaced with the degenerate exponential functions in (2). This question led us to the introduction of the concepts like λ-linear functionals, λ-differential operators, and λ-Sheffer sequences. In fact, for \(g(t)\) and \(f(t)\) as before, the sequence \(S_{n,\lambda }(x)\) is the λ-Sheffer sequence for \((g(t),f(t))\) if and only if

$$ \frac{1}{g(\overline{f}(t))}e_{\lambda }^{x} \bigl(\overline{f}(t) \bigr)= \sum_{n=0}^{\infty }S_{n,\lambda }(x) \frac{t^{n}}{n!}. $$

We noted that the λ-umbral calculus has some advantage over the traditional umbral calculus when dealing with λ-Sheffer sequences. This was illustrated with the problem of representing the degenerate Bernoulli polynomial in terms of the degenerate falling factorials. Moreover, the introduction of the λ-umbral calculus is natural in view of the recent regained interests of many mathematicians in the study of degenerate versions of some special polynomials and numbers, which was initiated by Carlitz.

In this paper, in order to study the degenerate poly-Bernoulli polynomials, which is a λ-Sheffer sequence, we used three different formulas, namely a formula about representing a λ-Sheffer sequence by another, a formula coming from the generating functions of λ-Sheffer sequences, and a formula arising from the definitions for λ-Sheffer sequences. Then we represented, among other things, the degenerate poly-Bernoulli polynomials by higher-order degenerate Bernoulli polynomials and also by higher-order degenerate derangement polynomials.

It is one of our future projects to continue to investigate the degenerate special numbers and polynomials by using the recently developed λ-umbral calculus.