Representations of modified type 2 degenerate poly-Bernoulli polynomials

1.   Introduction

The degenerate Stirling, Bernoulli and Euler polynomials and numbers were first introduced by Carlitz ^[5,6] with intriguing results of arithmetic and combinatorial identities. In recent years, much attention has been paid both to providing degenerate versions of special polynomials and numbers, and finding their relations and connections to known degenerate polynomials, as well as transcendental functions such as degenerate gamma functions and degenerate Laplace transforms ^[18]. In general, degenerate versions of special functions are provided by modifying generating functions in terms of the degenerate exponential functions and the degenerate logarithm function, (see ^{[1,7,11,12,14,16,20,21,25]} and the references therein). To investigate their arithmetic and combinatorial properties and relations, the research is usually performed by combinatorial methods, umbral calculus techniques, $p$-adic analysis, and differential equations. In particular, the relation between two $\lambda$-Sheffer sequences can be expressed by the representation formula (3.6) that is recently established ^[17] replacing linear functionals and differential operators by $\lambda$-functionals and $\lambda$-differential operators, respectively, in Rota's theory ^[26] of Sheffer sequences.

The aim of the paper is to introduce a new degenerate version of the type 2 poly-Bernoulli polynomials ^[21] called modified type 2 degenerate poly-Bernoulli polynomials (2.1) by modifying the generating function and present their properties as well as relations to other degenerate versions of polynomials. Also, we provide the computational examples to obtain explicit representations of the proposed polynomials for various $k$ and $\lambda$ based on two well known special polynomials: the degenerate Lah-Bell polynomials, and the higher-order degenerate derangement polynomials.

The rest of the paper is organized as follows: Section 2 introduces a new class of degenerate type 2 poly-Bernoulli polynomials and related numbers. Then, we present their properties and expressions in terms of the degenerate Stirling numbers of the second kind and the degenerate type 2 Bernoulli polynomials. Section 3 reviews the $\lambda$-linear functionals, related $\lambda$-differential operators, and $\lambda$-Sheffer polynomials, and presents the connection between the proposed polynomials and the known degenerate polynomials, such as the type 2 degenerate Bernoulli polynomials, the degenerate Hermite polynomials, the degenerate derangement polynomials, the higher-order degenerate Euler polynomials, and the degenerate Lah-Bell polynomials. Section 4 presents several specific examples to confirm the computed results between the proposed polynomials and the existing degenerate type polynomials, the degenerate Lah-Bell polynomials and the degenerate derangement polynomials. Section 5 concludes the paper. Before closing this section, we recall several definitions to introduce our new type of poly-Bernoulli polynomials.

Throughout this paper, we denote $\mathbb{N}$ the set of all natural numbers and $\mathbb{Z}$ be the set of all integers. The Bernoulli polynomials $B_n(x)$ and the Bernoulli numbers $B_n$ for $\mathbb{N}\cup\{0\}$ are respectively given by the generating functions (see ^[2,16,21] for details):

The degenerate Bernoulli polynomials $B_{n, \lambda}(x)$ and the degenerate Bernoulli numbers $B_{n, \lambda}$ are respectively defined by (see ^[5,6] for detail)

Here, the degenerate exponential functions $e_{\lambda}^x(t)$ for $\lambda \in \mathbb{R}\backslash \{0\}$ are given by

where $(x)_{n, \lambda}$ is the $\lambda$-falling factorial sequence defined by (see ^[21])

The degenerate Bernoulli polynomials $B_{n, \lambda}^{(r)}(x)$ of order $r \geq 1$ are defined by (see ^[16])

The degenerate type 2-Bernoulli polynomials $\beta_{n, \lambda}(x)$ in ^[16] are considered as

and the related extension, the type 2 degenerate Bernoulli polynomials ${\beta}_{n, \lambda}^{(r)}(x)$ of order $r$ is introduced (see ^[19])

The type 2 degenerate Euler polynomials $\mathcal{E}_{n, \lambda}(x)$ are introduced in ^[12] based on the following generating function:

Next, we briefly recall some basic functions used throughout the paper. The polylogarithm functions $Li_k(t)$ for $k\in \mathbb{Z}$ are defined by (see ^[13,19])

Note that $Li_k(t)$ for $k = 1$ satisfy

The degenerate polylogarithm functions $Li_{k, \lambda}(t)$ for $k\in \mathbb{Z}$ are given by (see ^[20])

The degenerate Stirling numbers $S_{2, \lambda}(n, m)$ of the second kind are defined by (see ^[23,24])

Note that (see ^[23])

where $(t)_{n}$ is the falling factorial sequence defined by (see ^[16])

2.   Modified type 2 degenerate poly-Bernoulli polynomials

In this section, we introduce a new type of degenerate poly-Bernoulli polynomials and numbers and discuss their properties and representations in relation to the degenerate type 2-Bernoulli polynomials.

Definition 2.1. The modified type 2 degenerate poly-Bernoulli polynomials $\mathfrak{B}^{(k)}_{n, \lambda}(x)$ for $k\in\mathbb{Z}$ are defined by the generating function

In the case of $x = 0$, we call ${\mathfrak{B}^{(k)}_{n, \lambda} }: = {\mathfrak{B}^{(k)}_{n, \lambda} }(0)$ the modified type 2 degenerate poly-Bernoulli numbers.

Remark 2.2. By noting that ${ Li_{1, \lambda}(1-e_\lambda(-t))} = t$ for $k = 1$, $\mathfrak{B}^{(1)}_{n, \lambda}(x)$ yield the degenerate type 2-Bernoulli polynomials $\beta_{n, \lambda}(x)$ listed in (1.6), we can show that $\lim_{\lambda\to0} \mathfrak{B}^{(1)}_{n, \lambda}(x) = \beta_{n} (x)$ for $k = 1$, where $\beta_{n} (x)$ are type 2-Bernoulli polynomials ^[13] that satisfy

We next state the expression for ${\mathfrak{B}^{(k)}_{n, \lambda} }(x)$ as the sum of the products of the modified type 2 degenerate poly-Bernoulli numbers and $\lambda$-falling factorial sequence.

Theorem 2.3. Let $k$ be any integer. Then the following identity holds true for all $n\geq 0$,

where $(x)_{n, \lambda}$ is the $\lambda$-falling factorial sequence in (1.4).

Proof. From Definition 2.1, we can consider the generating series of the polynomials ${\mathfrak{B}^{(k)}_{n, \lambda} }$ as the indicated product:

By the comparison of coefficients, we have the desired identity.

Remark 2.4. Since $\frac{d}{dt} e_\lambda(-t) = \frac{-1}{1-\lambda t}e_{\lambda}(-t)$ and $1-\lambda t = e^{\lambda}_{\lambda}(-t)$, we have

for $k\geq 2$. By the multiple integrals of $Li_{1, \lambda}\left(1-e_\lambda(-t)\right)$, the function $Li_{k, \lambda}\left(1-e_\lambda(-t)\right)$ is computed, and ${\mathfrak{B}^{(k)}_{n, \lambda} }(x)$ can be expressed as

in which $k-1$ times of integrals are performed by the product with $\frac{e_\lambda^{1-\lambda}(-t_{k-1})}{1-e_\lambda(-t_{k-1})}$ each time. For example, when $k = 2$, we have

Thus, by comparing coefficients in both sides of (2.2), we have that the identity holds

Next, we can express the modified type 2 degenerate poly-Bernoulli polynomials in terms of the degenerate Stirling numbers of the second kind and the degenerate type 2-Bernoulli polynomials (1.6).

Theorem 2.5. For $n\geq0$ and $\lambda \in \mathbb{R}\backslash\{0\}$, the following identity holds:

where $S_{2, \lambda}(n, m)$ are the degenerate Stirling numbers of the second kind defined in (1.10).

Proof. We first note that the degenerate polylogarithm function (1.9) for $0 < \lambda < 1$ satisfies

From (2.3) and Definition 2.1, we can see that

Comparison of the coefficients on both sides concludes the desired result for $\lambda\neq 0$.

Using Theorem 2.3 and Eq (2.3), we have the following result.

Theorem 2.6. For $n\geq 0$ and $k\in\mathbb{Z}$, the following relationholds:

Proof. First, notice that, from (2.3), we have

On the other hand, by the virtue of the result of Theorem 2.3, it follows that

which shows the desired result by comparison of the coefficients of (2.6) and (2.7).

3.   Representations of Modified Type 2 Degenerate poly-Bernoulli Polynomials

In this section, we recall a family of $\lambda$-linear functionals on the space of polynomials, $\lambda$-differential operators related to the family of $\lambda$-linear functionals, and $\lambda$-Sheffer sequences. The details on these concepts and definitions can be found in ^{[2,3,4,8,10,17]} and the references therein.

Let $\mathcal{F}$ be the algebra of exponential formal power series in $t$ over the field $\mathbb{C}$ of complex numbers

and let $\mathbb{P}$ be the algebra of polynomials in $x$ over $\mathbb{C}$, i.e.,

Recall that $\left((x)_{n, \lambda}\right)_{n\in\mathbb{N}}$ form a basis for the $\mathbb{C}$-vector space $\mathbb{P}$. Then, the $\lambda$-linear functional $\langle f(t)| \cdot\rangle_{\lambda}$ on $\mathbb{P}$ for $f(t) = \sum_{n = 0}^\infty a_n \frac{t^n}{n!}\in \mathcal{F}$ is defined by

and satisfies

where $\delta_{n, k}$ is the Kronecker delta.

The order $o(f(t))$ of the formal power series for a nontrivial $f(t)$ is the smallest integer $k$ for which $a_k$ does not vanish. In particular, $f(t)$ is called a delta series if $o(f(t)) = 1$, while $f(t)$ is called an invertible series if $o(f(t)) = 0$, (see ^[17] for details).

In ^[17], the $\lambda$- differential operator $(t^k)_{\lambda}$ of order $k$ on $\mathbb{P}$ is defined by

where $(n)_{k} = n(n-1)(n-2)\cdots (n-k+1)$ for $n\geq 1$ and $(n)_{0} = 1$. In general, for $f(t) = \sum_{k = 0}^\infty a_k \frac{t^k}{k!}\in \mathcal{F}$, the $\lambda$-differential operator $\big(f(t)\big)_{\lambda}$ is defined by

Or equivalently, $\big(f(t)\big)_{\lambda}$ can be expressed by

In particular, when $f(t) = e_\lambda^{x}(t)$ where $e_\lambda^{x}(t) = \sum_{n = 0}^\infty (x)_{n, \lambda}\frac{t^n}{n!}$, $(f(t))_{\lambda}$ satisfies

For $f(t)$ a delta series and $g(t)$ an invertible series, i.e., $o\big(f(t)\big) = 1$ and $o\big(g(t)\big) = 0$, there exists a unique sequence ${s}_{n, \lambda}(x)$ of polynomials $deg\big({s}_{n, \lambda}(x)\big) = n$ satisfying the orthogonality condition

Here, ${s}_{n, \lambda}(x)$ is called the $\lambda$-Sheffer sequence for $\big(g(t), f(t)\big)$ and is denoted by ${s}_{n, \lambda}(x)\sim \big(g(t), f(t)\big)_{\lambda}$.

We recall that ${s}_{n, \lambda}(x) \sim \left(g(t), f(t)\right)_{\lambda}$ if and only if

where $\bar{f}(t)$ is the compositional inverse of $f(t)$ and satisfies $\bar{f}\left(f(t)\right) = f\left(\bar{f}(t)\right) = t$.

For given $\lambda$-Sheffer sequences ${s}_{n, \lambda}(x)\sim \big(g(t), f(t)\big)_{\lambda}$ and ${r}_{n, \lambda}(x)\sim \big(h(t), \ell(t)\big)_{\lambda}$, the following relation holds

where ${c}_{n, k}$ is obtained by

Therefore, since $(x)_{n, \lambda}\sim(1, t)_\lambda$, any $\lambda$-Sheffer sequence ${s}_{n, \lambda}(x)\sim \left(g(t), f(t)\right)_{\lambda}$ satisfies

Remark 3.1. From (3.1), we have

so that (3.8) yields

Further, using $(x)_{\ell, \lambda} = \sum_{m = 0}^\ell \sum_{j = 0}^m S_{2, \lambda}(\ell, m)S_{1, \lambda}(m, j)(x)_{j, \lambda}$ in (3.9), the identity holds

where $S_{1, \lambda}(n, m)$ are the degenerate Stirling numbers of the first kind ^[24] given by

Now, we explore the representations of the modified type 2 degenerate poly-Bernoulli polynomials in terms of various known degenerate polynomials by using (3.6). We first note that from (3.5), the corresponding $\lambda$-Sheffer sequence for ${\mathfrak{B}^{(k)}_{n, \lambda} } (x)$ is given by

Theorem 3.2. Let $\beta_{n, \lambda}(x)$ be the degenerate type 2-Bernoulli polynomials (1.6). Then for any $n\geq0$ and integer $k\in\mathbb{Z}$, we have

Proof. From (1.6), (2.3), (3.1), (3.2), and Definition 2.1, the following computations are established.

which yields the result.

Next, we consider the type 2 degenerate Bernoulli polynomials of order $r$ defined in (1.7), which satisfy the following $\lambda$-Sheffer sequence

Theorem 3.3. For any $n\geq0$ and integer $k\in\mathbb{Z}$, we have

Proof. Let ${\mathfrak{B}^{(k)}_{n, \lambda} } (x) = \sum\limits_{m = 0}^n c_{n, m} \beta_{m, \lambda}^{(r)} (x)$, {$c_{n, m} \in \mathbb{C}$.} Then, by (3.8) with (3.7), we can obtain

which yields the conclusion from $\langle \left.\frac{ {\rm{Li}}_{k, \lambda}(1-e_\lambda(-t))}{e_{\lambda}(t)-e_\lambda^{-1}(t)}\right| (x)_{n-m-\ell, \lambda} \rangle_{\lambda} = {\mathfrak{B}^{(k)}_{n-m-\ell, \lambda}}$.

We also consider the degenerate Hermite polynomials ^[9], which are defined by

where $H_{n, \lambda} (x)$ can be considered to be the $\lambda$-Sheffer sequence for $\left(e_{\lambda}(\frac{1}{4}t^2), \frac{t}{2}\right)$, or equivalently,

Let $\mathbb{P}_n = \{p(x)\in \mathbb{C}[x] |deg p(x)\leq n \}$ be an $(n+1)$ dimensional vector space over $\mathbb{C}$. Since the degenerate Hermite polynomials form a basis of the vector space $\mathbb{P}_n$, for every polynomial $p(x)$ of degree $n$ we have

where $c_{n, k}$ satisfies

Thus, we have an explicit form of $c_{n, k}$.

Theorem 3.4. For any $n\geq0$ and integer $k\in\mathbb{Z}$, we have

Proof. Taking $p(x) = {\mathfrak{B}^{(k)}_{m, \lambda} (x)}$ in (3.11), we have

From (1.3), (3.7) and (3.12), we get

and yield the conclusive result.

Next, let $d^{(r)}_{n, \lambda}(x)$ be the degenerate derangement polynomials ^[22] of order $r\in \mathbb{N}$ given by

These polynomials form a $\lambda$-Sheffer sequence such that

Then, we have the following representation of ${\mathfrak{B}^{(k)}_{n, \lambda} } (x)$.

Theorem 3.5. For any $n\geq0$ and integer $k\in\mathbb{Z}$, we have

Proof. Let $p(x) = {\mathfrak{B}^{(k)}_{n, \lambda} (x)}\in \mathbb{P}_n$. Then we can express that

where $c_{n, \ell}$ satisfy

which provides the conclusion.

Remark 3.6. In the case of $r = 1$ in the previous result, the modified type 2 degenerate poly-Bernoulli polynomials have the relation with the degenerate derangement polynomials of order $1$ as

Recalling that the degenerate Bernoulli polynomials ^[6] of order $r$ are given by

we obtain the following relation.

Theorem 3.7. For any $n\geq0$ and integer $k\in\mathbb{Z}$, we have

Proof. Let $p(x) = {\mathfrak{B}^{(k)}_{n, \lambda} (x)}\in \mathbb{P}_n$. Then we can express $p(x)$ in terms of $B^{(r)}_{n, \lambda}(x)$

where $B^{(r)}_{n, \lambda}(x)$ satisfy

and $c_{n, \ell}$ is computed by

In the above computation, the following relation is applied

Applying the property of (3.1) in the last equation of (3.13) completes the proof.

Taking into account the degenerate Euler polynomials ^[6] of order $r$

we have the following result.

Theorem 3.8. For any $n\geq0$ and integer $k\in\mathbb{Z}$, we have

Proof. We note that the corresponding $\lambda$-Sheffer sequence for $E^{(r)}_{n, \lambda}(x)$ is given by

Then, we have

where $c_{n, \ell}$ satisfies

Thus, the desired identity can be obtained by the property of (3.1).

We finally consider the degenerate Lah-Bell polynomials ^[15] that are defined by

and the unsigned Lah numbers $L(n, k)$ defined by

having the generating function

Theorem 3.9. For any $n\geq0$ and integer $k\in\mathbb{Z}$, ${\mathfrak{B}^{(k)}_{n, \lambda} } (x)$ satisfy

Proof. If we let $p(x) = {\mathfrak{B}^{(k)}_{n, \lambda} (x)}\in \mathbb{P}_n$, then we have

By noting that the corresponding $\lambda$-Sheffer sequence is given by

the followings are established for $c_{n, \ell}$:

in which the identity for the unsigned Lah numbers is applied:

Therefore, $c_{n, \ell}$ is explicitly computed in a the similar way.

4.   Illustrative examples

In this section, we present two examples that show the explicit representations of the modified type 2 degenerate poly-Bernoulli polynomials in terms of the degenerate Lah-Bell polynomials and the degenerate derangement polynomials of order $r\in \mathbb{N}$. We compute the combinatorial results $c_{n, \ell}$ presented in Theorems 3.5 and 3.9 to confirm the connections.

To do this, we first calculate ${\mathfrak{B}^{(0)}_{n, \lambda}(x)}$ for $k = 0, -1$ and $n = 0, 1, 2, 3, 4$. For $k = 0$, ${\mathfrak{B}^{(0)}_{n, \lambda}(x)}$ can be obtained using ${ Li_{0, \lambda}(t)} = t(1-t)^{\lambda-1}$, which are listed below:

For $k = -1$, ${\mathfrak{B}^{(-1)}_{n, \lambda}(x)}$ are computed as follows:

Figure 1 illustrates the shapes of ${\mathfrak{B}^{(k)}_{n, \lambda}(x)}, \; k = -1$ for $n = 0, 1, 2, 3, 4$ when $\lambda = 1/10$ and $\lambda = 1$.

We express ${\mathfrak{B}^{(k)}_{n, \lambda}(x)}$, $k = 0, -1, \; n = 0, 1, 2, 3, 4$ by the degenerate Lah-Bell polynomials $B^L_{n, \lambda} (x)$ for various $\lambda = \frac{1}{100}, \frac{1}{10}, {1}$ in the following example.

Example 4.1 (Illustration of Theorem 3.9). The degenerate Lah-Bell polynomials $B^L_{n, \lambda} (x)$ for $n = 0, 1, 2, 3, 4$ are explicitly given by

Figure 2 shows the graphs of ${ B^L_{n, \lambda} }(x)$ for $n = 0, 1, 2, 3, 4$ when $\lambda = 1/10, 1$.

To confirm the established identity in Theorem 3.9, we compute $\sum\limits_{\ell = 0}^n c_{n, \ell} B^L_{\ell, \lambda}(x)$, where

Then, we present that each linear combination of $B^L_{n, \lambda} (x)$ with $c_{n, \ell}$ is identically equal to ${\mathfrak{B}^{(k)}_{n, \lambda}(x)}$, $n = 1, 2, 3, 4$ for $\lambda = 1/100, 1/10, 1$ and $k = 0, -1$:

Case (Ⅰ): For $k = 0$ and $\lambda = \frac{1}{100}$,

Case (Ⅱ): For $k = 0$, $\lambda = 1$,

Case (Ⅲ): For $k = -1$, $\lambda = 1/10$,

Case (Ⅳ): For $k = -1$, $\lambda = 1,$

In the next example, we express ${\mathfrak{B}^{(k)}_{n, \lambda}(x)}$, $k = 0, -1, \; n = 0, 1, 2, 3, 4$ by the degenerate derangement polynomials $d^{(r)}_{n, \lambda}(x)$ for $\lambda = \frac{1}{10}, 1, 10$, as selected cases of the results in Theorem 3.5.

Example 4.2 (Illustration of Theorem 3.5). The degenerate derangement polynomials of order $r = 2$ for $n = 0, 1, 2, 3, 4$ are listed as:

We plot the graphs of $d^{(r)}_{n, \lambda}(x)$ of order $2$ for $n = 0, 1, 2, 3, 4$ when $\lambda = 1/10, 1$ in Figure 3.

To confirm the identity presented in Theorem 3.5, we compute $\sum\limits_{\ell = 0}^n c_{n, \ell} d^{(2)}_{n, \lambda}(x)$, where

Then, the following presentations support the analysis that each linear combination of $d^{(2)}_{n, \lambda}(x)$ with computed $c_{n, \ell}$ is exactly equal to ${\mathfrak{B}^{(k)}_{n, \lambda}(x)}$, $n = 1, 2, 3, 4$ when $\lambda = 1, 10$ and $k = 0$:

Case (Ⅰ): For $k = 0$, $\lambda = 10$,

Case (Ⅱ): For $k = 0$, $\lambda = 1$,

5.   Conclusions

In this paper, we propose a new variant of type 2 poly-Bernoulli polynomials and numbers using the generating function that is a combination of the degenerate exponential functions with the degenerate polylogarithm function. Then, we derive explicit expressions of the proposed polynomials in terms of the degenerate type 2-Bernoulli polynomials and the degenerate Stirling numbers of the second kind. Furthermore, using $\lambda$-linear functionals and $\lambda$-Sheffer sequences, we present connections of the modified type 2 degenerate poly-Bernoulli polynomials among existing $\lambda$-Sheffer polynomials: the degenerate type 2-Bernoulli polynomials, the higher-order type 2 degenerate Bernoulli polynomials, the degenerate Hermite polynomials, the higher-order degenerate Bernoulli polynomials, the higher-order degenerate derangement polynomials, the degenerate Euler polynomials, and the higher-order degenerate Lah-Bell polynomials. Finally, to confirm the presented combinatorial results, we explicitly compute the connection constants and show combinations with the degenerate Lah-Bell polynomials and the higher-order degenerate derangement polynomials.

Acknowledgments

This work was supported by the Dongguk University Research Fund of 2020 and the first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (No. 2020R1F1A1A0107156412).

Conflict of interest

The authors declare no conflict of interest.