Theory and Modern Applications

# On $$(h,q)$$-Daehee numbers and polynomials

## Abstract

The p-adic q-integral (sometimes called q-Volkenborn integration) was defined by Kim. From p-adic q-integral equations, we can derive various q-extensions of Bernoulli polynomials and numbers. DS Kim and T Kim studied Daehee polynomials and numbers and their applications. Kim et al. introduced the q-analogue of Daehee numbers and polynomials which are called q-Daehee numbers and polynomials. Lim considered the modified q-Daehee numbers and polynomials which are different from the q-Daehee numbers and polynomials of Kim et al. In this paper, we consider $$(h,q)$$-Daehee numbers and polynomials and give some interesting identities. In case $$h=0$$, we cover the q-analogue of Daehee numbers and polynomials of Kim et al. In case $$h=1$$, we modify q-Daehee numbers and polynomials. We can find out various $$(h,q)$$-related numbers and polynomials which are studied by many authors.

## 1 Introduction

Let p be a fixed prime number. Throughout this paper, $$\mathbb{Z}_{p}$$, $$\mathbb{Q}_{p}$$ and $$\mathbb{C}_{p}$$ will respectively denote the ring of p-adic rational integers, the field of p-adic rational numbers and the completion of algebraic closure of $$\mathbb{Q}_{p}$$. The p-adic norm is defined $$|p|_{p}=\frac{1}{p}$$.

When one talks of q-extension, q is variously considered as an indeterminate, complex $$q\in\mathbb{C}$$, or p-adic number $$q\in\mathbb {C}_{p}$$. If $$q\in\mathbb{C}$$, one normally assumes that $$|q|<1$$. If $$q\in \mathbb{C}_{p}$$, then we assume that $$|q-1|_{p}< p^{-\frac{1}{p-1}}$$ so that $$q^{x}=\exp(x\log q)$$ for each $$x\in\mathbb{Z}_{p}$$. Throughout this paper, we use the notation

$$[x]_{q}=\frac{1-q^{x}}{1-q}.$$

Note that $$\lim_{q\rightarrow1}[x]_{q}=x$$ for each $$x\in\mathbb{Z}_{p}$$.

Let $$UD(\mathbb{Z}_{p})$$ be the space of a uniformly differentiable function on $$\mathbb{Z}_{p}$$. For $$f\in UD(\mathbb{Z}_{p})$$, the p-adic q-integral on $$\mathbb{Z}_{p}$$ is defined by Kim as follows:

$$I_{q}(f)=\int_{\mathbb{Z}_{p}}f(x)\,d\mu_{q}(x)=\lim_{N\rightarrow\infty }\frac{1}{[p^{N}]_{q}}\sum _{x=0}^{p^{N}-1}f(x)q^{x} \quad(\mbox{see [1, 2]}).$$
(1)

Using this integration, the q-Daehee polynomials $$D_{n,q}(x)$$ are defined and studied by Kim et al. (see [3]), their generating function is as follows:

$$\frac{1-q+\frac{1-q}{\log q}\log(1+t)}{1-q-qt}(1+t)^{x}=\sum _{n=0}^{\infty}D_{n,q}(x)\frac{t^{n}}{n!}.$$
(2)

The generating function of the modified q-Daehee polynomials are defined and studied by Lim (see [4]).

$$F_{q}(x,t)=\frac{q-1}{\log q}\frac{\log(1+t)}{t}(1+t)^{x}= \sum_{n=0}^{\infty}D_{n}(x|q) \frac{t^{n}}{n!} \quad(\mbox{see [1--16]}).$$
(3)

From (1), we have the following integral identity:

$$qI_{q}(f_{1})-I_{q}(f)= \frac{q-1}{\log q}f'(0)+(q-1)f(0),$$
(4)

where $$f_{1}(x)=f(x+1)$$ and $$\frac{d}{dx}f(x)=f'(x)$$.

In a special case, for $$h\in\mathbb{Z}_{+}$$ ($$=\mathbb{N}\cup\{0\}$$), we apply $$f(x)=q^{-hx}e^{tx}$$ on (4), we have

$$\int_{\mathbb{Z}_{p}}q^{-hx}e^{xt}\,d\mu_{q}(x)=\frac{q^{h-1}(q-1)}{\log q}\frac{t-(h-1)\log{q}}{e^{t}-q^{h-1}}.$$
(5)

For $$h\in\mathbb{Z}_{+}$$, we define the $$(h,q)$$-Bernoulli number $$B^{(h)}_{n}(q)$$ as follows:

$$\sum_{n=0}^{\infty}B^{(h)}_{n}(q) \frac{t^{n}}{n!}=\frac {q^{h-1}(q-1)}{\log q}\frac{t-(h-1)\log{q}}{e^{t}-q^{h-1}}.$$
(6)

Indeed if $$q\rightarrow1$$, we have $$\lim_{q\rightarrow 1}B^{(h)}_{n}(q)=B_{n}$$. So we call this $$B^{(h)}_{n}(q)$$ the nth $$(h,q)$$-Bernoulli number. And we define $$(h,q)$$-Bernoulli polynomials and the generating function to be

$$\frac{q^{h-1}(q-1)}{\log q}\frac{t-(h-1)\log{q}}{e^{t}-q^{h-1}}e^{xt}=\sum _{n=0}^{\infty}B^{(h)}_{n}(x|q) \frac{t^{n}}{n!}.$$
(7)

When $$x=0$$, $$B^{(h)}_{n}(0|q)=B^{(h)}_{n}(q)$$ are the nth $$(h,q)$$-Bernoulli numbers.

From (4) and (7), we have

$$B^{(h)}_{n}(x|q)=\int_{\mathbb{Z}_{p}}q^{-hy}(x+y)^{n}\,d \mu_{q}(y).$$

From (7) we note that

$$B^{(h)}_{n}(x|q)=\sum _{l=0}^{n}\binom{n}{l}B^{(h)}_{l}(q)x^{n-l}.$$
(8)

For the case $$|t|_{p}\leq p^{-\frac{1}{p-1}}$$, the Daehee polynomials are defined as follows (see [3]):

$$\sum_{n=0}^{\infty}D_{n}(x) \frac{t^{n}}{n!}=\frac{\log(1+t)}{t}(1+t)^{x}.$$
(9)

From (2) and (3), if $$q\rightarrow1$$, we have

$$\lim_{q\rightarrow1}D_{n,q}(x)=D_{n}(x)$$

and

$$\lim_{q\rightarrow1}D_{n}(x|q)=D_{n}(x).$$

The p-adic q-integral (or q-Volkenborn integration) was defined by Kim (see [1, 2]). From p-adic q-integral equations, we can derive various q-extensions of Bernoulli polynomials and numbers (see [1â€“24]). In [20], DS Kim and T Kim studied Daehee polynomials and numbers and their applications. In [3], Kim et al. introduced the q-analogue of Daehee numbers and polynomials which are called q-Daehee numbers and polynomials. Lim considered in [4] the modified q-Daehee numbers and polynomials which are different from the q-Daehee numbers and polynomials of Kim et al. In this paper, we consider $$(h,q)$$-Daehee numbers and polynomials and give some interesting identities. In case $$h=0$$, we cover the q-analogue of Daehee numbers and polynomials of Kim et al. (see [3]). In case $$h=1$$, we have modified q-Daehee numbers and polynomials in [4]. We can find out various $$(h,q)$$-related numbers and polynomials in [10, 13, 14].

## 2 $$(h,q)$$-Daehee numbers and polynomials

Let us now consider the p-adic q-integral representation as follows: for each $$h\in\mathbb{Z}_{+}$$,

$$\int_{\mathbb{Z}_{p}}q^{-hy}(x+y)_{n}\,d \mu_{q}(y)\quad \bigl(n\in\mathbb{Z}_{+}=\mathbb {N}\cup\{0\} \bigr),$$
(10)

where $$(x)_{n}$$ is known as the Pochhammer symbol (or decreasing factorial) defined by

$$(x)_{n}=x(x-1)\cdots(x-n+1)=\sum _{k=0}^{n}S_{1}(n,k)x^{k},$$
(11)

and here $$S_{1}(n,k)$$ is the Stirling number of the first kind (see [3, 20]).

From (10) we have

\begin{aligned}[b] \sum_{n=0}^{\infty} \biggl(\int_{\mathbb{Z}_{p}}q^{-hy}(y)_{n}\,d\mu _{q}(y) \biggr)\frac{t^{n}}{n!}&=\int_{\mathbb{Z}_{p}}q^{-hy} \Biggl(\sum_{n=0}^{\infty}\binom{y}{n}t^{n} \Biggr)\,d\mu_{q}(y) \\ &=\int_{\mathbb{Z}_{p}}q^{-hy}(1+t)^{y}\,d\mu_{q}(y), \end{aligned}
(12)

where $$t\in\mathbb{C}_{p}$$ with $$|t|_{p}< p^{-\frac{1}{p-1}}$$.

For $$|t|_{p}< p^{-\frac{1}{p-1}}$$, from (4) we have

$$\int_{\mathbb{Z}_{p}}q^{-hy}(1+t)^{y}\,d\mu_{q}(y)=\frac{q^{h-1}(q-1)}{\log q}\frac{\log{\frac{1+t}{q^{h-1}}}}{1+t-q^{h-1}}.$$
(13)

Let

$$F^{(h)}_{q}(t)=\frac{q^{h-1}(q-1)}{\log q} \frac{\log{\frac {1+t}{q^{h-1}}}}{1+t-q^{h-1}}=\sum_{n=0}^{\infty }D^{(h)}_{n}(q) \frac{t^{n}}{n!}.$$
(14)

Here, the numbers $$D^{(h)}_{n}(q)$$ are called the nth $$(h,q)$$-Daehee numbers of the first kind. Moreover, we have

$$D^{(h)}_{n}(q)=\int_{\mathbb{Z}_{p}}q^{-hy}(y)_{n}\,d \mu_{q}(y).$$
(15)

From (14) and (15), if $$h=0$$, $$D^{(0)}_{n}(q)$$ is just the q-Daehee numbers which are defined by Kim et al. in [3]. If $$h=1$$, $$D^{(1)}_{n}(q)$$ is just the modified q-Daehee numbers which are studied in [4].

On the other hand, we can derive $$(h,q)$$-Daehee polynomials

\begin{aligned}[b] \sum_{n=0}^{\infty} \biggl(\int_{\mathbb{Z}_{p}}q^{-hy}(x+y)_{n}\,d\mu _{q}(y) \biggr)\frac{t^{n}}{n!}&=\int_{\mathbb{Z}_{p}}q^{-hy} \Biggl(\sum_{n=0}^{\infty}\binom{x+y}{n}t^{n} \Biggr)\,d\mu_{q}(y) \\ &=\int_{\mathbb{Z}_{p}}q^{-hy}(1+t)^{x+y}\,d\mu_{q}(y) \\ &=\frac{q^{h-1}(q-1)}{\log q}\frac{\log{(1+t)}-(h-1)\log {q}}{1+t-q^{h-1}}(1+t)^{x} \\ &=\sum_{n=0}^{\infty}D^{(h)}_{n}(x|q) \frac{t^{n}}{n!}, \end{aligned}
(16)

where $$t\in\mathbb{C}_{p}$$ with $$|t|_{p}< p^{-\frac{1}{p-1}}$$.

When $$x=0$$, $$D^{(h)}_{n}(0|q)=D^{(h)}_{n}(q)$$ is called the nth $$(h,q)$$-Daehee number.

Notice that $$F^{(h)}_{q}(0,t)$$ seems to be a new q-extension of the generating function for Daehee numbers of the first kind. Therefore, from (9) and the following fact, we get

$$\lim_{q\rightarrow1}F^{(h)}_{q}(t)= \frac{\log(1+t)}{t}.$$

From (11) and (12), we have

$$D^{(h)}_{n}(x|q)=\int _{\mathbb{Z}_{p}}q^{-hy}(x+y)_{n}\,d\mu_{q}(y) =\sum_{k=0}^{n}S_{1}(n,k)B^{(h)}_{k}(x|q),$$
(17)

where $$B^{(h)}_{k}(x|q)$$ are the $$(h,q)$$-Bernoulli polynomials introduced in (7).

Thus we have the following theorem, which relates $$(h,q)$$-Bernoulli polynomials and $$(h,q)$$-Daehee polynomials.

### Theorem 1

For $$n,m\in\mathbb{Z}_{+}$$, we have the following equalities:

$$D^{(h)}_{n}(x|q)=\sum_{k=0}^{n}S_{1}(n,k)B^{(h)}_{k}(x|q)$$

and

$$D^{(h)}_{n}(q)=\sum_{k=0}^{n}S_{1}(n,k)B^{(h)}_{k}(q).$$

From the generating function of the $$(h,q)$$-Daehee polynomials in $$D^{(h)}_{n}(x|q)$$ in (14), by replacing t to $$e^{t}-1$$, we have

\begin{aligned}[b] \sum_{n=0}^{\infty}D^{(h)}_{n}(x|q) \frac{(e^{t}-1)^{n}}{n!}&=\frac {q^{h-1}(q-1)}{\log q}\frac{t-(h-1)\log{q}}{e^{t}-q^{h-1}}e^{xt} \\ &=\sum_{n=0}^{\infty}B^{(h)}_{n}(x|q) \frac{t^{n}}{n!}. \end{aligned}
(18)

On the other hand,

$$\sum_{n=0}^{\infty}D^{(h)}_{n}(x|q) \frac{(e^{t}-1)^{n}}{n!}=\sum_{m=0}^{\infty}D^{(h)}_{m}(x|q) \sum_{n=0}^{\infty}S_{2}(n,m) \frac{t^{n}}{n!}.$$
(19)

Here, $$S_{2}(n,m)$$ is the Stirling number of the second kind defined by the following generating series:

$$\sum_{n=m}^{\infty}S_{2}(n,m) \frac{t^{n}}{n!}=\frac {(e^{t}-1)^{m}}{m!} \quad\textit{cf. }\mbox{[3, 20]}.$$
(20)

Thus by comparing the coefficients of $$t^{n}$$, we have

$$B^{(h)}_{n}(x|q)=\sum_{m=0}^{n}D^{(h)}_{m}(x|q)S_{2}(n,m).$$

Therefore, we obtain the following theorem.

### Theorem 2

For $$n,m\in\mathbb{Z}_{+}$$, we have the following identity:

$$B^{(h)}_{n}(x|q)=\sum_{m=0}^{n}D^{(h)}_{m}(x|q)S_{2}(n,m).$$

The increasing factorial sequence is known as

$$x^{(n)}=x(x+1) (x+2)\cdots(x+n-1)\quad (n\in\mathbb{Z}_{+}).$$

Let us define the $$(h,q)$$-Daehee numbers of the second kind as follows:

$$\widehat{D}^{(h)}_{n}(q)=\int _{\mathbb{Z}_{p}}q^{-hy}(-y)_{n}\,d\mu_{q}(y) \quad (n\in\mathbb{Z}_{+}).$$
(21)

It is easy to observe that

$$x^{(n)}=(-1)^{n}(-x)_{n}=\sum _{k=0}^{n}S_{1}(n,k) (-1)^{n-k}x^{k}.$$
(22)

From (21) and (22), we have

\begin{aligned}[b] \widehat{D}^{(h)}_{n}(q)&=\int _{\mathbb{Z}_{p}}q^{-hy}(-y)_{n}\,d\mu_{q}(y) \\ &=\int_{\mathbb{Z}_{p}}q^{-hy}y^{(n)}(-1)^{n}\,d \mu_{q}(y) \\ &=\sum_{k=0}^{n}S_{1}(n,k) (-1)^{k}B^{(h)}_{k}(q). \end{aligned}
(23)

Thus, we state the following theorem, which relates $$(h,q)$$-Daehee numbers and $$(h,q)$$-Bernoulli numbers.

### Theorem 3

The following holds true:

$$\widehat{D}^{(h)}_{n}(q)=\sum_{k=0}^{n}S_{1}(n,k) (-1)^{k}B^{(h)}_{k}(q).$$

Let us now consider the generating function of $$(h,q)$$-Daehee numbers of the second kind as follows:

\begin{aligned}[b] \sum_{n=0}^{\infty} \widehat{D}^{(h)}_{n}(q)\frac{t^{n}}{n!}&=\sum _{n=0}^{\infty} \biggl(\int_{\mathbb{Z}_{p}}q^{-hy}(-y)_{n}\,d \mu _{q}(y) \biggr)\frac{t^{n}}{n!} \\ &=\int_{\mathbb{Z}_{p}}q^{-hy} \Biggl(\sum _{n=0}^{\infty}\binom {-y}{n}t^{n} \Biggr)\,d\mu_{q}(y) \\ &=\int_{\mathbb{Z}_{p}}q^{-hy}(1+t)^{-y}\,d\mu_{q}(y). \end{aligned}
(24)

From (4) and (24), we have the generating function for $$(h,q)$$-Daehee numbers of the second kind as follows:

$$\int_{\mathbb{Z}_{p}}q^{-hy}(1+t)^{-y}\,d\mu_{q}(y)=\frac{q^{h-1}(q-1)}{\log q}\frac{\log q-\log(1+t)}{1+t-q^{h-1}}.$$
(25)

Let us consider the $$(h,q)$$-Daehee polynomials of the second kind as follows:

\begin{aligned}[b] \sum_{n=0}^{\infty} \widehat{D}^{(h)}_{n}(x|q)\frac{t^{n}}{n!}&=\sum _{n=0}^{\infty}\int_{\mathbb{Z}_{p}}q^{-hy}(x-y)_{n}\,d \mu_{q}(y)\frac {t^{n}}{n!} \\ &=\int_{\mathbb{Z}_{p}}q^{-hy}(1+t)^{x-y}\,d\mu_{q}(y) \\ &=\frac{q^{h-1}(q-1)}{\log q}\frac{\log q-\log(1+t)}{1+t-q^{h}}(1+t)^{x}. \end{aligned}
(26)

From the $$(h,q)$$-Bernoulli polynomials in (7),

\begin{aligned}[b] q^{h}\sum_{n=0}^{\infty}(-1)^{n}B^{(h)}_{n} \bigl(x|q^{-1} \bigr)\frac {t^{n}}{n!}&=q^{h}\frac{q^{1-h}(q^{-1}-1)}{\log q^{-1}} \frac{-t-\log {q^{1-h}}}{e^{-t}-q^{1-h}}e^{-xt} \\ &=\frac{q^{h-1}(q-1)}{\log q}\frac{t-\log {q^{h-1}}}{e^{t}-q^{h-1}}e^{(1-x)t} \\ &=\sum_{n=0}^{\infty}B^{(h)}_{n}(1-x|q) \frac{t^{n}}{n!}. \end{aligned}
(27)

Thus, we have

$$q^{h}(-1)^{n}B^{(h)}_{n} \bigl(x|q^{-1} \bigr)=B^{(h)}_{n}(1-x|q).$$
(28)

From (28), the value at $$x=1$$, we have

$$q^{h}(-1)^{n}B^{(h)}_{n} \bigl(1|q^{-1} \bigr)=B^{(h)}_{n}(q).$$

On the other hand, we note that

\begin{aligned}[b] (-x)_{n}&=(-1)^{n}x^{(n)} =\sum_{l=0}^{n}S_{1}(n,l) (-x)^{l} =(-1)^{n}\sum_{l=0}^{n}\bigl|S_{1}(n,l)\bigr|x^{l}, \end{aligned}
(29)

where $$n\geq0$$ and $$|S_{1}(n,k)|$$ is the unsigned Stirling number of the first kind.

From (28) and (29),

\begin{aligned}[b] \widehat{D}^{(h)}_{n}(x|q)&= \sum_{l=0}^{n}\bigl|S_{1}(n,l)\bigr|(-1)^{l} \int_{\mathbb{Z}_{p}}q^{-hy}(-x+y)^{l}\,d \mu_{q}(y) \\ &=\sum_{l=0}^{n}\bigl|S_{1}(n,l)\bigr|(-1)^{l}B^{(h)}_{l}(-x|q) \\ &=q^{-h}\sum_{l=0}^{n}\bigl|S_{1}(n,l)\bigr|B^{(h)}_{l} \bigl(x+1|q^{-1} \bigr). \end{aligned}
(30)

Thus, we have the following identity.

### Theorem 4

For $$n\in\mathbb{Z}_{+}$$, the following is true:

$$\widehat{D}^{(h)}_{n}(x|q)=q^{-h}\sum _{l=0}^{n}\bigl|S_{1}(n,l)\bigr|B^{(h)}_{l} \bigl(x+1|q^{-1} \bigr).$$

On the other hand, we can check easily the following:

$$(x+y)_{n}=(-1)^{n}(-x-y+n-1)_{n}$$
(31)

and

$$\frac{(x+y)_{n}}{n!}=(-1)^{n}\binom{-x+y+n-1}{n}.$$
(32)

From (14), (26), (31) and (32), we have

\begin{aligned}[b] (-1)^{n}\frac{D^{(h)}_{n}(x|q)}{n!}&=\int _{\mathbb{Z}_{p}}q^{-hy}\binom {-x-y+n-1}{n}\,d\mu_{q}(y) \\ &=\sum_{m=0}^{n}\binom{n-1}{n-m}\int _{\mathbb{Z}_{p}}q^{-hy}\binom {-x-y}{m}\,d\mu_{q}(y) \\ &=\sum_{m=1}^{n}\binom{n-1}{m-1} \frac{\widehat{D}^{(h)}_{m}(-x|q)}{m!} \end{aligned}
(33)

and

\begin{aligned}[b] (-1)^{n}\frac{\widehat{D}^{(h)}_{n}(x|q)}{n!}&=(-1)^{n} \int_{\mathbb {Z}_{p}}q^{-hy}\binom{-x+y}{n}\,d\mu_{q}(y) \\ &=\int_{\mathbb{Z}_{p}}q^{-hy}\binom{-x+y+n-1}{n}\,d\mu_{q}(y) \\ &=\sum_{m=0}^{n}\binom{n-1}{n-m}\int _{\mathbb{Z}_{p}}q^{-hy}\binom {-x+y}{m}\,d\mu_{q}(y) \\ &=\sum_{m=1}^{n}\binom{n-1}{m-1} \frac{D^{(h)}_{m}(-x|q)}{m!}. \end{aligned}
(34)

Therefore, we get the following theorem, which relates $$(h,q)$$-Daehee polynomials of the first and the second kind.

### Theorem 5

For $$n\in\mathbb{N}$$, the following equalities hold true:

$$(-1)^{n}\frac{D^{(h)}_{n}(x|q)}{n!}=\sum_{m=1}^{n} \binom {n-1}{m-1}\frac{\widehat{D}^{(h)}_{m}(-x|q)}{m!}$$

and

$$(-1)^{n}\frac{\widehat{D}^{(h)}_{n}(x|q)}{n!}=\sum_{m=1}^{n} \binom {n-1}{m-1}\frac{D^{(h)}_{m}(-x|q)}{m!}.$$

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## Acknowledgements

Authors wish to express their sincere gratitude to the referees for their valuable suggestions and comments.

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Correspondence to Dongkyu Lim.

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Do, Y., Lim, D. On $$(h,q)$$-Daehee numbers and polynomials. Adv Differ Equ 2015, 107 (2015). https://doi.org/10.1186/s13662-015-0445-3