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 [1–24]). 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

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