# On the twisted Daehee polynomials with q-parameter

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## Abstract

The n th twisted Daehee numbers with q-parameter are closely related to higher-order Bernoulli numbers and Bernoulli numbers of the second kind. In this paper, we give a p-adic integral representation of the twisted Daehee polynomials with q-parameter, and we derive some interesting properties related to the n th twisted Daehee polynomials with q-parameter.

## 1 Introduction

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

When one talks of q-extension, q is variously considered as an indeterminate, a complex $q∈C$, or a p-adic number $q∈ C p$. If $q∈C$, one normally assumes that $|q|<1$. If $q∈ C p$, then we assume that $| q − 1 | p < p − 1 p − 1$ so that $q x =exp(xlogq)$ for each $x∈ Z p$. Throughout this paper, we use the notation

$[ x ] q = 1 − q x 1 − q .$

Note that $lim q → 1 [ x ] q =x$ for each $x∈ Z p$.

Let $UD( Z p )$ be the space of uniformly differentiable functions on $Z p$. For $f∈UD( Z p )$, the p-adic invariant integral on $Z p$ is defined by Kim as follows:

$I(f)= ∫ Z p f(x)d μ 0 (x)= lim n → ∞ 1 p n ∑ x = 0 p n − 1 f(x)(see [1–3]).$
(1.1)

Let $f 1$ be the translation of f with $f 1 (x)=f(x+1)$. Then, by (1.1), we get

(1.2)

As is well known, the Stirling number of the first kind is defined by

$( x ) n =x(x−1)⋯(x−n+1)= ∑ l = 0 n S 1 (n,l) x l ,$
(1.3)

and the Stirling number of the second kind is given by the generating function to be

$( e t − 1 ) m =m! ∑ l = m ∞ S 2 (l,m) t l l !$
(1.4)

(see ).

Unsigned Stirling numbers of the first kind are given by

$x n ̲ =x(x+1)⋯(x+n−1)= ∑ l = 0 n | S 1 ( n , l ) | x l .$
(1.5)

Note that if we replace x to −x in (1.3), then

$( − x ) n = ( − 1 ) n x n ̲ = ∑ l = 0 n S 1 ( n , l ) ( − 1 ) l x l = ( − 1 ) n ∑ l = 0 n | S 1 ( n , l ) | x l .$
(1.6)

Hence $S 1 (n,l)=| S 1 (n,l)| ( − 1 ) n − l$.

For $r∈N$, the Bernoulli polynomials of order r are defined by the generating function to be

$( t e t − 1 ) r e x t = ∑ n = 0 ∞ B n ( r ) (x) t n n ! (see [1, 4, 7–18]).$
(1.7)

When $x=0$, $B n ( r ) = B n ( r ) (0)$ are called the Bernoulli numbers of order r, and in the special case, $r=1$, $B n ( 1 ) (x)= B n (x)$ are called the ordinary Bernoulli polynomials.

For $n∈N$, let $T p$ be the p-adic locally constant space defined by

$T p = ⋃ n ≥ 1 C p n = lim n → ∞ C p n ,$

where $C p n ={ω| ω p n =1}$ is the cyclic group of order $p n$.

We assume that q is an indeterminate in $C p$ with $| 1 − q | p < p − 1 p − 1$. Then we define the q-analog of a falling factorial sequence as follows:

$( x ) n , q =x(x−q)(x−2q)⋯ ( x − ( n − 1 ) q ) (n≥1), ( x ) 0 , q =1.$

Note that

$lim q → 1 ( x ) n , q = ( x ) n = ∑ l = 0 n S 1 (n,l) x l .$

Recently, DS Kim and T Kim introduced the Daehee polynomials as follows:

$D n (x)= ∫ Z p ( x + y ) n d μ 0 (y)(n≥0)(see [2, 9, 19]).$
(1.8)

When $x=0$, $D n = D n (0)$ are called the nth Daehee numbers. From (1.8), we can derive the generating function to be

$( log ( 1 + t ) t ) ( 1 + t ) x = ∑ n = 0 ∞ D n (x) t n n ! (see ).$
(1.9)

In addition, DS Kim et al. consider the Daehee polynomials with q-parameter, which are defined by the generating function to be

$∑ n = 0 ∞ D n , q t n n ! = ( 1 + q t ) x q log ( 1 + q t ) q ( ( 1 + q t ) 1 q − 1 ) (see [20, 21]).$
(1.10)

When $x=0$, $D n , q = D n , q (0)$ are called the Daehee numbers with q-parameter.

From the viewpoint of a generalization of the Daehee polynomials with q-parameter, we consider the twisted Daehee polynomials with q-parameter, defined to be

$∑ n = 0 ∞ D n , ξ , q t n n ! = ( 1 + q ξ t ) x q log ( 1 + q ξ t ) q ( ( 1 + q ξ t ) 1 q − 1 ) ,$
(1.11)

where $t,q∈ C p$ with $| t | p < | q | p p − 1 p − 1$ and $ξ∈ T p$.

In this paper, we give a p-adic integral representation of the twisted Daehee polynomials with q-parameter, which is called the Witt-type formula for the twisted Daehee polynomials with q-parameter. We can derive some interesting properties related to the n th twisted Daehee polynomials with q-parameter.

## 2 Witt-type formula for the n th twisted Daehee polynomials with q-parameter

First, we consider the following integral representation associated with falling factorial sequences:

(2.1)

By (2.1),

$∑ n = 0 ∞ ξ n ∫ Z p ( x + y ) n , q d μ 0 ( y ) t n n ! = ∑ n = 0 ∞ ξ n q n ∫ Z p ( x + y q ) n d μ 0 ( y ) t n n ! = ∫ Z p ( 1 + q ξ t ) x + y q d μ 0 ( y ) ,$
(2.2)

where $t,q∈ C p$ with $| t | p < | q | p p − 1 p − 1$. For $t∈ C p$ with $| t | p < | q | p p − 1 p − 1$, put $f(x)= ( 1 + q ξ t ) x + y q$. By (1.1), we get

$∫ Z p ( 1 + q ξ t ) x + y q d μ 0 ( y ) = ( 1 + q ξ t ) x q log ( 1 + q ξ t ) q ( ( 1 + q ξ t ) 1 q − 1 ) = ∑ n = 0 ∞ D n , ξ , q ( x ) t n n ! .$
(2.3)

By (2.2) and (2.3), we obtain the following theorem.

Theorem 2.1 For $n≥0$, we have

$D n , ξ , q (x)= ξ n ∫ Z p ( x + y ) n , q d μ 0 (y).$

In (2.3), by replacing t by $1 ξ q ( e ξ t −1)$, we have

$∑ n = 0 ∞ D n , ξ , q (x) 1 ξ n q n ( e ξ t − 1 ) n n ! = e ξ t x q ξ t q e ξ t q − 1 = ∑ n = 0 ∞ B n (x) ξ n q n t n n !$
(2.4)

and

$∑ n = 0 ∞ D n , ξ , q ( x ) ξ n q n 1 n ! ( e ξ t − 1 ) n = ∑ n = 0 ∞ D n , ξ , q ( x ) ξ n q n ∑ m = n ∞ ξ m S 2 ( m , n ) t m m ! = ∑ m = 0 ∞ ∑ n = 0 m D n , ξ , q ( x ) ξ n q n ξ m S 2 ( m , n ) t m m ! .$
(2.5)

By (2.4) and (2.5), we obtain the following corollary.

Corollary 2.2 For $n≥0$, we have

$B n (x)= ∑ m = 0 n D m , ξ , q (x) ξ − m q n − m S 2 (n,m).$

By Theorem 2.1,

$D n , ξ , q ( x ) = ξ n ∫ Z p ( x + y ) n , q d μ 0 ( y ) = ξ n q n ∑ l = 0 n 1 q l S 1 ( n , l ) ∫ Z p ( x + y ) l d μ 0 ( y ) .$
(2.6)

By (1.2), we can derive easily that

$∫ Z p e ( x + y ) t d μ 0 ( y ) = t e t − 1 e x t = ∑ n = 0 ∞ B n ( x ) t n n ! = ∑ l = 0 ∞ ∫ Z p ( x + y ) l d μ 0 ( y ) t l l ! ,$
(2.7)

and so

$B n (x)= ∫ Z p ( x + y ) n d μ 0 (y).$
(2.8)

By (1.6), (2.7), and (2.8), we obtain the following corollary.

Corollary 2.3 For $n≥0$, we have

$D n , ξ , q (x)= ξ n ∑ l = 0 n q n − l S 1 (n,l) B l (x)= ξ n ∑ l = 0 n | S 1 ( n , l ) | ( − q ) n − l B l (x).$

From now on, we consider twisted Daehee polynomials of order $k∈N$ with q-parameter. Twisted Daehee polynomials of order $k∈N$ with q-parameter are defined by the multivariant p-adic invariant integral on $Z p$:

$D n , ξ , q ( k ) (x)= ξ n ∫ Z p ⋯ ∫ Z p ( x 1 + ⋯ + x k + x ) n , q d μ 0 ( x 1 )⋯d μ 0 ( x k ),$
(2.9)

where n is a nonnegative integer and $k∈N$. In the special case, $x=0$, $D n , ξ , q ( k ) = D n , ξ , q ( k ) (0)$ are called the Daehee numbers of order k with q-parameter.

From (2.9), we can derive the generating function of $D n , ξ , q ( k ) (x)$ as follows:

$∑ n = 0 ∞ D n , ξ , q ( k ) ( x ) t n n ! = ∑ n = 0 ∞ ξ n q n ∫ Z p ⋯ ∫ Z p ( x 1 + ⋯ + x k + x q n ) d μ 0 ( x 1 ) ⋯ d μ 0 ( x k ) t n = ∫ Z p ⋯ ∫ Z p ( 1 + q ξ t ) x 1 + ⋯ + x k + x q d μ 0 ( x 1 ) ⋯ d μ 0 ( x k ) = ( 1 + q ξ t ) x q ∫ Z p ⋯ ∫ Z p ( 1 + q ξ t ) x 1 + ⋯ + x k q d μ 0 ( x 1 ) ⋯ d μ 0 ( x k ) = ( 1 + q ξ t ) x q ( log ( 1 + q ξ t ) q ( ( 1 + q ξ t ) 1 q − 1 ) ) k .$
(2.10)

Note that, by (2.9),

$D n , ξ , q ( k ) (x)= ξ n q n ∑ m = 0 n S 1 ( n , m ) q m ∫ Z p ⋯ ∫ Z p ( x 1 + ⋯ + x k + x ) m d μ 0 ( x 1 )⋯d μ 0 ( x k ).$
(2.11)

Since

$∫ Z p ⋯ ∫ Z p e ( x 1 + ⋯ + x k + x ) t d μ 0 ( x 1 )⋯d μ 0 ( x k )= ( t e t − 1 ) k e x t = ∑ n = 0 ∞ B n ( k ) (x) t n n ! ,$

we can derive easily

$B n ( k ) (x)= ∫ Z p ⋯ ∫ Z p ( x 1 + ⋯ + x k + x ) n d μ 0 ( x 1 )⋯d μ 0 ( x k ).$
(2.12)

Thus, by (2.11) and (2.12), we have

$D n , ξ , q ( k ) ( x ) = ξ n q n ∑ m = 0 n S 1 ( n , m ) q m B m ( k ) ( x ) = ξ n ∑ m = 0 n q n − m S 1 ( n , m ) B m ( k ) ( x ) = ξ n ∑ m = 0 n | S 1 ( n , m ) | ( − q ) n − m B m ( k ) ( x ) .$
(2.13)

In (2.10), by replacing t by $1 q ξ ( e ξ t −1)$, we get

$∑ n = 0 ∞ D n , ξ , q ( k ) (x) ( e ξ t − 1 ) n ξ n q n n ! = e ξ t x q ( ξ t q e ξ t q − 1 ) k = ∑ n = 0 ∞ ξ n B n ( k ) ( x ) q n t n n !$
(2.14)

and

$∑ n = 0 ∞ D n , ξ , q ( k ) ( x ) ξ n q n 1 n ! ( e ξ t − 1 ) n = ∑ n = 0 ∞ D n , ξ , q ( k ) ( x ) ξ n q n ∑ l = n ∞ S 2 ( l , n ) ξ l t l l ! = ∑ m = 0 ∞ ( ξ m ∑ n = 0 m D n , ξ , q ( k ) ( x ) ξ n q n S 2 ( m , n ) ) t m m ! .$
(2.15)

By (2.13), (2.14), and (2.15), we obtain the following theorem.

Theorem 2.4 For $n≥0$ and $k∈N$, we have

$D n , ξ , q ( k ) (x)= ξ n ∑ m = 0 n q n − m S 1 (n,m) B m ( k ) (x)= ξ n ∑ m = 0 n | S 1 ( n , m ) | ( − q ) n − m B m ( k ) (x)$

and

$B n ( k ) (x)= ∑ m = 0 n D m , ξ , q ( k ) (x) ξ − m q n − m S 2 (n,m).$

Now, we consider the twisted Daehee polynomials of the second kind with q-parameter as follows:

$D ˆ n , ξ , q (x)= ξ n ∫ Z p ( − y + x ) n , q d μ 0 (y)(n≥0).$
(2.16)

In the special case $x=0$, $D ˆ n , ξ , q (0)= D ˆ n , ξ , q$ are called the twisted Daehee numbers of the second kind with q-parameter.

By (2.16), we have

$D ˆ n , ξ , q (x)= ξ n q n ∫ Z p ( − y + x q ) n d μ 0 (y),$
(2.17)

and so we can derive the generating function of $D ˆ n , ξ , q (x)$ by (1.1) as follows:

$∑ n = 0 ∞ D ˆ n , ξ , q ( x ) t n n ! = ∑ n = 0 ∞ q n ξ n ∫ Z p ( − y + x q ) n d μ 0 ( y ) t n n ! = ∑ n = 0 ∞ q n ξ n ∫ Z p ( − y + x q n ) d μ 0 ( y ) t n = ∫ Z p ( 1 + q ξ t ) − y + x q d μ 0 ( y ) = ( 1 + q ξ t ) x q log ( 1 + q ξ t ) q ( ( 1 + q ξ t ) 1 q − 1 ) ( 1 + q ξ t ) 1 q .$
(2.18)

From (1.3), (1.6), and (2.17), we get

$D ˆ n , ξ , q ( x ) = q n ξ n ∫ Z p ( − y + x q ) n d μ 0 ( y ) = q n ξ n ∫ Z p ∑ l = 0 n S 1 ( n , l ) q l ( − y + x ) l d μ 0 ( y ) = ξ n ∑ l = 0 n S 1 ( n , l ) ( − 1 ) l ∫ Z p ( y − x ) l d μ 0 ( y ) q n − l = ξ n ∑ l = 0 n S 1 ( n , l ) ( − 1 ) l B l ( − x ) q n − l = ( − ξ ) n ∑ l = 0 n | S 1 ( n , l ) | B l ( − x ) q n − l .$
(2.19)

By (1.10), it is easy to show that $B n (−x)= ( − 1 ) n B n (x+1)$. Thus, from (2.19), we have the following theorem.

Theorem 2.5 For $n≥0$, we have

$D ˆ n , ξ , q (x)= ξ n ∑ l = 0 n S 1 (n,l) ( − 1 ) l B l (−x) q n − l = ξ n ∑ l = 0 n | S 1 ( n , l ) | B l (x+1) ( − q ) n − l .$

By replacing t by $1 q ξ ( e ξ t −1)$ in (2.18), we have

$∑ n = 0 ∞ D ˆ n , ξ , q (x) 1 q n ξ n ( e ξ t − 1 ) n n ! = e ξ t q ( x + 1 ) ξ t q e ξ t q − 1 = ∑ n = 0 ∞ ξ n B n ( x + 1 ) q n t n n !$
(2.20)

and

$∑ n = 0 ∞ D ˆ n , ξ , q ( x ) 1 q n ξ n ( e ξ t − 1 ) n n ! = ∑ n = 0 ∞ D ˆ n , ξ , q ( x ) q n ξ n ∑ m = n ∞ S 2 ( m , n ) ( ξ t ) m m ! = ∑ n = 0 ∞ ( ∑ m = 0 n D ˆ m , ξ , q ( x ) S 2 ( n , m ) q − m ξ n − m ) t n n ! .$
(2.21)

By (2.20) and (2.21), we obtain the following theorem.

Theorem 2.6 For $n≥0$, we have

$B n (x+1)= ∑ m = 0 n q n − m ξ − m D ˆ m , ξ , q (x) S 2 (n,m).$

Now, we consider higher-order twisted Daehee polynomials of the second kind with q-parameter. Higher-order twisted Daehee polynomials of the second kind with q-parameter are defined by the multivariant p-adic invariant integral on $Z p$:

$D ˆ n , ξ , q ( k ) (x)= ξ n ∫ Z p ⋯ ∫ Z p ( − x 1 − ⋯ − x k + x ) n , q d μ 0 ( x 1 )⋯d μ 0 ( x k ),$
(2.22)

where n is a nonnegative integer and $k∈N$. In the special case, $x=0$, $D ˆ n , ξ , q ( k ) = D ˆ n , ξ , q ( k ) (0)$ are called the higher-order twisted Daehee numbers of the second kind with q-parameter.

From (2.22), we can derive the generating function of $D ˆ n , ξ , q ( k ) (x)$ as follows:

$∑ n = 0 ∞ D ˆ n , ξ , q ( k ) ( x ) t n n ! = ∑ n = 0 ∞ ξ n q n ∫ Z p ⋯ ∫ Z p ( − x 1 − ⋯ − x k + x q n ) d μ 0 ( x 1 ) ⋯ d μ 0 ( x k ) t n = ∫ Z p ⋯ ∫ Z p ( 1 + q ξ t ) − x 1 − ⋯ − x k + x q d μ 0 ( x 1 ) ⋯ d μ 0 ( x k ) = ( 1 + q ξ t ) x + k q ( log ( 1 + q ξ t ) q ( ( 1 + q ξ t ) 1 q − 1 ) ) k .$
(2.23)

By (2.22),

$D ˆ n , ξ , q ( k ) ( x ) = ξ n q n ∑ m = 0 n S 1 ( n , m ) q m ∫ Z p ⋯ ∫ Z p ( − x 1 − ⋯ − x k + x ) m d μ 0 ( x 1 ) ⋯ d μ 0 ( x k ) = ξ n q n ∑ m = 0 n S 1 ( n , m ) ( − q ) m ∫ Z p ⋯ ∫ Z p ( x 1 + ⋯ + x k − x ) m d μ 0 ( x 1 ) ⋯ d μ 0 ( x k ) = ξ n q n ∑ m = 0 n S 1 ( n , m ) ( − q ) m B m ( k ) ( − x ) = ξ n ∑ m = 0 n q n − m | S 1 ( n , m ) | B m ( k ) ( − x ) .$
(2.24)

From (1.10), we know that $B n ( k ) (−x)= ( − 1 ) n B n ( k ) (k+x)$. Hence, by (2.24), we obtain the following theorem.

Theorem 2.7 For $n≥0$, we have

$D ˆ n , ξ , q ( k ) (x)= ξ n ∑ m = 0 n ( − 1 ) m q n − m S 1 (n,m) B m ( k ) (−x)= ξ n ∑ m = 0 n ( − 1 ) m q n − m | S 1 ( n , m ) | B m ( k ) (x+k).$

In (2.23), by replacing t by $1 q ξ ( e ξ t −1)$, we get

$∑ n = 0 ∞ D ˆ n , ξ , q ( k ) (x) ( e ξ t − 1 ) n ξ n q n n ! = e ξ t q ( x + k ) ( ξ t q e ξ t q − 1 ) k = ∑ n = 0 ∞ ξ n B n ( k ) ( x + k ) q n t n n !$
(2.25)

and

$∑ n = 0 ∞ D ˆ n , ξ , q ( k ) ( x ) ξ n q n 1 n ! ( e ξ t − 1 ) n = ∑ n = 0 ∞ D ˆ n , ξ , q ( k ) ( x ) ξ n q n ∑ l = n ∞ S 2 ( l , n ) ξ l t l l ! = ∑ n = 0 ∞ ( ξ n ∑ m = 0 n D ˆ m , ξ , q ( k ) ( x ) ξ m q m S 2 ( n , m ) ) t n n ! .$
(2.26)

By (2.25) and (2.26), we obtain the following theorem.

Theorem 2.8 For $n≥0$ and $k∈N$, we have

$B n ( k ) (x+k)= ∑ m = 0 n D ˆ m , ξ , q ( k ) (x) ξ − m q n − m S 2 (n,m).$

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## Acknowledgements

The author is grateful for the valuable comments and suggestions of the referees. This paper was supported by the Sehan University Research Fund in 2014.

## Author information

Correspondence to Jin-Woo Park.

### Competing interests

The author declares that they have no competing interests.

### Author’s contributions

The author contributed to the manuscript and typed, read, and approved the final manuscript.

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