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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 qC, or a p-adic number q C p . If qC, 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 fUD( 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

I( f 1 )=I(f)+ f (0),where  f (0)= d f ( x ) d x | x = 0 .
(1.2)

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

( x ) n =x(x1)(xn+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 [46]).

Unsigned Stirling numbers of the first kind are given by

x n ̲ =x(x+1)(x+n1)= 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 rN, 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 nN, 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(xq)(x2q) ( x ( n 1 ) q ) (n1), ( 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)(n0)(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 [9]).
(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:

ξ n Z p ( x + y ) n , q d μ 0 (y),where n Z + =N{0} and ξ T p .
(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 n0, 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 n0, 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 n0, 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 kN with q-parameter. Twisted Daehee polynomials of order kN 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 kN. 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 n0 and kN, 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)(n0).
(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 n0, 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 n0, 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 kN. 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 n0, 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 n0 and kN, 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.

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Correspondence to Jin-Woo Park.

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Keywords

  • Bernoulli polynomials
  • Daehee polynomials with q-parameter
  • p-adic invariant integral