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Theory and Modern Applications

On the twisted Daehee polynomials with q-parameter

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

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(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 [4–6]).

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 [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 Î¾âˆˆ 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 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).

References

  1. Kim T: On q -analogye of the p -adic log gamma functions and related integral. J. Number Theory 1999, 76(2):320-329. 10.1006/jnth.1999.2373

    Article  MathSciNet  Google Scholar 

  2. Kim T: An invariant p -adic integral associated with Daehee numbers. Integral Transforms Spec. Funct. 2002, 13(1):65-69. 10.1080/10652460212889

    Article  MathSciNet  Google Scholar 

  3. Kim T: q -Volkenborn integration. Russ. J. Math. Phys. 2002, 9(3):288-299.

    MathSciNet  Google Scholar 

  4. Comtet L: Advanced Combinatorics. Reidel, Dordrecht; 1974.

    Book  Google Scholar 

  5. Kim T, Kim DS, Mansour T, Rim SH, Schork M: Umbral calculus and Sheffer sequences of polynomials. J. Math. Phys. 2013., 52(8): Article ID 083504

    Google Scholar 

  6. Roman S: The Umbral Calculus. Dover, New York; 2005.

    Google Scholar 

  7. Dolgy DV, Kim T, Lee B, Lee SH: Some new identities on the twisted Bernoulli and Euler polynomials. J. Comput. Anal. Appl. 2013, 15(3):441-451.

    MathSciNet  Google Scholar 

  8. Jeong JH, Jin JH, Park JW, Rim SH: On the twisted weak q -Euler numbers and polynomials with weight 0. Proc. Jangjeon Math. Soc. 2013, 16(2):157-163.

    MathSciNet  Google Scholar 

  9. Kim DS, Kim T: Daehee numbers and polynomials. Appl. Math. Sci. 2013, 7(120):5969-5976.

    MathSciNet  Google Scholar 

  10. Kim YH, Hwang KW: Symmetry of power sum and twisted Bernoulli polynomials. Adv. Stud. Contemp. Math. 2009, 18(2):43-48.

    MathSciNet  Google Scholar 

  11. Luo QL: Some recursion formulae and relations for Bernoulli numbers and Euler numbers of higher order. Adv. Stud. Contemp. Math. 2005, 10(1):63-70.

    MathSciNet  Google Scholar 

  12. Ozden H, Cangul IN, Simsek Y: Remarks on q -Bernoulli numbers associated with Daehee numbers. Adv. Stud. Contemp. Math. 2009, 18(1):41-48.

    MathSciNet  Google Scholar 

  13. Simsek Y: Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions. Adv. Stud. Contemp. Math. 2008, 16(2):251-278.

    MathSciNet  Google Scholar 

  14. Simsek Y: Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their applications. Fixed Point Theory Appl. 2013., 2013: Article ID 87

    Google Scholar 

  15. Simsek Y: On p -adic twisted q - L -function related to generalized twisted Bernoulli numbers. Russ. J. Math. Phys. 2006, 13(3):340-348. 10.1134/S1061920806030095

    Article  MathSciNet  Google Scholar 

  16. Araci S: Novel identities involving Genocchi numbers and polynomials arising from applications of umbral calculus. Appl. Math. Comput. 2014, 233: 599-607.

    Article  MathSciNet  Google Scholar 

  17. Araci S, Acikgoz M, Sen E: On the von Staudt-Clausen’s theorem associated with q -Genocchi numbers. Appl. Math. Comput. 2014, 247: 780-785.

    Article  MathSciNet  Google Scholar 

  18. Araci S, Bagdasaryan A, Özel C, Srivastava HM: New symmetric identities involving q -zeta type functions. Appl. Math. Inf. Sci. 2014, 8(6):2803-2808. 10.12785/amis/080616

    Article  MathSciNet  Google Scholar 

  19. Park JW, Rim SH, Kim J: The twisted Daehee numbers and polynomials. Adv. Differ. Equ. 2014., 2014: Article ID 1

    Google Scholar 

  20. Kim DS, Kim T, Kwon HI, Seo JJ: Daehee polynomials with q -parameter. Adv. Stud. Theor. Phys. 2014, 8(13):561-569.

    Google Scholar 

  21. Kim T, Lee SH, Mansour T, Seo JJ: A note on q -Daehee polynomials and numbers. Adv. Stud. Contemp. Math. 2014, 24(2):155-160.

    Google Scholar 

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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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Park, JW. On the twisted Daehee polynomials with q-parameter. Adv Differ Equ 2014, 304 (2014). https://doi.org/10.1186/1687-1847-2014-304

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