Skip to main content

Some identities of special q-polynomials

Abstract

In this paper, we investigate some identities of q-extensions of special polynomials which are derived from the fermonic q-integral on Z p and the bosonic q-integral onĀ  Z p .

1 Introduction

Let p be a fixed odd prime number. Throughout this paper, Z p , Q p , and C p will denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of Q p , respectively. Let q be an indeterminate in C p with |1āˆ’q | p < p āˆ’ 1 p āˆ’ 1 and UD( Z p ) be the space of all uniformly differentiable functions on Z p . The q-analog of x is defined as [ x ] q = 1 āˆ’ q x 1 āˆ’ q . Note that lim q ā†’ 1 [ x ] q =x. For fāˆˆUD( Z p ), the bosonic p-adic q-integral on Z p is defined by Kim to be

I q (f)= āˆ« Z p f(x)d Ī¼ q (x)= lim N ā†’ āˆž 1 [ p N ] q āˆ‘ x = 0 p N āˆ’ 1 f(x) q x (see [1, 2])
(1.1)

and the fermionic p-adic q-integral on Z p is also defined by Kim to be

I āˆ’ q (f)= āˆ« Z p f(x)d Ī¼ āˆ’ q (x)= lim N ā†’ āˆž 1 [ p N ] āˆ’ q āˆ‘ x = 0 p N āˆ’ 1 f(x) ( āˆ’ q ) x (see [1ā€“3]).
(1.2)

From (1.1) and (1.2), we have

q I q ( f 1 )āˆ’ I q (f)=(qāˆ’1)f(0)+ q āˆ’ 1 log q f ā€² (0)
(1.3)

and

q I āˆ’ q ( f 1 )+ I āˆ’ q (f)= [ 2 ] q f(0)(see [1ā€“3]).
(1.4)

As is well known, the q-analog of the Bernoulli polynomials is given by the generating function to be

q āˆ’ 1 + q āˆ’ 1 log q t q e t āˆ’ 1 e x t = āˆ‘ n = 0 āˆž B n , q (x) t n n ! (see [1, 2, 4ā€“20]),
(1.5)

and the q-analog of the Euler polynomials is given by

[ 2 ] q q e t + 1 e x t = āˆ‘ n = 0 āˆž E n , q (x) t n n ! (see [1, 2, 4ā€“21]).
(1.6)

The higher-order q-Daehee polynomials are given by

q āˆ’ 1 + q āˆ’ 1 log q log ( 1 + t ) q ( 1 + t ) āˆ’ 1 ( 1 + t ) x = āˆ‘ n = 0 āˆž D n , q (x) t n n ! ,
(1.7)

where tāˆˆ C p with |t | p < p āˆ’ 1 p āˆ’ 1 .

Now, we define the q-analog of the Changhee polynomials, which are given by the generating function to be

( [ 2 ] q q t + [ 2 ] q ) ( 1 + t ) x = āˆ‘ n = 0 āˆž Ch n , q (x) t n n ! .
(1.8)

In this paper, we investigate some properties for the q-analog of several special polynomials which are derived from the bosonic or fermionic p-adic q-integral on Z p .

2 Some special q-polynomials

In this section, we assume that tāˆˆ C p with |t | p < p āˆ’ 1 p āˆ’ 1 . Now, we define the higher-order q-Bernoulli numbers,

( q āˆ’ 1 + q āˆ’ 1 log q t q e t āˆ’ 1 ) r e x t = āˆ‘ n = 0 āˆž B n , q ( r ) (x) t n n ! .
(2.1)

When x=0, B n , q ( r ) = B n , q ( r ) (0) are called the higher-order q-Bernoulli numbers.

We also consider the higher-order q-Daehee polynomials as follows:

( q āˆ’ 1 + q āˆ’ 1 log q log ( 1 + t ) q ( 1 + t ) āˆ’ 1 ) r ( 1 + t ) x = āˆ‘ n = 0 āˆž D n , q ( r ) (x) t n n ! .
(2.2)

When x=0, D n , q ( r ) = D n , q ( r ) (0) are called the higher-order q-Daehee numbers.

From (1.3), we can derive the following equation:

āˆ« Z p ā‹Æ āˆ« Z p ( 1 + t ) x 1 + ā‹Æ + x r + x d Ī¼ q ( x 1 ) ā‹Æ d Ī¼ q ( x r ) = ( q āˆ’ 1 + q āˆ’ 1 log q log ( 1 + t ) q ( 1 + t ) āˆ’ 1 ) r ( 1 + t ) x = āˆ‘ n = 0 āˆž D n , q ( r ) ( x ) t n n ! .
(2.3)

Thus, by (2.3), we get

āˆ« Z p ā‹Æ āˆ« Z p ( x 1 + ā‹Æ + x r + x n ) d Ī¼ q ( x 1 )ā‹Æd Ī¼ q ( x r )= D n , q ( r ) ( x ) n ! (nā‰„0).
(2.4)

By replacing t by e t āˆ’1 in (2.2), we get

āˆ‘ n = 0 āˆž D n , q ( r ) (x) ( e t āˆ’ 1 ) n n ! = ( q āˆ’ 1 + q āˆ’ 1 log q t q e t āˆ’ 1 ) r e x t = āˆ‘ n = 0 āˆž B n , q ( r ) (x) t n n !
(2.5)

and

āˆ‘ n = 0 āˆž D n , q ( r ) ( x ) 1 n ! ( e t āˆ’ 1 ) n = āˆ‘ n = 0 āˆž D n , q ( r ) ( x ) 1 n ! n ! āˆ‘ m = n āˆž S 2 ( m , n ) t m m ! = āˆ‘ m = 0 āˆž ( āˆ‘ n = 0 m D n , q ( r ) ( x ) S 2 ( m , n ) ) t m m ! .
(2.6)

Thus, by (2.5) and (2.6), we get

B n , q ( r ) (x)= āˆ‘ m = 0 n D m , q ( r ) (x) S 2 (n,m).
(2.7)

Therefore, by (2.4) and (2.7), we obtain the following theorem.

Theorem 1 For nā‰„0, we have

B n , q ( r ) (x)= āˆ‘ m = 0 n D m , q ( r ) (x) S 2 (n,m)

and

āˆ« Z p ā‹Æ āˆ« Z p ( x 1 + ā‹Æ + x r + x n ) d Ī¼ q ( x 1 ) ā‹Æ d Ī¼ q ( x r ) = D n , q ( r ) ( x ) n ! ,

where S 2 (n,m) is the Stirling number of the second kind.

From (2.1), by replacing t by log(1+t), we obtain

( q āˆ’ 1 + q āˆ’ 1 log q log ( 1 + t ) q ( 1 + t ) āˆ’ 1 ) r ( 1 + t ) x = āˆ‘ n = 0 āˆž B n , q ( r ) ( x ) 1 n ! ( log ( 1 + t ) ) n = āˆ‘ n = 0 āˆž B n , q ( r ) ( x ) 1 n ! n ! āˆ‘ m = n āˆž S 1 ( m , n ) t m m ! = āˆ‘ m = 0 āˆž ( āˆ‘ n = 0 m S 1 ( m , n ) B n , q ( r ) ( x ) ) t m m ! ,
(2.8)

where S 1 (n,m) is the Stirling number of the first kind.

Therefore, by (2.2) and (2.8), we obtain the following theorem.

Theorem 2 For nā‰„0, we have

D n , q ( r ) (x)= āˆ‘ m = 0 n S 1 (n,m) B m , q ( r ) (x).

Now, we define the higher-order q-Changhee polynomials as follows:

( [ 2 ] q q t + [ 2 ] q ) r ( 1 + t ) x = āˆ‘ n = 0 āˆž Ch n , q ( r ) (x) t n n ! .
(2.9)

When x=0, Ch n , q ( r ) = Ch n , q ( r ) (0) are called the higher-order q-Changhee numbers.

From (1.4), we note that

āˆ« Z p ā‹Æ āˆ« Z p ( 1 + t ) x 1 + ā‹Æ + x r + x d Ī¼ āˆ’ q ( x 1 )ā‹Æd Ī¼ āˆ’ q ( x r )= ( [ 2 ] q q t + [ 2 ] q ) r ( 1 + t ) x .
(2.10)

Thus, by (2.10), we get

āˆ« Z p ā‹Æ āˆ« Z p ( x 1 + ā‹Æ + x r + x n ) d Ī¼ āˆ’ q ( x 1 )ā‹Æd Ī¼ āˆ’ q ( x r )= Ch n , q ( r ) ( x ) n ! .
(2.11)

In view of (1.6), we define the higher-order q-Euler polynomials which are given by the generating function to be

( [ 2 ] q q e t + 1 ) r e x t = āˆ‘ n = 0 āˆž E n , q ( r ) (x) t n n ! .
(2.12)

From (2.10), we note that

āˆ« Z p ā‹Æ āˆ« Z p ( 1 + t ) x 1 + ā‹Æ + x r + x d Ī¼ āˆ’ q ( x 1 ) ā‹Æ d Ī¼ āˆ’ q ( x r ) = ( [ 2 ] q q e log ( 1 + t ) + 1 ) r e x log ( 1 + t ) = āˆ‘ n = 0 āˆž E n , q ( r ) ( x ) 1 n ! ( log ( 1 + t ) ) n = āˆ‘ n = 0 āˆž E n , q ( r ) ( x ) āˆ‘ m = n āˆž S 1 ( m , n ) t m m ! = āˆ‘ m = 0 āˆž ( āˆ‘ n = 0 m E n , q ( r ) ( x ) S 1 ( m , n ) ) t m m ! .
(2.13)

Therefore, by (2.11) and (2.13), we obtain the following theorem.

Theorem 3 For nā‰„0, we have

āˆ« Z p ā‹Æ āˆ« Z p ( x 1 + ā‹Æ + x r + x n ) d Ī¼ āˆ’ q ( x 1 ) ā‹Æ d Ī¼ āˆ’ q ( x r ) = Ch n , q ( r ) ( x ) n ! = 1 n ! āˆ‘ m = 0 n E m , q ( r ) ( x ) S 1 ( n , m ) .

By replacing t by e t āˆ’1 in (2.9), we get

āˆ‘ n = 0 āˆž Ch n , q ( r ) (x) ( e t āˆ’ 1 ) n n ! = ( [ 2 ] q q e t + 1 ) r e x t
(2.14)

and

āˆ‘ n = 0 āˆž Ch n , q ( r ) ( x ) 1 n ! ( e t āˆ’ 1 ) n = āˆ‘ n = 0 āˆž Ch n , q ( r ) ( x ) āˆ‘ m = n āˆž S 2 ( m , n ) t m m ! = āˆ‘ m = 0 āˆž ( āˆ‘ n = 0 m Ch n , q ( r ) ( x ) S 2 ( m , n ) ) t m n ! .
(2.15)

Therefore, by (2.12), (2.14), and (2.15), we obtain the following theorem.

Theorem 4 For mā‰„0, we have

E m , q ( r ) (x)= āˆ‘ n = 0 m Ch n , q ( r ) (x) S 2 (m,n).

Now, we consider the q-analog of the higher-order Cauchy polynomials, which are defined by the generating function to be

( q ( 1 + t ) āˆ’ 1 ( q āˆ’ 1 ) + q āˆ’ 1 log q log ( 1 + t ) ) r ( 1 + t ) x = āˆ‘ n = 0 āˆž C n , q ( r ) (x) t n n ! .
(2.16)

When x=0, C n , q ( r ) = C n , q ( r ) (0) are called the higher-order q-Cauchy numbers. Indeed,

lim q ā†’ 1 ( q ( 1 + t ) āˆ’ 1 q āˆ’ 1 + q āˆ’ 1 log q log ( 1 + t ) ) r ( 1 + t ) x = ( t log ( 1 + t ) ) r ( 1 + t ) x = āˆ‘ n = 0 āˆž C n ( r ) ( x ) t n n ! ,
(2.17)

where C n ( r ) (x) are called the higher-order Cauchy polynomials.

We observe that

( 1 + t ) x = ( q ( 1 + t ) āˆ’ 1 q āˆ’ 1 + q āˆ’ 1 log q log ( 1 + t ) ) r ( 1 + t ) x ( q āˆ’ 1 + q āˆ’ 1 log q log ( 1 + t ) q ( 1 + t ) āˆ’ 1 ) r = ( āˆ‘ l = 0 āˆž C l , q ( r ) ( x ) t l l ! ) ( āˆ‘ m = 0 āˆž D m , q ( r ) t m m ! ) = āˆ‘ n = 0 āˆž ( āˆ‘ l = 0 n ( n l ) C l , q ( r ) ( x ) D n āˆ’ l , q ( r ) ) t n n !
(2.18)

and

( 1 + t ) x = āˆ‘ n = 0 āˆž ( x ) n t n n ! .
(2.19)

By (2.18) and (2.19), we get

( x ) n = āˆ‘ l = 0 n ( n l ) C l , q ( r ) (x) D n āˆ’ l , q ( r ) .
(2.20)

Therefore, by (2.20), we obtain the following theorem.

Theorem 5 For nā‰„0, we have

( x n ) = 1 n ! āˆ‘ l = 0 n ( n l ) C l , q ( r ) (x) D n āˆ’ l , q ( r ) .

For nāˆˆNāˆŖ{0}, we define the q-analog of the Bernoulli-Euler mixed-type polynomials of order (r,s) as follows:

B E n , q ( r , s ) (x)= āˆ« Z p ā‹Æ āˆ« Z p E n , q ( s ) (x+ y 1 +ā‹Æ+ y r )d Ī¼ q ( y 1 )ā‹Æd Ī¼ q ( y r ).
(2.21)

Then, by (2.21), we get

āˆ‘ n = 0 āˆž B E n , q ( r , s ) ( x ) t n n ! = āˆ« Z p ā‹Æ āˆ« Z p āˆ‘ n = 0 āˆž E n , q ( s ) ( x + y 1 + ā‹Æ + y r ) t n n ! d Ī¼ q ( y 1 ) ā‹Æ d Ī¼ q ( y r ) = ( [ 2 ] q q e t + 1 ) s āˆ« Z p ā‹Æ āˆ« Z p e ( x + y 1 + ā‹Æ + y r ) t d Ī¼ q ( y 1 ) ā‹Æ d Ī¼ q ( y r ) = ( [ 2 ] q q e t + 1 ) s ( q āˆ’ 1 + q āˆ’ 1 log q t q e t āˆ’ 1 ) r e x t .
(2.22)

It is easy to show that

( [ 2 ] q q e t + 1 ) s ( q āˆ’ 1 + q āˆ’ 1 log q t q e t āˆ’ 1 ) r e x t = āˆ‘ n = 0 āˆž ( āˆ‘ l = 0 n ( n l ) E l , q ( s ) B n āˆ’ l , q ( r ) ( x ) ) t n n ! .
(2.23)

Therefore, by (2.22) and (2.23), we obtain the following theorem.

Theorem 6 For nā‰„0, we have

B E n , q ( r , s ) (x)= āˆ‘ l = 0 n ( n l ) E l , q ( s ) B n āˆ’ l , q ( r ) (x).

By replacing t by log(1+t) in (2.22), we get

āˆ‘ n = 0 āˆž B E n , q ( r , s ) ( x ) ( log ( 1 + t ) ) n n ! = ( [ 2 ] q q t + [ 2 ] q ) s ( q āˆ’ 1 + q āˆ’ 1 log q log ( 1 + t ) q ( 1 + t ) āˆ’ 1 ) r ( 1 + t ) x = āˆ‘ n = 0 āˆž { āˆ‘ m = 0 n ( n m ) D m , q ( r ) ( x ) Ch n āˆ’ m , q ( s ) } t n n !
(2.24)

and

āˆ‘ m = 0 āˆž B E m , q ( r , s ) ( x ) ( log ( 1 + t ) ) m m ! = āˆ‘ n = 0 āˆž { āˆ‘ m = 0 n B E m , q ( r , s ) ( x ) S 1 ( n , m ) } t n n ! .
(2.25)

Therefore, by (2.24) and (2.25), we obtain the following theorem.

Theorem 7 For nā‰„0, we have

āˆ‘ m = 0 n ( n m ) D m , q ( r ) (x) Ch n āˆ’ m , q ( s ) = āˆ‘ m = 0 n B E m , q ( r , s ) (x) S 1 (n,m).

Let us consider the q-analog of the Daehee-Changhee mixed-type polynomials of order (r,s) as follows: for nā‰„0,

D C n , q ( r , s ) (x)= āˆ« Z p ā‹Æ āˆ« Z p D n , q ( r ) (x+ y 1 +ā‹Æ+ y s )d Ī¼ āˆ’ q ( y 1 )ā‹Æd Ī¼ āˆ’ q ( y s ).
(2.26)

Thus, by (2.26), we get

āˆ‘ n = 0 āˆž D C n , q ( r , s ) ( x ) t n n ! = āˆ« Z p ā‹Æ āˆ« Z p āˆ‘ n = 0 āˆž D n , q ( r ) ( x + y 1 + ā‹Æ + y s ) t n n ! d Ī¼ āˆ’ q ( y 1 ) ā‹Æ d Ī¼ āˆ’ q ( y s ) = ( q āˆ’ 1 + q āˆ’ 1 log q log ( 1 + t ) q ( 1 + t ) āˆ’ 1 ) r āˆ« Z p ā‹Æ āˆ« Z p ( 1 + t ) x + y 1 + ā‹Æ + y s d Ī¼ āˆ’ q ( y 1 ) ā‹Æ d Ī¼ āˆ’ q ( y s ) = ( q āˆ’ 1 + q āˆ’ 1 log q log ( 1 + t ) q ( 1 + t ) āˆ’ 1 ) r ( [ 2 ] q q t + [ 2 ] q ) s ( 1 + t ) x = ( āˆ‘ m = 0 āˆž D m , q ( r ) t m m ! ) ( āˆ‘ l = 0 āˆž Ch l , q ( s ) ( x ) t l l ! ) = āˆ‘ n = 0 āˆž { āˆ‘ m = 0 n ( n m ) D m , q ( r ) Ch n āˆ’ m , q ( s ) ( x ) } t n n !
(2.27)

and

āˆ‘ n = 0 āˆž D C n , q ( r , s ) ( x ) ( e t āˆ’ 1 ) n n ! = ( q āˆ’ 1 + q āˆ’ 1 log q t q e t āˆ’ 1 ) r ( [ 2 ] q q e t + 1 ) s e x t = āˆ‘ n = 0 āˆž { āˆ‘ m = 0 n ( n m ) B m , q ( r ) E n āˆ’ m , q ( s ) ( x ) } t n n ! .
(2.28)

Now, we observe that

āˆ‘ n = 0 āˆž D C n , q ( r , s ) ( x ) ( e t āˆ’ 1 ) n n ! = āˆ‘ n = 0 āˆž D C n , q ( r , s ) ( x ) 1 n ! n ! āˆ‘ m = n āˆž S 2 ( m , n ) t m m ! = āˆ‘ m = 0 āˆž { āˆ‘ n = 0 m D C n , q ( r , s ) ( x ) S 2 ( m , n ) } t m m ! .
(2.29)

Therefore, by (2.27), (2.28), and (2.29), we obtain the following theorem.

Theorem 8 For nā‰„0, we have

D C n , q ( r , s ) (x)= āˆ‘ m = 0 n ( n m ) D m , q ( r ) Ch n āˆ’ m , q ( s ) (x)

and

āˆ‘ m = 0 n ( n m ) B m , q ( r ) E n āˆ’ m , q ( s ) (x)= āˆ‘ m = 0 n D C m , q ( r , s ) (x) S 2 (n,m).

Now, we consider the q-extension of the Cauchy-Changhee mixed-type polynomials of order (r,s) as follows: for nā‰„0,

C C n , q ( r , s ) (x)= āˆ« Z p ā‹Æ āˆ« Z p C n , q ( r ) (x+ y 1 +ā‹Æ+ y s )d Ī¼ āˆ’ q ( y 1 )ā‹Æd Ī¼ āˆ’ q ( y r ).
(2.30)

Thus, by (2.30), we get

āˆ‘ n = 0 āˆž C C n , q ( r , s ) ( x ) t n n ! = āˆ« Z p ā‹Æ āˆ« Z p āˆ‘ n = 0 āˆž C n , q ( r ) ( x + y 1 + ā‹Æ + y s ) t n n d Ī¼ āˆ’ q ( y 1 ) ā‹Æ d Ī¼ āˆ’ q ( y s ) = ( q ( 1 + t ) āˆ’ 1 q āˆ’ 1 + q āˆ’ 1 log q log ( 1 + t ) ) r ( [ 2 ] q q t + [ 2 ] q ) s ( 1 + t ) x = āˆ‘ n = 0 āˆž { āˆ‘ m = 0 n ( n m ) C m , q ( r ) Ch n āˆ’ m , q ( s ) ( x ) } t n n ! ,
(2.31)
āˆ‘ n = 0 āˆž C C n , q ( r , s ) ( x ) ( e t āˆ’ 1 ) n n ! = ( q e t āˆ’ 1 q āˆ’ 1 + q āˆ’ 1 log q t ) r ( [ 2 ] q q e t + 1 ) s e t x = ( āˆ‘ m = 0 āˆž B m , q ( āˆ’ r ) t m m ! ) ( āˆ‘ l = 0 āˆž E l , q ( s ) ( x ) t l l ! ) = āˆ‘ n = 0 āˆž ( āˆ‘ m = 0 n ( n m ) B m , q ( āˆ’ r ) E n āˆ’ m , q ( s ) ( x ) ) t n n ! .
(2.32)

Note that

āˆ‘ n = 0 āˆž C C n , q ( r , s ) (x) ( e t āˆ’ 1 ) n n ! = āˆ‘ n = 0 āˆž ( āˆ‘ m = 0 n C C m , q ( r , s ) ( x ) S 2 ( n , m ) ) t n n ! .
(2.33)

Therefore, by (2.31), (2.32), and (2.33), we obtain the following theorem.

Theorem 9 For nā‰„0, we have

C C n , q ( r , s ) (x)= āˆ‘ m = 0 n ( n m ) C m , q ( r ) Ch n āˆ’ m , q ( s ) (x)

and

āˆ‘ m = 0 n ( n m ) B m , q ( āˆ’ r ) E n āˆ’ m , q ( s ) (x)= āˆ‘ m = 0 n C C m , q ( r , s ) (x) S 2 (n,m).

Finally, we define the q-extension of the Cauchy-Daehee mixed-type polynomials of order (r,s) as follows:

C D n , q ( r , s ) (x)= āˆ« Z p ā‹Æ āˆ« Z p C n , q ( r ) (x+ y 1 +ā‹Æ+ y r )d Ī¼ q ( x 1 )ā‹Æd Ī¼ q ( x r ).
(2.34)

Thus, by (2.34), we get

āˆ‘ n = 0 āˆž C D n , q ( r , s ) ( x ) t n n ! = āˆ« Z p ā‹Æ āˆ« Z p āˆ‘ n = 0 āˆž C n , q ( r ) ( x + y 1 + ā‹Æ + y s ) t n n ! d Ī¼ q ( y 1 ) ā‹Æ d Ī¼ q ( y s ) = ( q ( 1 + t ) āˆ’ 1 q āˆ’ 1 + q āˆ’ 1 log q log ( 1 + t ) ) r āˆ« Z p ā‹Æ āˆ« Z p ( 1 + t ) x + y 1 + ā‹Æ + y s d Ī¼ q ( y 1 ) ā‹Æ d Ī¼ q ( y s ) = ( q ( 1 + t ) āˆ’ 1 q āˆ’ 1 + q āˆ’ 1 log q log ( 1 + t ) ) r ( q āˆ’ 1 + q āˆ’ 1 log q log ( 1 + t ) q ( 1 + t ) āˆ’ 1 ) s ( 1 + t ) x = { āˆ‘ n = 0 āˆž C n , q ( r āˆ’ s ) ( x ) t n n ! if  r > s , āˆ‘ n = 0 āˆž D n , q ( s āˆ’ r ) ( x ) t n n ! if  r < s , āˆ‘ n = 0 āˆž ( x ) n t n n ! if  r = s .
(2.35)

Therefore, by (2.35), we obtain the following equation:

C D n , q ( r , s ) (x)={ C n , q ( r āˆ’ s ) ( x ) if  r > s , D n , q ( s āˆ’ r ) ( x ) if  r < s , ( x ) n if  r = s .

References

  1. Choi J, Kim T, Kim YH: A note on the modified q -Euler numbers and polynomials with weight. Proc. Jangjeon Math. Soc. 2011,14(4):399ā€“402. MR 2894491 (2012k:05045)

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  2. Choi J, Kim T: Arithmetic properties for the q -Bernoulli numbers and polynomials. Proc. Jangjeon Math. Soc. 2012,15(2):137ā€“143. MR 2954135

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  3. Araci S, Acikgoz M, Sen E: On the extended Kimā€™s p -adic q -deformed fermionic integrals in the p -adic integer ring. J.Ā Number Theory 2013,133(10):3348ā€“3361. MR 3071817 10.1016/j.jnt.2013.04.007

    ArticleĀ  MathSciNetĀ  MATHĀ  Google ScholarĀ 

  4. Gaboury S, Tremblay R, FugĆØre B-J: Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials. Proc. Jangjeon Math. Soc. 2014,17(1):115ā€“123. MR 3184467

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  5. Jeong J-H, Jin J-H, Park J-W, Rim S-H: On the twisted weak q -Euler numbers and polynomials with weight 0. Proc. Jangjeon Math. Soc. 2013,16(2):157ā€“163. MR 3097729

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  6. Kim H-M, Kim DS, Kim T, Lee S-H, Dolgy DV, Lee B: Identities for the Bernoulli and Euler numbers arising from the p -adic integral on Z p . Proc. Jangjeon Math. Soc. 2012,15(2):155ā€“161. MR 2954137

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  7. Kim D, Kim M-S, Kim T: Higher-order twisted q -Euler polynomials and numbers. Proc. Jangjeon Math. Soc. 2010,13(2):265ā€“277. MR 2676691 (2011f:11027)

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  8. Kim DS, Lee N, Na J, Park KH: Abundant symmetry for higher-order Bernoulli polynomials (II). Proc. Jangjeon Math. Soc. 2013,16(3):359ā€“378. MR 3100091

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  9. Kim DS, Kim T, Kwon HI, Seo J-J: Identities of some special mixed-type polynomials. Adv. Stud. Theor. Phys. 2014,8(17):745ā€“754.

    Google ScholarĀ 

  10. Kim DS, Dolgy DV, Kim T, Rim S-H: Identities involving Bernoulli and Euler polynomials arising from Chebyshev polynomials. Proc. Jangjeon Math. Soc. 2012,15(4):361ā€“370. MR 3050107

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  11. Kim DS, Kim T, Lee S-H, Seo J-J: Higher-order Daehee numbers and polynomials. Int. J. Math. Anal., Ruse 2014,8(5ā€“8):273ā€“283.

    MathSciNetĀ  Google ScholarĀ 

  12. Kim DS: Symmetry identities for generalized twisted Euler polynomials twisted by unramified roots of unity. Proc. Jangjeon Math. Soc. 2012,15(3):303ā€“316. MR 2978431

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  13. Kim DS, Kim T, Seo J-J: A note on Changhee Polynomials and numbers. Adv. Stud. Theor. Phys. 2013,7(20):993ā€“1003.

    Google ScholarĀ 

  14. Kim DS, Kim T, Seo J-J: Higher-order Daehee polynomials of the first kind with umbral calculus. Adv. Stud. Contemp. Math. (Kyungshang) 2014,24(1):5ā€“18. MR 3157404

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  15. Kim DS, Kim T: Daehee numbers and polynomials. Appl. Math. Sci., Ruse 2013,7(117ā€“120):5969ā€“5976. MR 3141903

    MathSciNetĀ  Google ScholarĀ 

  16. Kim T, Rim S-H: On Changhee-Barnesā€™ q -Euler numbers and polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 2004,9(2):81ā€“86. MR 2090111 (2005f:11028)

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  17. Ryoo CS: A note on the Frobenius-Euler polynomials. Proc. Jangjeon Math. Soc. 2011,14(4):495ā€“501. MR 2894498

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  18. Seo J-J, Rim S-H, Lee S-H, Dolgy DV, Kim T: q -Bernoulli numbers and polynomials related to p -adic invariant integral on Z p . Proc. Jangjeon Math. Soc. 2013,16(3):321ā€“326. MR 3100087

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  19. Seo J-J, Rim S-H, Kim T, Lee S-H: Sums products of generalized Daehee numbers. Proc. Jangjeon Math. Soc. 2014,17(1):1ā€“9. MR 3184457

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

  20. Simsek Y, Rim S-H, Jang L-C, Kang D-J, Seo J-J: A note on q -Daehee sums. J. Anal. Comput. 2005,1(2):151ā€“160. MRĀ 2475196

    MathSciNetĀ  Google ScholarĀ 

  21. Jolany H, Sharifi H, Alikelaye RE: Some results for the Apostol-Genocchi polynomials of higher order. Bull. Malays. Math. Soc. 2013,36(2):465ā€“479. MR 3030964

    MathSciNetĀ  MATHĀ  Google ScholarĀ 

Download references

Acknowledgements

This paper is supported by grant No.Ā 14-11-00022 of Russian Scientific fund.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Taekyun Kim.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authorsā€™ contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

Rights and permissions

Open Access Ā This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.

The images or other third party material in this article are included in the articleā€™s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the articleā€™s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dolgy, D.V., Kim, D.S., Kim, T. et al. Some identities of special q-polynomials. J Inequal Appl 2014, 438 (2014). https://doi.org/10.1186/1029-242X-2014-438

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2014-438

Keywords