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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 on  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 |1q | p < p 1 p 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 q x 1 q . Note that lim q 1 [ x ] q =x. For fUD( Z p ), the bosonic p-adic q-integral on Z p is defined by Kim to be

I q (f)= Z p f(x)d μ q (x)= lim N 1 [ p N ] q x = 0 p N 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 q (f)= Z p f(x)d μ q (x)= lim N 1 [ p N ] q x = 0 p N 1 f(x) ( q ) x (see [1–3]).
(1.2)

From (1.1) and (1.2), we have

q I q ( f 1 ) I q (f)=(q1)f(0)+ q 1 log q f (0)
(1.3)

and

q I q ( f 1 )+ I q (f)= [ 2 ] q f(0)(see [1–3]).
(1.4)

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

q 1 + q 1 log q t q e t 1 e x t = n = 0 B n , q (x) t n n ! (see [1, 2, 4–20]),
(1.5)

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

[ 2 ] q q e t + 1 e x t = n = 0 E n , q (x) t n n ! (see [1, 2, 4–21]).
(1.6)

The higher-order q-Daehee polynomials are given by

q 1 + q 1 log q log ( 1 + t ) q ( 1 + t ) 1 ( 1 + t ) x = n = 0 D n , q (x) t n n ! ,
(1.7)

where t C p with |t | p < p 1 p 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 = n = 0 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 t C p with |t | p < p 1 p 1 . Now, we define the higher-order q-Bernoulli numbers,

( q 1 + q 1 log q t q e t 1 ) r e x t = n = 0 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 1 + q 1 log q log ( 1 + t ) q ( 1 + t ) 1 ) r ( 1 + t ) x = n = 0 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:

Z p Z p ( 1 + t ) x 1 + + x r + x d μ q ( x 1 ) d μ q ( x r ) = ( q 1 + q 1 log q log ( 1 + t ) q ( 1 + t ) 1 ) r ( 1 + t ) x = n = 0 D n , q ( r ) ( x ) t n n ! .
(2.3)

Thus, by (2.3), we get

Z p Z p ( x 1 + + x r + x n ) d μ q ( x 1 )d μ q ( x r )= D n , q ( r ) ( x ) n ! (n0).
(2.4)

By replacing t by e t 1 in (2.2), we get

n = 0 D n , q ( r ) (x) ( e t 1 ) n n ! = ( q 1 + q 1 log q t q e t 1 ) r e x t = n = 0 B n , q ( r ) (x) t n n !
(2.5)

and

n = 0 D n , q ( r ) ( x ) 1 n ! ( e t 1 ) n = n = 0 D n , q ( r ) ( x ) 1 n ! n ! m = n S 2 ( m , n ) t m m ! = m = 0 ( 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)= 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 n0, we have

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

and

Z p Z p ( x 1 + + x r + x n ) d μ q ( x 1 ) d μ 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 1 + q 1 log q log ( 1 + t ) q ( 1 + t ) 1 ) r ( 1 + t ) x = n = 0 B n , q ( r ) ( x ) 1 n ! ( log ( 1 + t ) ) n = n = 0 B n , q ( r ) ( x ) 1 n ! n ! m = n S 1 ( m , n ) t m m ! = m = 0 ( 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 n0, we have

D n , q ( r ) (x)= 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 = n = 0 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

Z p Z p ( 1 + t ) x 1 + + x r + x d μ q ( x 1 )d μ q ( x r )= ( [ 2 ] q q t + [ 2 ] q ) r ( 1 + t ) x .
(2.10)

Thus, by (2.10), we get

Z p Z p ( x 1 + + x r + x n ) d μ q ( x 1 )d μ 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 = n = 0 E n , q ( r ) (x) t n n ! .
(2.12)

From (2.10), we note that

Z p Z p ( 1 + t ) x 1 + + x r + x d μ q ( x 1 ) d μ q ( x r ) = ( [ 2 ] q q e log ( 1 + t ) + 1 ) r e x log ( 1 + t ) = n = 0 E n , q ( r ) ( x ) 1 n ! ( log ( 1 + t ) ) n = n = 0 E n , q ( r ) ( x ) m = n S 1 ( m , n ) t m m ! = m = 0 ( 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 n0, we have

Z p Z p ( x 1 + + x r + x n ) d μ q ( x 1 ) d μ q ( x r ) = Ch n , q ( r ) ( x ) n ! = 1 n ! m = 0 n E m , q ( r ) ( x ) S 1 ( n , m ) .

By replacing t by e t 1 in (2.9), we get

n = 0 Ch n , q ( r ) (x) ( e t 1 ) n n ! = ( [ 2 ] q q e t + 1 ) r e x t
(2.14)

and

n = 0 Ch n , q ( r ) ( x ) 1 n ! ( e t 1 ) n = n = 0 Ch n , q ( r ) ( x ) m = n S 2 ( m , n ) t m m ! = m = 0 ( 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 m0, we have

E m , q ( r ) (x)= 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 ) 1 ( q 1 ) + q 1 log q log ( 1 + t ) ) r ( 1 + t ) x = n = 0 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 1 ( q ( 1 + t ) 1 q 1 + q 1 log q log ( 1 + t ) ) r ( 1 + t ) x = ( t log ( 1 + t ) ) r ( 1 + t ) x = n = 0 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 ) 1 q 1 + q 1 log q log ( 1 + t ) ) r ( 1 + t ) x ( q 1 + q 1 log q log ( 1 + t ) q ( 1 + t ) 1 ) r = ( l = 0 C l , q ( r ) ( x ) t l l ! ) ( m = 0 D m , q ( r ) t m m ! ) = n = 0 ( l = 0 n ( n l ) C l , q ( r ) ( x ) D n l , q ( r ) ) t n n !
(2.18)

and

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

By (2.18) and (2.19), we get

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

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

Theorem 5 For n0, we have

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

For nN{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)= Z p Z p E n , q ( s ) (x+ y 1 ++ y r )d μ q ( y 1 )d μ q ( y r ).
(2.21)

Then, by (2.21), we get

n = 0 B E n , q ( r , s ) ( x ) t n n ! = Z p Z p n = 0 E n , q ( s ) ( x + y 1 + + y r ) t n n ! d μ q ( y 1 ) d μ q ( y r ) = ( [ 2 ] q q e t + 1 ) s Z p Z p e ( x + y 1 + + y r ) t d μ q ( y 1 ) d μ q ( y r ) = ( [ 2 ] q q e t + 1 ) s ( q 1 + q 1 log q t q e t 1 ) r e x t .
(2.22)

It is easy to show that

( [ 2 ] q q e t + 1 ) s ( q 1 + q 1 log q t q e t 1 ) r e x t = n = 0 ( l = 0 n ( n l ) E l , q ( s ) B n l , q ( r ) ( x ) ) t n n ! .
(2.23)

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

Theorem 6 For n0, we have

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

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

n = 0 B E n , q ( r , s ) ( x ) ( log ( 1 + t ) ) n n ! = ( [ 2 ] q q t + [ 2 ] q ) s ( q 1 + q 1 log q log ( 1 + t ) q ( 1 + t ) 1 ) r ( 1 + t ) x = n = 0 { m = 0 n ( n m ) D m , q ( r ) ( x ) Ch n m , q ( s ) } t n n !
(2.24)

and

m = 0 B E m , q ( r , s ) ( x ) ( log ( 1 + t ) ) m m ! = n = 0 { 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 n0, we have

m = 0 n ( n m ) D m , q ( r ) (x) Ch n m , q ( s ) = 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 n0,

D C n , q ( r , s ) (x)= Z p Z p D n , q ( r ) (x+ y 1 ++ y s )d μ q ( y 1 )d μ q ( y s ).
(2.26)

Thus, by (2.26), we get

n = 0 D C n , q ( r , s ) ( x ) t n n ! = Z p Z p n = 0 D n , q ( r ) ( x + y 1 + + y s ) t n n ! d μ q ( y 1 ) d μ q ( y s ) = ( q 1 + q 1 log q log ( 1 + t ) q ( 1 + t ) 1 ) r Z p Z p ( 1 + t ) x + y 1 + + y s d μ q ( y 1 ) d μ q ( y s ) = ( q 1 + q 1 log q log ( 1 + t ) q ( 1 + t ) 1 ) r ( [ 2 ] q q t + [ 2 ] q ) s ( 1 + t ) x = ( m = 0 D m , q ( r ) t m m ! ) ( l = 0 Ch l , q ( s ) ( x ) t l l ! ) = n = 0 { m = 0 n ( n m ) D m , q ( r ) Ch n m , q ( s ) ( x ) } t n n !
(2.27)

and

n = 0 D C n , q ( r , s ) ( x ) ( e t 1 ) n n ! = ( q 1 + q 1 log q t q e t 1 ) r ( [ 2 ] q q e t + 1 ) s e x t = n = 0 { m = 0 n ( n m ) B m , q ( r ) E n m , q ( s ) ( x ) } t n n ! .
(2.28)

Now, we observe that

n = 0 D C n , q ( r , s ) ( x ) ( e t 1 ) n n ! = n = 0 D C n , q ( r , s ) ( x ) 1 n ! n ! m = n S 2 ( m , n ) t m m ! = m = 0 { 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 n0, we have

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

and

m = 0 n ( n m ) B m , q ( r ) E n m , q ( s ) (x)= 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 n0,

C C n , q ( r , s ) (x)= Z p Z p C n , q ( r ) (x+ y 1 ++ y s )d μ q ( y 1 )d μ q ( y r ).
(2.30)

Thus, by (2.30), we get

n = 0 C C n , q ( r , s ) ( x ) t n n ! = Z p Z p n = 0 C n , q ( r ) ( x + y 1 + + y s ) t n n d μ q ( y 1 ) d μ q ( y s ) = ( q ( 1 + t ) 1 q 1 + q 1 log q log ( 1 + t ) ) r ( [ 2 ] q q t + [ 2 ] q ) s ( 1 + t ) x = n = 0 { m = 0 n ( n m ) C m , q ( r ) Ch n m , q ( s ) ( x ) } t n n ! ,
(2.31)
n = 0 C C n , q ( r , s ) ( x ) ( e t 1 ) n n ! = ( q e t 1 q 1 + q 1 log q t ) r ( [ 2 ] q q e t + 1 ) s e t x = ( m = 0 B m , q ( r ) t m m ! ) ( l = 0 E l , q ( s ) ( x ) t l l ! ) = n = 0 ( m = 0 n ( n m ) B m , q ( r ) E n m , q ( s ) ( x ) ) t n n ! .
(2.32)

Note that

n = 0 C C n , q ( r , s ) (x) ( e t 1 ) n n ! = n = 0 ( 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 n0, we have

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

and

m = 0 n ( n m ) B m , q ( r ) E n m , q ( s ) (x)= 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)= Z p Z p C n , q ( r ) (x+ y 1 ++ y r )d μ q ( x 1 )d μ q ( x r ).
(2.34)

Thus, by (2.34), we get

n = 0 C D n , q ( r , s ) ( x ) t n n ! = Z p Z p n = 0 C n , q ( r ) ( x + y 1 + + y s ) t n n ! d μ q ( y 1 ) d μ q ( y s ) = ( q ( 1 + t ) 1 q 1 + q 1 log q log ( 1 + t ) ) r Z p Z p ( 1 + t ) x + y 1 + + y s d μ q ( y 1 ) d μ q ( y s ) = ( q ( 1 + t ) 1 q 1 + q 1 log q log ( 1 + t ) ) r ( q 1 + q 1 log q log ( 1 + t ) q ( 1 + t ) 1 ) s ( 1 + t ) x = { n = 0 C n , q ( r s ) ( x ) t n n ! if  r > s , n = 0 D n , q ( s r ) ( x ) t n n ! if  r < s , n = 0 ( 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 s ) ( x ) if  r > s , D n , q ( s r ) ( x ) if  r < s , ( x ) n if  r = s .

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Acknowledgements

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

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Correspondence to Taekyun Kim.

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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

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Keywords

  • Prime Number
  • Rational Number
  • Differentiable Function
  • Algebraic Closure
  • Bernoulli Polynomial