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On the modified q-Euler polynomials with weight

Abstract

In this paper, we construct a new q-extension of Euler numbers and polynomials with weight related to fermionic p-adic q-integral on Z p and give new explicit formulas related to these numbers and polynomials.

Throughout this paper Z p , Q p and C p will respectively denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of Q p . Let ν p be the normalized exponential valuation of C p with | p | p = p ν p ( p ) = 1 p .

In this paper, we assume that q C p with | 1 q | p < p 1 p 1 so that q x =exp(xlogq) for x Z p . The q-number of x is denoted by [ x ] q = 1 q x 1 q . Note that lim q 1 [ x ] q =x. Let d be a fixed integer bigger than 0, and let p be a fixed prime number and (d,p)=1. We set

X d = lim N Z / d p N Z , X = 0 < a < d p ( a , p ) = 1 ( a + d p Z p ) , a + d p N Z p = { x X x a ( mod d p N ) } ,

where aZ lies in 0a<d p N (see [122]).

Let C( Z p ) be the space of continuous functions on Z p . For fC( Z p ), the fermionic p-adic q-integral on Z p is defined by Kim as

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 [8–22]).

As is well known, Euler polynomials are defined by the generating function to be

2 e t + 1 e x t = e E ( x ) t = n = 0 E n (x) t n n ! (see [11–13, 15, 20–22])

with the usual convention about replacing E n (x) by E n (x). In the special case, x=0, E n (0)= E n are called the nth Euler numbers.

In [13, 20, 23], Kim defined the q-Euler numbers as follows:

E 0 , q =1,q ( q E + 1 ) n + E n , q ={ [ 2 ] q , if  n = 0 , 0 , if  n 0 ,
(1)

with the usual convection of replacing E n by E n , q . From (1), we also derive

E n , q = [ 2 ] q ( 1 q ) n l = 0 n ( n l ) ( 1 ) l 1 + q l + 1 (see [20, 23]).

By using an invariant p-adic q-integral on Z p , a q-extension of ordinary Euler polynomials, called q-Euler polynomials, is considered and investigated by Kim [14, 15, 18]. For x Z p , q-Euler polynomials are defined as follows:

E n , q (x)= Z p [ x + y ] q n d μ q (y).
(2)

By (2), the following relation holds:

E n , q (x)= k = 0 n ( n k ) [ x ] q n k q k x E k , q .

Recently, Kim considered the modified q-Euler polynomials which are slightly different from Kim’s q-Euler polynomials as follows:

ϵ n , q (x)= Z p q x [ x + y ] q n d μ q (y)for nN,

and he showed that

ϵ n , q (x)= [ 2 ] q ( 1 q ) n l = 0 n ( n l ) q x l 1 + q l
(3)

(see [22]). In the special case, x=0, ϵ n , q (0)= ϵ n , q are called the nth modified q-Euler numbers, and it is showed that

ϵ n , q = [ 2 ] q ( 1 q ) n l = 0 n ( n l ) 1 1 + q l .
(4)

And in [24], authors defined modified q-Euler polynomials with weight α ϵ n , q ( α ) (x) as follows:

ϵ n , q ( α ) (x)= Z p q x [ x + y ] q α n d μ q α (y)

and proved that

ϵ n , q ( α ) (x)= [ 2 ] q ( 1 q α ) n l = 0 n ( n l ) ( 1 ) l q α l 1 + q α l .
(5)

In the special case, x=0, ϵ n , q ( α ) (0)= ϵ n , q ( α ) are called the nth modified q-Euler numbers with weight α, and it is showed that

ϵ n , q ( α ) = [ 2 ] q ( 1 q α ) n l = 0 n ( n l ) ( 1 ) l q α l 1 1 + q α l = [ 2 ] q m = 0 ( 1 ) m [ m + x ] q α n .
(6)

In this paper, we construct a new q-extension of Euler numbers and polynomials with weight related to fermionic p-adic q-integral on Z p and give new explicit formulas related to these numbers and polynomials.

1 A new approach of modified q-Euler polynomials

Let us consider the following modified q-Euler numbers:

ϵ ˜ n , q ( x ) = Z p q y ( x + [ y ] q ) n d μ q ( y ) = l = 0 n ( n l ) x n l ϵ l , q = l = 0 n k = 0 l ( n l ) ( l k ) [ 2 ] q ( 1 q ) l x n l 1 + q k ,

where

ϵ ˜ n , q (0)= ϵ n , q = [ 2 ] q ( 1 q ) n l = 0 n ( n l ) 1 1 + q l .
(7)

Thus, by (7),

( 1 q ) n ϵ n , q = [ 2 ] q l = 0 n ( n l ) 1 1 + q l .

Consider the equation

n = 0 ( 1 q ) n ϵ n , q t n n ! = [ 2 ] q n = 0 l = 0 n ( n l ) 1 1 + q l t n n ! = [ 2 ] q ( m = 0 t m m ! ) ( l = 0 1 1 + q l t l l ! ) = [ 2 ] q e t ( l = 0 1 1 + q l t l l ! ) .

Since

e ( 1 q ) x t n = 0 ( 1 q ) n ϵ n , q t n n ! = ( l = 0 ( 1 q ) l x l t l l ! ) ( n = 0 ( 1 q ) n ϵ n , q t n n ! ) = m = 0 ( 1 q ) m n = 0 m ( m n ) ϵ n , q x m n t m m ! = m = 0 ( 1 q ) m ϵ ˜ m , q ( x ) t m m !
(8)

and

e ( 1 q ) x t [ 2 ] q e t ( l = 0 1 1 + q l t l l ! ) = [ 2 ] q e ( ( 1 q ) x + 1 ) t ( l = 0 1 1 + q l t l l ! ) = [ 2 ] q ( m = 0 ( ( 1 q ) x + 1 ) m t m m ! ) ( l = 0 1 1 + q l t l l ! ) = [ 2 ] q n = 0 l = 0 n ( n l ) ( ( 1 q ) x + 1 ) n l 1 + q l t n n ! ,
(9)

by (8) and (9), we get

( 1 q ) n ϵ ˜ n , q ( x ) = [ 2 ] q l = 0 n ( n l ) ( ( 1 q ) x + 1 ) n l 1 + q l = [ 2 ] q l = 0 n ( n l ) 1 1 + q l j = 0 n l ( n l j ) ( 1 q ) j x j .

Thus, we have the following result.

Theorem 1.1 For n1,

ϵ ˜ n , q ( x ) = [ 2 ] q ( 1 q ) n l = 0 n ( n l ) ( ( 1 q ) x + 1 ) n l 1 + q l = [ 2 ] q ( 1 q ) n l = 0 n j = 0 n l ( n l ) ( n l j ) ( 1 q ) j 1 + q l x j .

2 A new approach of q-Euler polynomials with weight α

Let us consider the following modified q-Euler polynomials with weight α:

ϵ ˜ n , q ( α ) ( x ) = Z p q y ( x + [ y ] q α ) n d μ q α ( y ) = l = 0 n ( n k ) x n l ϵ k , q ( α ) = k = 0 n l = 0 k ( n k ) ( k l ) [ 2 ] q α ( 1 q ) n ( 1 ) l 1 + q α + l x n k ,

where

ϵ ˜ n , q ( α ) (0)= ϵ n , q ( α ) = [ 2 ] q ( 1 q α ) n l = 0 n ( n l ) ( 1 ) l q α l 1 + q α l .
(10)

Thus, by (10), we have

( 1 q α ) n ϵ n , q ( α ) = [ 2 ] q l = 0 n ( n l ) ( 1 ) l q α l 1 + q α l .

Consider the equation

n = 0 ( 1 q α ) n ϵ n , q ( α ) t n n ! = [ 2 ] q n = 0 l = 0 n ( n l ) ( 1 ) l q α l 1 + q α l t n n ! = [ 2 ] q ( m = 0 t m m ! ) ( l = 0 ( 1 ) l q α l 1 + q α l t l l ! ) = [ 2 ] q e t ( l = 0 ( q α ) l 1 + q α l t l l ! ) .

Since

e ( 1 q α ) x t n = 0 ( 1 q α ) n ϵ n , q ( α ) t n n ! = ( l = 0 ( 1 q α ) l x l t l l ! ) ( n = 0 ( 1 q α ) n ϵ n , q ( α ) t n n ! ) = m = 0 ( 1 q α ) m n = 0 m ( m n ) ϵ n , q ( α ) x m n t m m ! = m = 0 ( 1 q α ) m ϵ ˜ m , q ( α ) ( x ) t m m !
(11)

and

e ( 1 q α ) x t [ 2 ] q e t ( l = 0 ( q α ) l 1 + q α l t l l ! ) = [ 2 ] q e ( ( 1 q α ) x + 1 ) t ( l = 0 ( q α ) l 1 + q α l t l l ! ) = [ 2 ] q ( m = 0 ( ( 1 q α ) x + 1 ) m t m m ! ) ( l = 0 ( q α ) l 1 + q α l t l l ! ) = [ 2 ] q n = 0 l = 0 n ( n l ) ( ( 1 q α ) x + 1 ) n l 1 + q α l ( q α ) l t n n ! ,
(12)

by (11) and (12), we get

( 1 q α ) n ϵ ˜ n , q ( α ) ( x ) = [ 2 ] q l = 0 n ( n k ) ( ( 1 q α ) x + 1 ) n l 1 + q α l ( q α ) l = [ 2 ] q l = 0 n ( n l ) ( q α ) l 1 + q α l j = 0 n l ( n l j ) ( 1 q α ) j x j .

Thus, we have the following result.

Theorem 2.1 For n1,

ϵ ˜ n , q ( α ) ( x ) = [ 2 ] q ( 1 q α ) n l = 0 n ( n l ) ( q α ) l ( ( 1 q α ) x + 1 ) n l 1 + q α l = [ 2 ] q ( 1 q α ) n l = 0 n j = 0 n l ( n l ) ( n l j ) ( q α ) l ( 1 q α ) j 1 + q α l x j .

A systemic study of some families of the modified q-Euler polynomials with weight is presented by using the multivariate fermionic p-adic integral on Z p . The study of these modified q-Euler numbers and polynomials yields an interesting q-analogue of identities for Stirling numbers.

In recent years, many mathematicians and physicists have investigated zeta functions, multiple zeta functions, L-functions, and multiple q-Bernoulli numbers and polynomials mainly because of their interest and importance. These functions and polynomials are used not only in complex analysis and mathematical physics, but also in p-adic analysis and other areas. In particular, multiple zeta functions and multiple L-functions occur within the context of knot theory, quantum field theory, applied analysis and number theory (see [129]).

In our subsequent papers, we shall apply this p-adic mathematical theory to quantum statistical mechanics. Using p-adic quantum statistical mechanics, we can also derive a new partition function in the p-adic space and adopt this new partition function to quantum transport theory which is based on the projection technique related to the Liouville equation. We expect that a new quantum transport theory will explain diverse physical properties of the condensed matter system.

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Acknowledgements

The authors are grateful for the valuable comments and suggestions of the referees.

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Correspondence to Jin-Woo Park.

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The authors declare that they have no competing interests.

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All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.

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Rim, S., Park, J., Kwon, J. et al. On the modified q-Euler polynomials with weight. Adv Differ Equ 2013, 356 (2013). https://doi.org/10.1186/1687-1847-2013-356

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Keywords

  • modified q-Euler polynomials
  • modified q-Euler polynomials with weight
  • fermionic p-adic q-integral on Z p