# 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 $a∈Z$ lies in $0≤a (see ).

Let $C( Z p )$ be the space of continuous functions on $Z p$. For $f∈C( Z p )$, 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:

(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:

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 ). 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 , 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 $n≥1$,

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

## 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 $n≥1$,

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

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 ).

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.

## References

1. 1.

Araci S, Acikgoz M: A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials. Adv. Stud. Contemp. Math. 2012, 22(3):399-406.

2. 2.

Araci S, Acikgoz M, Şen E: On the extended Kim’s p -adic q -deformed fermionic integrals in the p -adic integer ring. J. Number Theory 2013, 133: 3348-3361. 10.1016/j.jnt.2013.04.007

3. 3.

Araci S, Acikgoz M, Jolany H: On p -adic interpolating function associated with modified Dirichlet’s type of twisted q -Euler numbers and polynomials with weight. J. Class. Anal. 2013, 1(1):35-48.

4. 4.

Bayad A: Special values of Lerch zeta function and their Fourier expansions. Adv. Stud. Contemp. Math. 2011, 21(1):1-4.

5. 5.

Carlitz L: q -Bernoulli numbers and polynomials. Duke Math. J. 1948, 15: 987-1000. 10.1215/S0012-7094-48-01588-9

6. 6.

Can M, Cenkci M, Kurt V, Simsek Y: Twisted Dedekind type sums associated with Barnes’ type multiple Frobenius-Euler l -functions. Adv. Stud. Contemp. Math. 2009, 18(2):135-160.

7. 7.

Cenkci M, Simsek Y, Kurt V: Multiple two-variable p -adic q - L -function and its behavior at $s=0$. Russ. J. Math. Phys. 2008, 15(4):447-459. 10.1134/S106192080804002X

8. 8.

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):399-402.

9. 9.

Choi J, Kim T, Kim YH, Lee B: On the $(w,q)$ -Euler numbers and polynomials with weight α . Proc. Jangjeon Math. Soc. 2012, 15(1):91-100.

10. 10.

Ding D, Yang J: Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials. Adv. Stud. Contemp. Math. 2010, 20(1):7-21.

11. 11.

Şen E, Acikgoz M, Araci S: q -Analogue of p -adic log Γ type functions associated with modified q -extension of Genocchi numbers withe weight α and β . Turk. J. Math. Anal. Number Theory 2013, 1: 9-12. 10.12691/tjant-1-1-3

12. 12.

Kim DS: Identities associated with generalized twisted Euler polynomials twisted by ramified roots of unity. Adv. Stud. Contemp. Math. 2012, 22(3):363-377.

13. 13.

Kim DS, Lee N, Na J, Pak HK: Identities of symmetry for higher-order Euler polynomials in three variables (I). Adv. Stud. Contemp. Math. 2012, 22(1):51-74.

14. 14.

Kim DS, Kim T, Kim YH, Lee SH: Some arithmetic properties of Bernoulli and Euler numbers. Adv. Stud. Contemp. Math. 2010, 22(4):467-480.

15. 15.

Kim T: Some identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials. Adv. Stud. Contemp. Math. 2010, 20(1):23-28.

16. 16.

Kim T: On the weighted q -Bernoulli numbers and polynomials. Adv. Stud. Contemp. Math. 2011, 21(2):207-215.

17. 17.

Kim T: Identities on the weighted q -Euler numbers and q -Bernstein polynomials. Adv. Stud. Contemp. Math. 2012, 22(1):7-12.

18. 18.

Kim T: An identity of the symmetry for the Frobenius-Euler polynomials associated with the fermionic p -adic invariant q -integrals on $Z p$ . Rocky Mt. J. Math. 2011, 41(1):239-247. 10.1216/RMJ-2011-41-1-239

19. 19.

Kim T: Some identities on the q -Euler polynomials of higher order and q -Stirling numbers by the fermionic p -adic integral on $Z p$ . Russ. J. Math. Phys. 2009, 16(4):484-491. 10.1134/S1061920809040037

20. 20.

Kim T: q -Generalized Euler numbers and polynomials. Russ. J. Math. Phys. 2006, 13(3):293-298. 10.1134/S1061920806030058

21. 21.

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

22. 22.

Kim T: The modified q -Euler numbers and polynomials. Adv. Stud. Contemp. Math. 2008, 16(2):161-170.

23. 23.

Rim SH, Jeong J: On the modified q -Euler numbers of higher order with weight. Adv. Stud. Contemp. Math. 2012, 22(1):93-98.

24. 24.

Rim SH, Jeong J: A note on the modified q -Euler numbers and polynomials with weight α . Int. Math. Forum 2011, 6(56):3245-3250.

25. 25.

Kim T: Symmetry of power sum polynomials and multivariated fermionic p -adic invariant integral on $Z p$ . Russ. J. Math. Phys. 2009, 16(1):93-96. 10.1134/S1061920809010063

26. 26.

Kim T: Identities involving Frobenius-Euler polynomials arising from non-linear differential equations. J. Number Theory 2012, 132(12):2854-2865. 10.1016/j.jnt.2012.05.033

27. 27.

Kim T, Choi JY, Sug JY: Extended q -Euler numbers and polynomials associated with fermionic p -adic q -integral on $Z p$ . Russ. J. Math. Phys. 2007, 14(2):160-163. 10.1134/S1061920807020045

28. 28.

Kim T, Kim DS, Bayad A, Rim SH: Identities on the Bernoulli and the Euler numbers and polynomials. Ars Comb. 2012, 107: 455-463.

29. 29.

Kim T, Kim DS, Dolgy DV, Rim SH: Some identities on the Euler numbers arising from Euler basis polynomials. Ars Comb. 2013, 109: 433-446.

## Acknowledgements

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

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### Corresponding author

Correspondence to Jin-Woo Park.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

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

• fermionic p-adic q-integral on $Z p$ 