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Interpolation Functions of -Extensions of Apostol's Type Euler Polynomials

Abstract

The main purpose of this paper is to present new -extensions of Apostol's type Euler polynomials using the fermionic -adic integral on . We define the --Euler polynomials and obtain the interpolation functions and the Hurwitz type zeta functions of these polynomials. We define -extensions of Apostol type's Euler polynomials of higher order using the multivariate fermionic -adic integral on . We have the interpolation functions of these --Euler polynomials. We also give -extensions of Apostol's type Euler polynomials of higher order and have the multiple Hurwitz type zeta functions of these --Euler polynomials.

1. Introduction, Definitions, and Notations

After Carlitz [1] gave -extensions of the classical Bernoulli numbers and polynomials, the -extensions of Bernoulli and Euler numbers and polynomials have been studied by several authors. Many authors have studied on various kinds of -analogues of the Euler numbers and polynomials (cf., [139]).T Kim [723] has published remarkable research results for -extensions of the Euler numbers and polynomials and their interpolation functions. In [13], T Kim presented a systematic study of some families of multiple -Euler numbers and polynomials. By using the -Volkenborn integration on , he constructed the -adic -Euler numbers and polynomials of higher order and gave the generating function of these numbers and the Euler --function. In [20], Kim studied some families of multiple -Genocchi and -Euler numbers using the multivariate -adic -Volkenborn integral on , and gave interesting identities related to these numbers. Recently, Kim [21] studied some families of -Euler numbers and polynomials of Nölund's type using multivariate fermionic -adic integral on .

Many authors have studied the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials, and their -extensions (cf., [1, 6, 25, 27, 28, 3341]). Choi et al. [6] studied some -extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order , and multiple Hurwitz zeta function. In [24], Kim et al. defined Apostol's type -Euler numbers and polynomials using the fermionic -adic -integral and obtained the generating functions of these numbers and polynomials, respectively. They also had the distribution relation for Apostol's type -Euler polynomials and obtained -zeta function associated with Apostol's type -Euler numbers and Hurwitz type -zeta function associated with Apostol's type -Euler polynomials for negative integers.

In this paper, we will present new -extensions of Apostol's type Euler polynomials using the fermionic -adic integral on , and then we give interpolation functions and the Hurwitz type zeta functions of these polynomials. We also give -extensions of Apostol's type Euler polynomials of higher order using the multivariate fermionic -adic integral on .

Let be a fixed odd prime number. Throughout this paper , , and will, respectively, denote the ring of -adic rational integers, the field of -adic rational numbers, the complex number field, and the completion of algebraic closure of . Let be the set of natural numbers and . Let be the normalized exponential valuation of with When one talks of -extension, is variously considered as an indeterminate, a complex number or a -adic number . If one normally assumes If then one assumes

Now we recall some -notations. The -basic natural numbers are defined by and the -factorial by . The -binomial coefficients are defined by

(1.1)

Note that , which is the binomial coefficient. The -shift factorial is given by

(1.2)

Note that . It is well known that the -binomial formulae are defined as

(1.3)

Since , it follows that

(1.4)

Hence it follows that

(1.5)

which converges to as

For a fixed odd positive integer with , let

(1.6)

where lies in . The distribution is defined by

(1.7)

Let be the set of uniformly differentiable functions on . For , the -adic invariant -integral is defined as

(1.8)

The fermionic -adic invariant -integral on is defined as

(1.9)

where . The fermionic -adic integral on is defined as

(1.10)

It follows that where . For , let . we have

(1.11)

For details, see [721].

The classical Euler numbers and the classical Euler polynomials are defined, respectively, as follows:

(1.12)

It is known that the classical Euler numbers and polynomials are interpolated by the Euler zeta function and Hurwitz type zeta function, respectively, as follows:

(1.13)

In Section 2, we define new -extensions of Apostol's type Euler polynomials using the fermionic -adic integral on which will be called the --Euler polynomials . Then we obtain the interpolation functions and the Hurwitz type zeta functions of these polynomials. In Section 3, we define -extensions of Apostol's type Euler polynomials of higher order using the multivariate fermionic -adic integral on . We have the interpolation functions of these higher-order --Euler polynomials. In Section 4, we also give -extensions of Apostol's type Euler polynomials of higher order and have the multiple Euler zeta functions of these --Euler polynomials.

2. -Extensions of Apostol's Type Euler Polynomials

First, we assume that with . In , the -Euler polynomials are defined by

(2.1)

and are called the -Euler numbers. Then it follows that

(2.2)

The generating functions of are defined as

(2.3)

By (2.3), the interpolation functions of the -Euler polynomials are obtained as follows:

(2.4)

Thus, we have the following theorem.

Theorem 2.1.

Assume with . Then one has

(2.5)

Differentiating at shows that

(2.6)

In , we assume that with . The -Euler polynomials are defined by

(2.7)

By (2.7), we have

(2.8)

For , the Hurwitz type zeta functions for the -Euler polynomials are given as

(2.9)

For , we have from (2.9) that

(2.10)

Now we give new -extensions of Apostol's type Euler polynomials. For , let be the cyclic group of order . Let be the p-adic locally constant space defined by

(2.11)

First, we assume that with . For , we define -Euler polynomials of Apostol's type using the fermionic -adic integral as follows:

(2.12)

and we will call them the --Euler polynomials. Then are defined as the --Euler numbers. From (2.12), we have

(2.13)

Let . From (2.12), we easily derive

(2.14)

On the other hand, we have

(2.15)

From (2.14) and (2.15), we obtain the following theorem.

Theorem 2.2.

Assume that with . For , let Then one has

(2.16)

In , we assume that with . Let with . We define the --Euler polynomials to be satisfied the following equation:

(2.17)

When we differentiate both sides of (2.17) at , we have

(2.18)

Hence we have the interpolation functions of the --Euler polynomials as follows:

(2.19)

For , we define the Hurwitz type zeta function of the --Euler polynomials as

(2.20)

where For , we have

(2.21)

3. -Extensions of Apostol's Type Euler Polynomials of Higher Order

In this section, we give the -extension of Apostol's type Euler polynomials of higher order using the multivariate fermionic -adic integral.

First, we assume that with . Let . We define the --Euler polynomials of order as follows:

(3.1)

Note that are called the --Euler number of order . Using the multivariate fermionic -adic integral, we obtain from (3.1) that

(3.2)

Let be the generating functions of defined by

(3.3)

By (2.12) and (3.3), we have

(3.4)

Thus we have the following theorem.

Theorem 3.1.

Assume that with . For and , let . Then one has

(3.5)

In , we assume that with and with for . We define the --Euler polynomial of order as follows:

(3.6)

From (3.6), we have

(3.7)

For , we define the multiple Hurwitz type zeta functions for --Euler polynomials as

(3.8)

where In the special case with , we have

(3.9)

4. -Extension of Apostol's Type Euler Polynomials of Higher Order

In this section, we give the -extension of --Euler polynomials of higher order using the multivariate fermionic -adic integral.

Assume that with . For , we define --Euler polynomials of order as follows:

(4.1)

Note that are called the --Euler numbers.

When , the --Euler polynomials are

(4.2)

where is the Gaussian binomial coefficient. From (4.2), we obtain the following theorem.

Theorem 4.1.

Assume that with . For and , let . Then one has

(4.3)

In , assume that with and with . Then we can define --Euler polynomials for as follows:

(4.4)

Differentiating both sides of (4.4) at , we have

(4.5)

From (4.5), we have

(4.6)

Then we have

(4.7)

For , we define the Hurwitz type zeta function of --Euler polynomials of order as

(4.8)

where

From (4.4) and (4.8), we easily see that

(4.9)

References

  1. 1.

    Govil NK, Gupta V: Convergence of -Meyer-König-Zeller-Durrmeyer operators Advanced Studies in Contemporary Mathematics 2009, 19: 97–108.

    MATH  MathSciNet  Google Scholar 

  2. 2.

    Cangul IN, Ozden H, Simsek Y: extension of twisted Euler polynomials and numbers Acta Mathematica Hungarica 2008,120(3):281–299. 10.1007/s10474-008-7139-1

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Carlitz L: -Bernoulli and Eulerian numbers Transactions of the American Mathematical Society 1954, 76: 332–350.

    MATH  MathSciNet  Google Scholar 

  4. 4.

    Cenkci M: The -adic generalized twisted -Euler--function and its applications. Advanced Studies in Contemporary Mathematics 2007,15(1):37–47.

    MATH  MathSciNet  Google Scholar 

  5. 5.

    Cenkci M, Can M: Some results on -analogue of the Lerch zeta function. Advanced Studies in Contemporary Mathematics 2006,12(2):213–223.

    MATH  MathSciNet  Google Scholar 

  6. 6.

    Choi J, Anderson PJ, Srivastava HM: Some -extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order , and the multiple Hurwitz zeta function. Applied Mathematics and Computation 2008,199(2):723–737. 10.1016/j.amc.2007.10.033

    MATH  MathSciNet  Article  Google Scholar 

  7. 7.

    Kim T: -Volkenborn integration. Russian Journal of Mathematical Physics 2002,9(3):288–299.

    MATH  MathSciNet  Google Scholar 

  8. 8.

    Kim T: On Euler-Barnes multiple zeta functions. Russian Journal of Mathematical Physics 2003,10(3):261–267.

    MATH  MathSciNet  Google Scholar 

  9. 9.

    Kim T: Analytic continuation of multiple -zeta functions and their values at negative integers. Russian Journal of Mathematical Physics 2004,11(1):71–76.

    MATH  MathSciNet  Google Scholar 

  10. 10.

    Kim T: -Riemann zeta function. International Journal of Mathematics and Mathematical Sciences 2004,2004(9–12):599–605.

    MATH  Article  Google Scholar 

  11. 11.

    Kim T: -generalized Euler numbers and polynomials. Russian Journal of Mathematical Physics 2006,13(3):293–298. 10.1134/S1061920806030058

    MATH  MathSciNet  Article  Google Scholar 

  12. 12.

    Kim T: Multiple -adic -function. Russian Journal of Mathematical Physics 2006,13(2):151–157. 10.1134/S1061920806020038

    MATH  MathSciNet  Article  Google Scholar 

  13. 13.

    Kim T: -Euler numbers and polynomials associated with -adic -integrals. Journal of Nonlinear Mathematical Physics 2007,14(1):15–27. 10.2991/jnmp.2007.14.1.3

    MATH  MathSciNet  Article  Google Scholar 

  14. 14.

    Kim T: On the -extension of Euler and Genocchi numbers. Journal of Mathematical Analysis and Applications 2007,326(2):1458–1465. 10.1016/j.jmaa.2006.03.037

    MATH  MathSciNet  Article  Google Scholar 

  15. 15.

    Kim T: A note on -adic -integral on associated with -Euler numbers. Advanced Studies in Contemporary Mathematics 2007,15(2):133–137.

    MATH  MathSciNet  Google Scholar 

  16. 16.

    Kim T: On the analogs of Euler numbers and polynomials associated with -adic -integral on at . Journal of Mathematical Analysis and Applications 2007,331(2):779–792. 10.1016/j.jmaa.2006.09.027

    MATH  MathSciNet  Article  Google Scholar 

  17. 17.

    Kim T: On the -extension of Euler and Genocchi numbers. Journal of Mathematical Analysis and Applications 2007,326(2):1458–1465. 10.1016/j.jmaa.2006.03.037

    MATH  MathSciNet  Article  Google Scholar 

  18. 18.

    Kim T: -extension of the Euler formula and trigonometric functions. Russian Journal of Mathematical Physics 2007,14(3):275–278. 10.1134/S1061920807030041

    MATH  MathSciNet  Article  Google Scholar 

  19. 19.

    Kim T: The modified -Euler numbers and polynomials. Advanced Studies in Contemporary Mathematics 2008,16(2):161–170.

    MATH  MathSciNet  Google Scholar 

  20. 20.

    Kim T: On the multiple -Genocchi and Euler numbers. Russian Journal of Mathematical Physics 2008,15(4):481–486. 10.1134/S1061920808040055

    MATH  MathSciNet  Article  Google Scholar 

  21. 21.

    Kim T: Some identities on the -Euler polynomials of higher order and -Stirling numbers by the fermionic -adic integral on . to appear in Russian Journal of Mathematical Physics to appear in Russian Journal of Mathematical Physics

  22. 22.

    Kim T, Choi JY, Sug JY: Extended -Euler numbers and polynomials associated with fermionic -adic -integral on . Russian Journal of Mathematical Physics 2007,14(2):160–163. 10.1134/S1061920807020045

    MATH  MathSciNet  Article  Google Scholar 

  23. 23.

    Kim T, Kim M-S, Jang L, Rim S-H: New -Euler numbers and polynomials associated with -adic -integrals. Advanced Studies in Contemporary Mathematics 2007,15(2):243–252.

    MATH  MathSciNet  Google Scholar 

  24. 24.

    Kim Y-H: On the -adic interpolation functions of the generalized twisted -Euler numbers. International Journal of Mathematics and Analysis 2009,3(18):897–904.

    MATH  MathSciNet  Google Scholar 

  25. 25.

    Kim Y-H, Kim W, Jang L-C: On the -extension of Apostol-Euler numbers and polynomials. Abstract and Applied Analysis 2008, 2008:-10.

    Google Scholar 

  26. 26.

    Kim Y-H, Hwang K-W, Kim T: Interpolation functions of the -Genocchi and the -Euler polynomials of higher order. to appear in Journal of Computational Analysis and Applications to appear in Journal of Computational Analysis and Applications

  27. 27.

    Luo Q-M: Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions. Taiwanese Journal of Mathematics 2006,10(4):917–925.

    MATH  MathSciNet  Google Scholar 

  28. 28.

    Luo Q-M, Srivastava HM: Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials. Journal of Mathematical Analysis and Applications 2005,308(1):290–302. 10.1016/j.jmaa.2005.01.020

    MATH  MathSciNet  Article  Google Scholar 

  29. 29.

    Ozden H, Cangul IN, Simsek Y: Multivariate interpolation functions of higher-order -Euler numbers and their applications. Abstract and Applied Analysis 2008, 2008:-16.

    Google Scholar 

  30. 30.

    Ozden H, Cangul IN, Simsek Y: Remarks on sum of products of -twisted Euler polynomials and numbers. Journal of Inequalities and Applications 2008, 2008:-8.

    Google Scholar 

  31. 31.

    Ozden H, Simsek Y: A new extension of -Euler numbers and polynomials related to their interpolation functions. Applied Mathematics Letters 2008,21(9):934–939. 10.1016/j.aml.2007.10.005

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Srivastava HM, Kim T, Simsek Y: -Bernoulli numbers and polynomials associated with multiple -zeta functions and basic -series. Russian Journal of Mathematical Physics 2005,12(2):241–268.

    MATH  MathSciNet  Google Scholar 

  33. 33.

    Cangul IN, Kurt V, Simsek Y, Pak HK, Rim S-H: An invariant -adic -integral associated with -Euler numbers and polynomials. Journal of Nonlinear Mathematical Physics 2007,14(1):8–14. 10.2991/jnmp.2007.14.1.2

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Cangul IN, Ozden H, Simsek Y: A new approach to -Genocchi numbers and their interpolation functions. Nonlinear Analysis: Theory, Methods & Applications. In press Nonlinear Analysis: Theory, Methods & Applications. In press

  35. 35.

    Simsek Y: Twisted -Bernoulli numbers and polynomials related to twisted -zeta function and -function. Journal of Mathematical Analysis and Applications 2006,324(2):790–804. 10.1016/j.jmaa.2005.12.057

    MATH  MathSciNet  Article  Google Scholar 

  36. 36.

    Simsek Y: Complete sums of products of -extension of Euler numbers and polynomials. preprint, 2007, http://arxiv1.library.cornell.edu/abs/0707.2849v1 preprint, 2007,

  37. 37.

    Simsek Y: -Hardy-Berndt type sums associated with -Genocchi type zeta and --functions. Nonlinear Analysis: Theory, Methods & Applications. In press Nonlinear Analysis: Theory, Methods & Applications. In press

  38. 38.

    Ozden H, Simsek Y: Interpolation function of the -extension of twisted Euler numbers. Computers & Mathematics with Applications 2008,56(4):898–908. 10.1016/j.camwa.2008.01.020

    MathSciNet  Article  MATH  Google Scholar 

  39. 39.

    Ozden H, Cangul IN, Simsek Y: On the behavior of two variable twisted -adic Euler --functions. Nonlinear Analysis: Theory, Methods & Applications. In press Nonlinear Analysis: Theory, Methods & Applications. In press

  40. 40.

    Apostol TM: On the Lerch zeta function. Pacific Journal of Mathematics 1951, 1: 161–167.

    MATH  MathSciNet  Article  Google Scholar 

  41. 41.

    Wang W, Jia C, Wang T: Some results on the Apostol-Bernoulli and Apostol-Euler polynomials. Computers & Mathematics with Applications 2008,55(6):1322–1332. 10.1016/j.camwa.2007.06.021

    MATH  MathSciNet  Article  Google Scholar 

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Acknowledgment

The present research has been conducted by the research grant of the Kwangwoon University in 2009.

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Correspondence to Young-Hee Kim or Taekyun Kim.

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Hwang, K., Kim, Y. & Kim, T. Interpolation Functions of -Extensions of Apostol's Type Euler Polynomials. J Inequal Appl 2009, 451217 (2009). https://doi.org/10.1155/2009/451217

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Keywords

  • Positive Integer
  • Zeta Function
  • Cyclic Group
  • Differentiable Function
  • Interpolation Function