Skip to main content

On the Fermionic -adic Integral Representation of Bernstein Polynomials Associated with Euler Numbers and Polynomials

Abstract

The purpose of this paper is to give some properties of several Bernstein type polynomials to represent the fermionic -adic integral on . From these properties, we derive some interesting identities on the Euler numbers and polynomials.

1. Introduction

Throughout this paper, let be an odd prime number. The symbol, , , and denote the ring of -adic integers, the field of -adic rational numbers, the complex number field and the completion of algebraic closure of , respectively.

Let be the set of natural numbers and . Let be the normalized exponential valuation of with . Note that .

When one talks of -extension, is variously considered as an indeterminate, a complex number , or -adic number . If , we normally assume , and if , we always assume .

We say that is uniformly differentiable function at a point and write , if the difference quotient has a limit as . For , the fermionic -adic -integral on is defined as

(1.1)

(see [1]). In the special case in (1.1), the integral

(1.2)

is called the fermionic -adic invariant integral on (see [2]). From (1.2), we note

(1.3)

where .

Moreover, for , let . Then we note that

(1.4)

It is well known that the Euler polynomials are defined by

(1.5)

(see [115]). In the special case, , and are called the th Euler numbers.

Let . Then, by (1.3), (1.4), and (1.5), we see that

(1.6)

Let denote the set of continuous functions on . For , Bernstein introduced the following well-known linear positive operator in the field of real numbers :

(1.7)

where (see [3, 4, 7, 10, 11, 14]). Here, is called the Bernstein operator of order for .

For , the Bernstein polynomial of degree is defined by

(1.8)

For example, , , , , , and for , .

In this paper, we study the properties of Bernstein polynomials in the -adic number field. For , we give some properties of several type Bernstein polynomials to represent the fermionic -adic invariant integral on . From those properties, we derive some interesting identities on the Euler polynomials.

2. Fermionic -adic Integral Representation of Bernstein Polynomials

By (1.5) and (1.6), we see that

(2.1)

We also have that

(2.2)

From (2.1) and (2.2), we note that . It is easy to show that

(2.3)

By (1.5), (1.6), (2.1), (2.2), and (2.3), we see that for ,

(2.4)

Therefore, we obtain the following theorem.

Theorem 2.1.

For , one has

(2.5)

Theorem 2.1 is important to derive our main result in this paper.

Taking the fermionic -adic integral on for one Bernstein polynomial in (1.8), we get

(2.6)

Therefore, we obtain the following proposition.

Proposition 2.2.

For , one is

(2.7)

It is known that . Thus, one has

(2.8)

By (2.8) and Theorem 2.1, we see that for ,

(2.9)

From (2.9), we obtain the following theorem.

Theorem 2.3.

For with , we have

(2.10)

By Proposition 2.2 and Theorem 2.3, we obtain the following corollary.

Corollary 2.4.

For with , we have

(2.11)

For with , fermionic -adic invariant integral for multiplication of two Bernstein polynomials on can be given by the following relation:

(2.12)

Therefore, we obtain the following theorem.

Theorem 2.5.

For with , one has

(2.13)

For , one has

(2.14)

Thus, we obtain the following proposition.

Proposition 2.6.

For , one has

(2.15)

By Theorem 2.5 and Proposition 2.6, we obtain the following corollary.

Corollary 2.7.

For with , one has

(2.16)

In the same manner, multiplication of three Bernstein polynomials can be given by the following relation:

(2.17)

where with .

For with , by the symmetry of Bernstein polynomals, we see that

(2.18)

Therefore, we obtain the following theorem.

Theorem 2.8.

For with , one has

(2.19)

By (2.17) and Theorem 2.8, we obtain the following corollary.

Corollary 2.9.

For with , one has

(2.20)

Using the above theorems and mathematical induction, we obtain the following theorem.

Theorem 2.10.

Let . For with , the multiplication of the sequence of Bernstein polynomials with different degrees under fermionic -adic invariant integral on can be given as

(2.21)

We also easily see that

(2.22)

By Theorem 2.10 and (2.22), we obtain the following corollary.

Corollary 2.11.

Let . For with , one has

(2.23)

Let with . By the definition of , we easily get

(2.24)

Therefore, we obtain the following theorem.

Theorem 2.12.

Let . For with , one has

(2.25)

By simple calculation, we easily get

(2.26)

where for . By Theorem 2.12 and (2.26), we obtain the following corollary.

Corollary 2.13.

Let . For with , one has

(2.27)

The fermionic -adic invariant integral of multiplication of Bernstein polynomials, the th degree Bernstein polynomials with and with multiplicity on , respectively, can be given by

(2.28)

where with .

Assume that . Then one has

(2.29)

Therefore, we obtain the following theorem.

Theorem 2.14.

Let .

(i)For with , one has

(2.30)

(ii)For , one has

(2.31)

By Theorem 2.14, we obtain the following corollary.

Corollary 2.15.

For with , one has

(2.32)

References

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

    MathSciNet  MATH  Google Scholar 

  2. T. Kim, Barnes-type multiple -zeta functions and -Euler polynomials, Journal of Physics A, vol. 43,no. 25, Article ID 255201, 11 pages, 2010.

    Google Scholar 

  3. Acikgoz M, Araci S: A study on the integral of the product of several type Bernstein polynomials. IST Transaction of Applied Mathematics-Modelling and Simulation. In press

  4. Bernstein S: Démonstration du théorème de Weierstrass, fondée sur le calcul des probabilities. Communications of the Kharkov Mathematical Society 1912, 13: 1–2.

    MATH  Google Scholar 

  5. Kim T, Choi J, Kim YH: On extended carlitz's type -Euler numbers and polynomials. Advanced Studies in Contemporary Mathematics 2010, 20(4):499–505.

    MathSciNet  MATH  Google Scholar 

  6. Govil NK, Gupta V: Convergence of -Meyer-König-Zeller-Durrmeyer operators. Advanced Studies in Contemporary Mathematics 2009, 19(1):97–108.

    MathSciNet  MATH  Google Scholar 

  7. Gupta V, Kim T, Choi J, Kim Y-H: Generating function for -Bernstein, -Meyer-König-Zeller and -beta basis. Automation Computers Applied Mathematics 2010, 19: 7–11.

    Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  9. Kim T: -Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russian Journal of Mathematical Physics 2008, 15(1):51–57.

    MathSciNet  Article  MATH  Google Scholar 

  10. Kim T, Choi J, Kim Y-H: Some identities on the -Bernstein polynomials, -Stirling numbers and -Bernoulli numbers. Advanced Studies in Contemporary Mathematics 2010, 20(3):335–341.

    MathSciNet  MATH  Google Scholar 

  11. Kim T, Jang L-C, Yi H: A note on the modified -bernstein polynomials. Discrete Dynamics in Nature and Society 2010, 2010:-12.

    Google Scholar 

  12. Kim T: Note on the Euler -zeta functions. Journal of Number Theory 2009, 129(7):1798–1804. 10.1016/j.jnt.2008.10.007

    MathSciNet  Article  MATH  Google Scholar 

  13. Kurt V: A further symmetric relation on the analogue of the Apostol-Bernoulli and the analogue of the Apostol-Genocchi polynomials. Applied Mathematical Sciences 2009, 3(53–56):2757–2764.

    MathSciNet  MATH  Google Scholar 

  14. Cangul IN, Kurt V, Ozden H, Simsek Y: On the higher-order --Genocchi numbers. Advanced Studies in Contemporary Mathematics 2009, 19(1):39–57.

    MathSciNet  MATH  Google Scholar 

  15. Jang L-C, Kim W-J, Simsek Y: A study on the -adic integral representation on associated with Bernstein and Bernoulli polynomials. Advances in Difference Equations 2010, 2010:-6.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to T Kim.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Kim, T., Choi, J., Kim, Y. et al. On the Fermionic -adic Integral Representation of Bernstein Polynomials Associated with Euler Numbers and Polynomials. J Inequal Appl 2010, 864247 (2010). https://doi.org/10.1155/2010/864247

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/2010/864247

Keywords

  • Continuous Function
  • Real Number
  • Simple Calculation
  • Differentiable Function
  • Positive Operator