Research Article | Open Access
Some Identities between the Extended -Bernstein Polynomials with Weight and -Bernoulli Polynomials with Weight (,)
Using bosonic -adic -integral on , we give some interesting relationships between -Bernoulli numbers with weight (,) and -Bernstein polynomials with weight . Also, using -Bernstein polynomials with two variables, we derive some interesting properties associated with -Bernoulli numbers with weight (,).
In many areas the Bernstein polynomials have many applications: in statistics, in computer applications, in approximations of functions, in numerical analysis, in -adic analysis, and in the solution of differential equations. In particular, recognition of human speech has long been a hot issue among artificial intelligence and signal processing researchers, and -Bernstein polynomials, which are mathematical operators, have been applied for pattern recognition, and a new method has thus been developed. For -Bernoulli numbers and polynomials, several results have been studied by Carlitz (see [1, 2]), Kim (see [3–7]), Ozden et al. (see ), and M.-S. Kim et al. (see ). Bernoulli numbers and polynomials possess many interesting properties and arise in many areas of mathematics, mathematical physics, and statistical physics. Recently, many mathematicians have studied in the area of Bernoulli numbers and polynomials. First, we introduce the -Bernoulli numbers and polynomials with weight as extended measure. And we investigate some identities due to the -Bernoulli numbers and polynomials with weight . Second, we consider the -adic analogue of the extended -Bernstein polynomials on and investigate some properties of several extended -Bernstein polynomials by using the bosonic -adic -integral. Finally, we investigate the relation of Bernstein polynomials and Bernoulli numbers and polynomials using -adic -integral on .
Throughout this paper, we use the following notations. By we denote the ring of -adic rational integers, denotes the field of -adic rational numbers, denotes the completion of algebraic closure of , denotes the set of natural numbers, denotes the ring of rational integers, denotes the field of rational numbers, denotes the set of complex numbers, and . Let be the normalized exponential valuation of with . When one talks of -extension, is considered in many ways such as an indeterminate, a complex number , or -adic number . If one normally assumes that . If , we normally assume that so that for . Throughout this paper we use the following notation: (see [1–22]). The -numbers satisfy many simple relations, all easily verified, as below:
For a fixed positive integer with , let where and (see ). We say that is a uniformly differential function at a point and denotes this property by if the difference quotients have a . For , let us begin with the expression representing a -analogue of the Riemann sum for . The integral of on is defined as the limit () of the sums (if exists). The -adic -integral (or -Volkenborn integrals of is defined by T. Kim (see ) as follows (see ). If we take in (6), then we easily see that From (6), we obtain where (cf. [1–19]).
Carlitz’s -Bernoulli numbers can be defined recursively by and by the value that with the usual convention of replacing .
Carliz’s -Bernoulli polynomials are also defined by
It is well known that where are called the th Carlitz’s -Bernoulli polynomials.
In this paper, we define the -Bernoulli numbers and polynomials with weight as below.
For and with , the -Bernoulli numbers with weight are defined by
For and with , the -Bernoulli polynomials with weight are also defined by
2. The Properties of the -Bernoulli Numbers and Polynomials with Weight
Our primary goal of this section is to find some properties of the -Bernoulli numbers with weight , respectively, , . This is a very important process to find the relationship between Bernoulli numbers and polynomials and Bernstein polynomials. In this section, by using the bosonic -adic -integral on , we obtain some properties.
For and with , -Bernoulli numbers with weight are represented as follows by simple calculus:
Note that for , where are the th Carliz’s -Bernoulli polynomials. In particular, , are the th Carliz’s -Bernoulli numbers.
For and with , since , we get the formula as below: with usual convention about replacing by . Also, by (6) and the bosonic -adic -integral on , we get the finite summation formula as below:
Theorem 1. For and with , one has and if , .
We also get from Theorem 1 that for ,
From (7) we get the following formula: and by comparison of coefficients we obtain as below:
From (8), we get the following formula: Specially, we consider when and , then we obtain the following Theorem.
Theorem 2. For and with , one has
By the bosonic -adic -integral on and (21), for
Theorem 3. For and with , one has
3. Properties of -Bernstein Polynomials
In this section, we introduce the -Bernstein polynomials and we get some properties to use. Let be the set of continuous functions on . Then, the classical Bernstein polynomials of degree for are defined by where is called the Bernstein operator and are called the Bernstein basis polynomials (or the Bernstein polynomials of degree ). In recent years, Gupta et al. have studied the generation function for -Bernstein polynomials (see ). Their generating function for is given by where . Observe that
Also, -Bernstein polynomials with weight , are studied by T. Kim (see ), and the formula is as below:
Let be continuous functions on . -Bernstein operator is defined as below:
From now on, we introduce some properties of -Bernstein polynomials with weight and -Bernstein operator. It is easy to show that these properties are true by some calculus are(1)(2), (3), (4)In the above property 4, we consider special cases , , as below.
If , then one has that .
From definition of with , and , for , we have that
It is possible to write as a linear combination of by using the degree evaluation formula and mathematical induction: By the same method, we get Continuing this process, we have the following:
Theorem 4. For , with and , one has
4. Relations of -Bernoulli Numbers and Polynomials with Weight and -Bernstein Polynomials
Therefore, we have the following theorem.
Theorem 5. For , with and , one has
Using , we get the other identities as below:
Specially, if , then If , then
Note that for , Also,
Hence, one has the theorem as below.
Theorem 6. For , with and , one has Also, from Theorem 6, we get the identity as below: Continuing this process, we obtain the following theorem.
Theorem 7. For , with and , one has where .
Hence, from Theorem 7, we get the corollary as below.
Corollary 8. For , with and , one has where .
5. Extension of -Bernstein Polynomials with Weight
In this section, we introduce the extended Kim’s -Bernstein polynomials of order and the extended Kim’s -Bernstein operator of order . Also, we define the extended Kim’s -Bernstein polynomials of order with weight and the extended Kim’s -Bernstein operator of order with weight and we investigate the properties of these. Finally, we investigate the relation of -Bernoulli numbers with weight and extension of -Bernstein polynomials with weight . We assume that with . Let be the set of continuous function on and . For and , the extended Kim’s -Bernstein polynomials of order are as below:
For , and , the extended Kim’s -Bernstein operator of order is as below:
For , , and , we define the extended Kim’s -Bernstein polynomials of order with weight and the extended Kim’s -Bernstein operator of order with weight as below:
Properties of and are same to and , respectively. So we will enumerate these properties without proof as below.
Theorem 9. For , with and , one has
We get from the -symmetric properties of the -Bernstein polynomials that for
Note that for
Note also that for
Theorem 10. For , with and , one has
Note that for and where .
Note also from the binomial theorem that and , where .
Theorem 11. For , with and , one has
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
- L. Carlitz, “q-Bernoulli numbers and polynomials,” Duke Mathematical Journal, vol. 15, no. 4, pp. 987–1000, 1948.
- L. Carlitz, “q-Bernoulli and Eulerian numbers,” Transactions of the American Mathematical Society, vol. 76, pp. 332–350, 1954.
- T. Kim, “An analogue of Bernoulli numbers and their applications,” Reports of the Faculty of Science and Engineering. Saga University. Mathematics, vol. 22, pp. 21–26, 1994.
- T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299, 2002.
- T. Kim, “q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients,” Russian Journal of Mathematical Physics, vol. 15, no. 1, pp. 51–57, 2008.
- T. Kim, “On the weighted q-Bernoulli numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 21, no. 2, pp. 207–215, 2011.
- T. Kim, “Some formulae for the q-Bernstein polynomials and q-deformed binomial distributions,” Journal of Computational Analysis and Applications, vol. 14, no. 5, pp. 917–933, 2012.
- H. Ozden, I. N. Cangul, and Y. Simsek, “Remarks on q-Bernoulli numbers associated with Daehee numbers,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 1, pp. 41–48, 2009.
- M.-S. Kim, T. Kim, B. Lee, and C. Ryoo, “Some Identities of Bernoulli numbers and polynomials associated with Bernstein polynomials,” Advances in Difference Equations, vol. 2010, Article ID 305018, 7 pages, 2010.
- A. Aral, V. Gupta, and R. P. Agrawal, Applications of q-Calculus in Operator Theory, Springer, New York, NY, USA, 2013.
- A. Bayad and T. Kim, “Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials,” Russian Journal of Mathematical Physics, vol. 18, no. 2, pp. 133–143, 2011.
- M. Cenkci and V. Kurt, “Congruences for generalized q-bernoulli polynomials,” Journal of Inequalities and Applications, vol. 2008, Article ID 270713, 19 pages, 2008.
- D. S. Kim, T. Kim, S. Lee, D. V. Dolgy, and S. Rim, “Some new identities on the Bernoulli and Euler numbers,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 856132, 10 pages, 2011.
- H. Y. Lee, N. S. Jung, and C. S. Ryoo, “Some identities of the twisted q-genocchi numbers and polynomials with weight α and q-bernstein polynomials with weight α,” Abstract and Applied Analysis, vol. 29, no. 5-6, Article ID 123483, pp. 1221–1228, 2011.
- H. Y. Lee, N. S. Jung, and C. S. Ryoo, “A numerical investigation of the roots of the second kind λ-Bernoulli polynomials,” Neural, Parallel and Scientific Computations, vol. 19, no. 3-4, pp. 295–306, 2011.
- B. A. Kupershmidt, “Reflection symmetries of q-bernoulli polynomials,” Journal of Nonlinear Mathematical Physics, vol. 12, no. 1, pp. 412–422, 2005.
- C. S. Ryoo, “Some Relations between q-Euler Numbers with Weight and q- Bernstein polynomials with Weight,” Applied Mathematical Science, vol. 6, no. 45, pp. 2227–2236, 2012.
- Y. Simsek, “Generating functions of the twisted Bernoulli Numbers and polynomials associated with their interpolation functions,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 251–278, 2008.
- V. Gupta, P. N. Agrawal, and D. K. Verma, “On discrete q-beta operators,” Annali dell'Universita di Ferrara, vol. 57, no. 1, pp. 39–66, 2011.
- Y. Simsek, V. Kurt, and D. Kim, “New approach to the complete sum of products of the twisted (h, q)-Bernoulli numbers and polynomials,” Journal of Nonlinear Mathematical Physics, vol. 14, no. 1, pp. 44–56, 2007.
- Y. Simsek, “Theorem on twisted L-function and twisted Bernoulli numbers,” Advanced Studies in Contemporary Mathematics, vol. 12, pp. 237–246, 2006.
- M. Acikgoz and Y. Simsek, “A new generating function of q-Bernstein-type polynomials and their interpolation function,” Abstract and Applied Analysis, vol. 2010, Article ID 769095, 12 pages, 2010.
Copyright © 2013 H. Y. Lee and C. S. Ryoo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.