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Applications on the Apostol-Daehee numbers and polynomials associated with special numbers, polynomials, and p-adic integrals
- Yilmaz Simsek^{1}Email authorView ORCID ID profile and
- Ahmet Yardimci^{1, 2}
https://doi.org/10.1186/s13662-016-1041-x
© The Author(s) 2016
Received: 29 September 2016
Accepted: 22 November 2016
Published: 29 November 2016
Abstract
In this paper, by using p-adic Volkenborn integral, and generating functions, we give some properties of the Bernstein basis functions, the Apostol-Daehee numbers and polynomials, Apostol-Bernoulli polynomials, some special numbers including the Stirling numbers, the Euler numbers, the Daehee numbers, and the Changhee numbers. By using an integral equation and functional equations of the generating functions and their partial differential equations (PDEs), we give a recurrence relation for the Apostol-Daehee polynomials. We also give some identities, relations, and integral representations for these numbers and polynomials. By using these relations, we compute these numbers and polynomials. We make further remarks and observations for special polynomials and numbers, which are used to study elementary word problems in engineering and in medicine.
Keywords
- Bernoulli numbers and polynomials
- Apostol-Bernoulli numbers and polynomials
- Daehee numbers and polynomials
- Apostol-Daehee numbers
- array polynomials
- Stirling numbers of the first kind and the second kind
- generating function
- functional equation
- derivative equation
- Bernstein basis functions
MSC
- 12D10
- 11B68
- 11S40
- 11S80
- 26C05
- 26C10
- 30B40
- 30C15
1 Introduction
The special numbers and polynomials have been used in various applications in such diverse areas as mathematics, probability and statistics, mathematical physics, and engineering. For example, due to the relative freedom of some basic operations including addition, subtraction, multiplication, polynomials can be seen almost ubiquitously in engineering. They are curves that represent properties or behavior of many engineering objects or devices. For example, polynomials are used in elementary word problems to complicated problems in the sciences, approximate or curve fit experimental data, calculate beam deflection under loading, represent some properties of gases, and perform computer aided geometric design in engineering. Polynomials are used as solutions of differential equations. Polynomials represent characteristics of linear dynamic system and we also know that a ratio of two polynomials represents a transfer function of a linear dynamic system. With the help of polynomials, one defines basis functions used in finite element computation and constructs parametric curves.
In order to give some results including identities, relations, and formulas for special numbers and polynomials, we use the p-adic Volkenborn integral and generating function methods. We need the following formulas, relations, generating functions, and notations for families of special numbers and polynomials. Throughout this paper, we use the following notations:
There are many methods and techniques for investigating and constructing generating functions for special polynomials and numbers. One of the most important techniques is the p-adic Volkenborn integral on \(\mathbb{Z}_{p}\). In [19], Kim constructed the p-adic q-Volkenborn integration. By using this integral, we derive some identities, and relations for the special polynomials. We now briefly give some definitions and properties of this integral.
The λ-Bernoulli numbers and polynomials have been studied in different sets. For instance on the set of complex numbers, we assume that \(\lambda\in\mathbb{C}\) and on set of p-adic numbers or p-adic integrals, we assume that \(\lambda\in\mathbb{Z}_{p}\).
Theorem 1
Proof of Theorem 1 was given by Schikhof [45].
Theorem 2
Proof of Theorem 2 was given by Kim et al. [13] and [16].
Remark 1
Many applications of the fermionic and bosonic p-adic integral on \(\mathbb{Z}_{p}\) have been given by T Kim and DS Kim first, Jang, Rim, Dolgy, Kwon, Seo, Lim and the others gave various novel identities, relations and formulas in some special numbers and polynomials (cf. [6–25, 35, 37, 38, 42, 45, 47], and the references cited therein).
A relation between the λ-Bernoulli polynomials \(\mathfrak{B}_{n}(x;\lambda)\), the Apostol-Daehee polynomials \(\mathfrak{D}_{n}(x;\lambda)\) and the Stirling numbers of the second kind is given by the following theorem.
Theorem 3
The proof of (1.20) was given by the first author in [43].
Observe that \(G ( \frac{e^{z}-1}{\lambda};\lambda ) \) is a generating function for the λ-Bernoulli numbers (cf. [22]). We also observe that \(G ( e^{z}-1;1 ) \) is a generating function for the Bernoulli numbers.
We summarize our results as follows.
In Section 2, we give some identities, relations, and formulas including the Apostol-Daehee numbers and polynomials of higher order, the Changhee numbers and polynomials and the Stirling numbers, the λ-Bernoulli polynomials, the λ-Apostol-Daehee polynomials and the Bernstein basis functions.
In Section 3, we give an integral representation for the Apostol-Daehee polynomials.
In Section 4, we introduce further remarks and observations on these numbers, polynomials, and their applications.
2 Identities
By using the above generating functions, we get some identities and relation. In [43] and [42], Simsek gave derivative formulas for the λ-Apostol-Daehee polynomials, \(\mathfrak{D}_{n}(x;\lambda)\). Here we give another derivative formula for these polynomials. We also give a relation between the λ-Bernoulli polynomials, the λ-Apostol-Daehee polynomials and the Bernstein polynomials.
Theorem 4
Proof
Comparing the coefficients of \(t^{n}\) on both sides of the above equation, we arrive at the desired result. □
Theorem 5
Proof
Theorem 6
Proof
Combining (1.20) with (2.2), we get the following theorem.
Theorem 7
Theorem 8
Proof
Theorem 9
3 Integral representation for the Apostol-Daehee polynomials
Theorem 10
Theorem 11
4 Further remark and observations on special polynomials
Polynomials appear in many branches of mathematics and science. For instance, polynomials are used to form polynomial equations, which encode a wide range of problems, from elementary world problems to complicated problems in the sciences, in settings ranging from basic chemistry and physics to economics and social science, in calculus and numerical analysis to approximate other functions (cf. [33, 48]). Therefore, many authors have studied and investigated special polynomials and special numbers. There are various applications of these polynomials and numbers in many branches of not only in mathematics and mathematical physics, but also in computer and in engineering science with real world problems including the combinatorial sums, combinatorial numbers such as the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers, the Changhee numbers and polynomials, the Daehee numbers and polynomials and the others. Especially, the Bernstein polynomials are also used in many branches of mathematics, particularly including statistics, probability, combinatorics, computer algorithm, discrete mathematics and CAGD. The Bernstein basis functions are applied in real world problems to construct the theory of the Bezier curves (cf. [33, 48], and the references therein).
Such special polynomials and numbers including the above-mentioned ones have found diverse applications in many research fields other than mathematics such as mathematical physics, computer science, and engineering. That is, in engineering, polynomials are used to model real phenomena. For instance, aerospace engineers use polynomials to model the projections of jet rockets. Scientists use polynomials in many formulas including gravity, temperature, and distance equations. In social science, economists need an understanding of polynomials to forecast future market patterns (cf. [33, 48], and the references therein). Polynomials are also used in analysis of ambulatory blood pressure measurements and also biostatistics problems (cf. [49]).
It is well known that there are many application of the p-adic integral on \(\mathbb{Z}_{p}\), one of the best known applications is to construct generating functions for special numbers and polynomials. The other applications are in p-adic analysis, in q-analysis, in quantum groups, in spectra of the q-deformed oscillator and in science (cf. [11, 45, 50]).
How can one give applications in investigating engineering and medicine related problems by using the Apostol-Daehee numbers and polynomials, and the Changhee numbers and polynomials with the p-integral on \(\mathbb{Z}_{p}\)?
Declarations
Acknowledgements
The paper was supported by the Scientific Research Project Administration of Akdeniz University. The authors would like to thank the reviewers for their comments, which improved the previous version of this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Acikgoz, M, Araci, S: On generating function of the Bernstein polynomials. AIP Conf. Proc. 1281, 1141-1143 (2010) View ArticleGoogle Scholar
- Charalambides, CA: Enumerative Combinatorics. Chapman & Hall/CRC, Boca Raton (2002) MATHGoogle Scholar
- Cigler, J: Fibonacci polynomials and central factorial numbers. Preprint Google Scholar
- Comtet, L: Advanced Combinatorics: The Art of Finite and Infinite Expansions. Reidel, Dordrecht (1974) (translated from the French by JW Nienhuys) View ArticleMATHGoogle Scholar
- El-Desouky, BS, Mustafa, A: New results on higher-order Daehee and Bernoulli numbers and polynomials. arXiv:1503.00104v1
- Do, Y, Lim, D: On \((h,q)\)-Daehee numbers and polynomials. Adv. Differ. Equ. 2015, 107 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Dolgy, DV, Kim, T, Kwon, HI, Seo, J-J: A note on degenerate Bell numbers and polynomials associated with p-adic integral on \(\mathbb{Z}_{p}\). Adv. Stud. Contemp. Math. 26(3), 457-466 (2016) MathSciNetGoogle Scholar
- Goldman, RN: Identities for the univariate and bivariate Bernstein basis functions. In: Paeth, A (ed.) Graphics Gems V, pp. 149-162. Academic Press, San Diego (1995) View ArticleGoogle Scholar
- Jang, LC, Kim, T: A new approach to q-Euler numbers and polynomials. J. Concr. Appl. Math. 6, 159-168 (2008) MathSciNetMATHGoogle Scholar
- Jang, LC, Pak, HK: Non-Archimedean integration associated with q-Bernoulli numbers. Proc. Jangjeon Math. Soc. 5(2), 125-129 (2002) MathSciNetMATHGoogle Scholar
- Khrennikov, A: p-Adic Valued Distributions in Mathematical Physics. Kluwer Academic, Dordrecht (1994) View ArticleMATHGoogle Scholar
- Kim, D-S, Kim, T, Seo, J-J, Komatsu, T: Barnes’ multiple Frobenius-Euler and poly-Bernoulli mixed-type polynomials. Adv. Differ. Equ. 2014, 92 (2014) MathSciNetView ArticleGoogle Scholar
- Kim, DS, Kim, T: Daehee numbers and polynomials. Appl. Math. Sci. (Ruse) 7(120), 5969-5976 (2013) MathSciNetView ArticleGoogle Scholar
- Kim, DS, Kim, T: Some new identities of Frobenius-Euler numbers and polynomials. J. Inequal. Appl. 2012, 307 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Kim, DS, Kim, T, Lee, S-H, Seo, J-J: A note on the lambda-Daehee polynomials. Int. J. Math. Anal. 7(62), 3069-3080 (2013) MathSciNetView ArticleGoogle Scholar
- Kim, DS, Kim, T, Seo, J-J: A note on Changhee numbers and polynomials. Adv. Stud. Theor. Phys. 7, 993-1003 (2013) View ArticleGoogle Scholar
- Kim, DS, Kim, T, Lee, SH, Seo, J-J: Higher-order Daehee numbers and polynomials. Int. J. Math. Anal. 8(6), 273-283 (2014) MathSciNetView ArticleGoogle Scholar
- Kim, DS, Kim, T, Kwon, HI: Identities of some special mixed-type polynomials. arXiv:1406.2124v1
- Kim, T: q-Volkenborn integration. Russ. J. Math. Phys. 19, 288-299 (2002) MathSciNetMATHGoogle Scholar
- Kim, T: q-Euler numbers and polynomials associated with p-adic q-integral and basic q-zeta function. Trends Math. Inf. Cent. Math. Sci. 9, 7-12 (2006) Google Scholar
- Kim, T: Some identities on the q-integral representation of the product of several q-Bernstein-type polynomials. Abstr. Appl. Anal. 2011, Article ID 634675 (2011) MathSciNetMATHGoogle Scholar
- Kim, T, Rim, S-H, Simsek, Y, Kim, D: On the analogs of Bernoulli and Euler numbers, related identities and zeta and L-functions. J. Korean Math. Soc. 45(2), 435-453 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Kim, T, Kim, DS, Kwon, HI, Dolgy, DV, Seo, J-J: Degenerate falling factorial polynomials. Adv. Stud. Contemp. Math. 26(3), 481-499 (2016) MathSciNetGoogle Scholar
- Lim, D: On the twisted modified q-Daehee numbers and polynomials. Adv. Stud. Theor. Phys. 9(4), 199-211 (2015) View ArticleGoogle Scholar
- Lim, D, Qi, F: On the Appell type λ-Changhee polynomials. J. Nonlinear Sci. Appl. 9, 1872-1876 (2016) MathSciNetMATHGoogle Scholar
- Lorentz, GG: Bernstein Polynomials. Chelsea, New York (1986) MATHGoogle Scholar
- Luo, QM, Srivastava, HM: Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind. Appl. Math. Comput. 217, 5702-5728 (2011) MathSciNetMATHGoogle Scholar
- Ozden, H, Simsek, Y: Modification and unification of the Apostol-type numbers and polynomials and their applications. Appl. Math. Comput. 235, 338-351 (2014) MathSciNetMATHGoogle Scholar
- Ozden, H, Cangul, IN, Simsek, Y: Remarks on q-Bernoulli numbers associated with Daehee numbers. Adv. Stud. Contemp. Math. 18(1), 41-48 (2009) MathSciNetMATHGoogle Scholar
- Ozden, H, Simsek, Y, Srivastava, HM: A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials. Comput. Math. Appl. 60, 2779-2787 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Park, J-W: On a q-analogue of \((h,q)\)-Daehee numbers and polynomials of higher order. J. Comput. Anal. Appl. 21(1), 769-777 (2016) MathSciNetMATHGoogle Scholar
- Park, J-W, Rim, S-H, Kwon, J: The hyper-geometric Daehee numbers and polynomials. Turk. J. Anal. Number Theory 1(1), 59-62 (2013) View ArticleGoogle Scholar
- Reinard, JC: Introduction to Communication Research. McGraw-Hill, New York (2001) Google Scholar
- Roman, S: The Umbral Calculus. Dover, New York (2005) MATHGoogle Scholar
- Simsek, Y: Special numbers on analytic functions. Appl. Math. (Irvine) 5, 1091-1098 (2014) View ArticleGoogle Scholar
- Simsek, Y: On q-deformed Stirling numbers. Int. J. Math. Comput. 15, 70-80 (2012) MathSciNetGoogle Scholar
- Simsek, Y: Complete sum of products of \((h,q)\)-extension of Euler polynomials and numbers. J. Differ. Equ. Appl. 16, 1331-1348 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Simsek, Y: Identities associated with generalized Stirling type numbers and Eulerian type polynomials. Math. Comput. Appl. 18, 251-263 (2013) MathSciNetGoogle Scholar
- Simsek, Y: Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their applications. Fixed Point Theory Appl. 2013, 87 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Simsek, Y: Functional equations from generating functions: a novel approach to deriving identities for the Bernstein basis functions. Fixed Point Theory Appl. 2013, 80 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Simsek, Y: Twisted \((h,q)\)-Bernoulli numbers and polynomials related to twisted \((h,q)\)-zeta function and L-function. J. Math. Anal. Appl. 324(2), 790-804 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Simsek, Y: Apostol type Daehee numbers and polynomials. Adv. Stud. Contemp. Math. 26(3), 555-566 (2016) Google Scholar
- Simsek, Y: Identities on the Changhee numbers and Apostol-Daehee polynomials. Adv. Stud. Contemp. Math. (2017, to appear) Google Scholar
- Simsek, Y, Acikgoz, M: A new generating function of (q-) Bernstein-type polynomials and their interpolation function. Abstr. Appl. Anal. 2010, Article ID 769095 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Schikhof, WH: Ultrametric Calculus: An Introduction to p-Adic Analysis. Cambridge Studies in Advanced Mathematics, vol. 4. Cambridge University Press, Cambridge (1984) MATHGoogle Scholar
- Srivastava, HM: Some generalizations and basic (or q-) extensions of the Bernoulli, Euler and Genocchi polynomials. Appl. Math. Inf. Sci. 5, 390-444 (2011) MathSciNetGoogle Scholar
- Srivastava, HM, Kim, T, Simsek, Y: q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series. Russ. J. Math. Phys. 12, 241-268 (2005) MathSciNetMATHGoogle Scholar
- https://en.wikipedia.org/wiki/Polynomial
- Zwinderman, AH, Cleophas, TA, Cleophas, TJ, van der Wall, EE: Polynomial analysis of ambulatory blood pressure measurements. Neth. Heart J. 9(2), 68-74 (2001) Google Scholar
- Vladimirov, VS, Volovich, IV, Zelenov, EI: p-Adic Analysis and Mathematical Physics. Series on Soviet and East European Mathematics, vol. 1. World Scientific, Singapore (1994) View ArticleMATHGoogle Scholar