 Applications of Apostol-type Numbers and Polynomials: Approach to Techniques of Computation Algorithms in Approximation and Interpolation FunctionsYilmaz Simsek ()
A chapter in Approximation and Computation in Science and Engineering, 2022, pp 783-860 from Springer Abstract: Abstract The purpose of this chapter is to survey and make a compilation that covers many families of the special numbers and polynomials including the Apostol-Bernoulli numbers and polynomials, the Apostol-Euler numbers and polynomials, the Apostol-Genocchi numbers and polynomials, the Fubini numbers, the Stirling numbers, the Frobenius-Euler polynomials, and the others, blending new results for of the polynomials Wn(x;λ), which were given in: Y. Simsek, Computation methods for combinatorial sums and Euler-type numbers related to new families of numbers, Math. Meth. Appl. Sci., 40 (2017), 2347–2361. Many well-known results of these polynomials are given in this chapter. Using these known and new results, a large number of new formulas and new relations are created. Some well-known relations among the polynomials Wn(x;λ), the Bernoulli and Euler polynomials of higher order, Apostol-type polynomials (Apostol-Bernoulli polynomials, Apostol-Euler polynomials, Apostol-Genocchi polynomials, etc.) are given. It has been presented in new relations related to these polynomials. Some open problems are raised from the results for the polynomials Wn(x;λ). Behaviors of the polynomials Wn(x;λ) under integral transforms are also examined in this chapter. Firstly, Laplace transform of the polynomials Wn(x;λ) is given. With the help of this transformation, new infinite series representations are found. Then, the behavior of the polynomials Wn(x;λ) under the Melin transform is also given with help of the works Kucukoglu et al. (Quaest Math 42(4):465–478, 2019) and Simsek (AIP Conf Proc 1978:040012-1–040012-4, 2018). With the aid of this transformation, some relationships with the family of zeta functions are also blended in detail with the previously well-known results using values from negative integers. Since these results are known to be used frequently in both approximation theory, number theory, analysis of functions, and mathematical physics, these results can potentially be used in these scientific areas. In addition, it has been tried to give a detailed perspective on the applications of the polynomials Wn(x;λ) with their generating functions in approximation theory. Firstly, with the help of computational algorithms, basic known information about numerical values and graphics of these polynomials are introduced. With the help of the algorithm given for these numerical values, the approach steps are tried to be given in detail. These details are then illustrated on graphics and shapes, so that the visual approach steps are made clearer. As a result, this chapter is compiled by blending, interpreting, and comparing the fundamental properties of the polynomials Wn(x;λ) and the numbers Wn(λ) with their generating functions and other special numbers and polynomials. Date: 2022 There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-3-030-84122-5_40 Ordering information: This item can be ordered from DOI: 10.1007/978-3-030-84122-5_40 Access Statistics for this chapter More chapters in Springer Optimization and Its Applications from Springer |
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