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Analysis of Apostol-Type Numbers and Polynomials with Their Approximations and Asymptotic Behavior

In: Approximation Theory and Analytic Inequalities

Author

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  • Yilmaz Simsek

    (University of Akdeniz, Faculty of Science, Department of Mathematics)

Abstract

In this chapter, using the methods and techniques of approximation of some classical polynomials and numbers including the Apostol–Bernoulli numbers and polynomials, we survey and investigate various properties of the Boole type combinatorial numbers and polynomials. By applying the p-adic q-integrals including the bosonic and fermionic p-adic integrals on p-adic integers, we study on generating functions for the generalized Boole type combinatorial numbers and polynomials attached to the Dirichlet character. These numbers and polynomials are related to the generalized Apostol–Bernoulli numbers and polynomials, the generalized Apostol–Euler numbers and polynomials, generalized Apostol–Daehee numbers and polynomials, and also generalized Apostol–Changhee numbers and polynomials. With the help of these generating functions, PDEs and their functional equation, many formulas, identities and relations involving the generalized Apostol–Daehee and Apostol–Changhee numbers and polynomials, the Stirling numbers, the Bernoulli numbers of the second kind, the generalized Bernoulli numbers and the generalized Euler numbers, and the Frobenius–Euler polynomials are given. Finally, by using asymptotic estimates for the Apostol–Bernoulli polynomials, asymptotic estimates for Boole type combinatorial numbers and polynomials are given.

Suggested Citation

  • Yilmaz Simsek, 2021. "Analysis of Apostol-Type Numbers and Polynomials with Their Approximations and Asymptotic Behavior," Springer Books, in: Themistocles M. Rassias (ed.), Approximation Theory and Analytic Inequalities, pages 435-486, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-60622-0_23
    DOI: 10.1007/978-3-030-60622-0_23
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