Şimşek Y.
Mathematics, vol.12, no.1, pp.1-20, 2024 (SCI-Expanded)
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Publication Type:
Article / Article
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Volume:
12
Issue:
1
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Publication Date:
2024
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Doi Number:
10.3390/math12010065
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Journal Name:
Mathematics
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Journal Indexes:
Science Citation Index Expanded (SCI-EXPANDED)
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Page Numbers:
pp.1-20
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Akdeniz University Affiliated:
Yes
Abstract
The purpose of this article is to give relations among the uniform B-splines, the Bernstein basis functions, and certain families of special numbers and polynomials with the aid of the generating functions method. We derive a relation between generating functions for the uniform B-splines and generating functions for the Bernstein basis functions. We derive some functional equations for these generating functions. Using the higher-order partial derivative equations of these generating functions, we derive both the generalized de Boor recursion relation and the higher-order derivative formula of uniform B-splines in terms of Bernstein basis functions. Using the functional equations of these generating functions, we derive the relations among the Bernstein basis functions, the uniform B-splines, the Apostol-Bernoulli numbers and polynomials, the Aposto–Euler numbers and polynomials, the Eulerian numbers and polynomials, and the Stirling numbers. Applying the p-adic integrals to these polynomials, we derive many novel formulas. Furthermore, by applying the Laplace transformation to these generating functions, we derive infinite series representations for the uniform B-splines and the Bernstein basis functions.