Atıf İçin Kopyala
Şimşek Y.
Mathematics, cilt.12, sa.1, ss.1-20, 2024 (SCI-Expanded)
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Yayın Türü:
Makale / Tam Makale
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Cilt numarası:
12
Sayı:
1
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Basım Tarihi:
2024
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Doi Numarası:
10.3390/math12010065
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Dergi Adı:
Mathematics
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Derginin Tarandığı İndeksler:
Science Citation Index Expanded (SCI-EXPANDED)
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Sayfa Sayıları:
ss.1-20
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Akdeniz Üniversitesi Adresli:
Evet
Özet
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.