Generating functions for B-Splines with knots in geometric or affine progression

Dişibüyük, Çetin; Budakçı, Gülter; Goldman, Ron; Oruç, Halil

doi:10.1007/s10092-013-0102-8

Cited by 3 publications

(3 citation statements)

References 6 publications

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“…In the literature, the uniform B-splines N 0,n (ω; p) are represented by many different notations. Some of these are presented as follows: N 0,n (ω), B n k (ω), B n (ω) (see [3,5,8,26]). By using (21), with the aid of the generating function method, the Goldman ( [5] Theorem 3) proved the following well-known Schoenberg's identity and the de Boor recurrence relation for the uniform B-splines, respectively:…”

Section: Lemmamentioning

confidence: 99%

Novel Formulas for B-Splines, Bernstein Basis Functions, and Special Numbers: Approach to Derivative and Functional Equations of Generating Functions

Simsek

2023

Mathematics

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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.

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Section: Lemmamentioning

confidence: 99%

Novel Formulas for B-Splines, Bernstein Basis Functions, and Special Numbers: Approach to Derivative and Functional Equations of Generating Functions

Simsek

2023

Mathematics

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“…Thus, (14), (15) and (17) imply that P i is of the form (12). To find the constants a i explicitly, we first use the condition (i) on (13) and we obtain f (x 0 ) = a 0 .…”

Section: Generalization Of Quantum Taylor's Formula and Quantum Binomialmentioning

confidence: 99%

“…On the other hand, the origins of the q-analysis started in the eighteenth century by Euler [13] and developed by Gauss, Ramanujan and Jacobi [23]. After Jacsons's pioneering articles [17][18][19][20][21][22], that systematically presented q-calculus, the number of studies in qand h-analysis have been grown rapidly in many areas of mathematics such as in number theory-combinatorics [4], in orthogonal polynomials [5,6], in ordinary and partial differential equations [1,28], in mathematical physics [2,26,27], in approximation theory [12] and also in physical disciplines such as quantum mechanics and relativity [30]. For the very detailed information about q-calculus, we refer to [24].…”

Section: Introductionmentioning

confidence: 99%

Generalized quantum exponential function and its applications

Silindir

Yantir

2019

Filomat

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This article aims to present (q, h)-analogue of exponential function which unifies, extends hand q-exponential functions in a convenient and efficient form. For this purpose, we introduce generalized quantum binomial which serves as an analogue of an ordinary polynomial. We state (q, h)-analogue of Taylor series and introduce generalized quantum exponential function which is determined by Taylor series in generalized quantum binomial. Furthermore, we prove existence and uniqueness theorem for a first order, linear, homogeneous IVP whose solution produces an infinite product form for generalized quantum exponential function. We conclude that both representations of generalized quantum exponential function are equivalent. We illustrate our results by ordinary and partial difference equations. Finally, we present a generic dynamic wave equation which admits generalized trigonometric, hyperbolic type of solutions and produces various kinds of partial differential/difference equations.

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A B-spline approach to q-Eulerian polynomials

Dişibüyük

Ulutaş

2020

Journal of Computational and Applied Mathematics

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Generating functions for B-Splines with knots in geometric or affine progression

Cited by 3 publications

References 6 publications

Novel Formulas for B-Splines, Bernstein Basis Functions, and Special Numbers: Approach to Derivative and Functional Equations of Generating Functions

Novel Formulas for B-Splines, Bernstein Basis Functions, and Special Numbers: Approach to Derivative and Functional Equations of Generating Functions

Generalized quantum exponential function and its applications

A B-spline approach to q-Eulerian polynomials

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