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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. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to present various identities and computation formulas for certain classes of Apostol-type numbers and polynomials. The results of this paper contain not only the λ-Apostol-Daehee numbers and polynomials, but also Simsek numbers and polynomials, the Stirling numbers of the first kind, the Daehee numbers, and the Chu-Vandermonde identity. Furthermore, we derive an infinite series representation for the λ-Apostol-Daehee polynomials. By using functional equations containing the generating functions for the Cauchy numbers and the Riemann integrals of the generating functions for the λ-Apostol-Daehee numbers and polynomials, we also derive some identities and formulas for these numbers and polynomials. Moreover, we give implementation of a computation formula for the λ-Apostol-Daehee polynomials in Mathematica by Wolfram language. By this implementation, we also present some plots of these polynomials in order to investigate their behaviour in some randomly selected special cases of their parameters. Finally, we conclude the paper with some comments and observations on our results. [ABSTRACT FROM AUTHOR]

In [M. M. Malakiyeh, S. Shojaee and S. Hamzehei-Javaran, Development of a direct time integration method based on Bezier curve and 5th-order Berstein basis function, Comput. Struct. 194 (2108) 15–31] an unconditionally stable implicit time-integration method using the Bezier curve was proposed for solving structural dynamic problems. In this study, a new class of the previous algorithm is presented by using the Bernstein polynomials and the Bezier curve as the interpolation functions for solving the equations of motion with the possibility of using large time steps. The spectral radius, period elongation, amplitude decay and overshooting of the present method are investigated and compared with some other methods. To show the high-performance, robustness and validity of this method, five numerical examples are presented. The theoretical analysis and numerical examples show that the proposed method has low dissipation in the lower modes and high dissipation in the higher modes in comparison with the other methods reported in the literature. [ABSTRACT FROM AUTHOR]

By using the Bernstein basis functions and Bernstein operator with their integral, we derive some combinatorial sums and identities including the Bernoulli numbers and polynomials, the Stirling numbers of the second kind and also some relations. Moreover, we give some inequalities involving combinatorial numbers and binomial coefficients. [ABSTRACT FROM AUTHOR]

The main concern of this article is to acquire some approximation properties of a new class of Bernstein polynomials based on B'ezier basis functions with shape parameter λ ∈ [-1,1]. We prove Korovkin type approximation theorem and estimate the degree of convergence in terms of the modulus of continuity, for the functions belong to Lipschitz type class and Peetre's K -functional, respectively. Additionally, with the help of Maple software, we present the comparison of the convergence of newly defined operators to the certain functions with some graphical illustrations and error estimation tables. Also, we conclude that the error estimation of our newly defined operators in some cases is better than classical Bernstein operators [3], Cai et al. [4] and Izgi [10]. [ABSTRACT FROM AUTHOR]

In order to investigate the fundamental properties of q-Bernstein basis functions, we give generating functions for these basis functions and their functional and differential equations. In [16], [15] and [17], we construct a novel collection of generating functions to derive many known and some new identities for the classical Bernstein basis functions. The main purpose of this paper is to construct analogous generating functions for the q-Bernstein basis functions. By using an approach similar to that of our methods in [16] as well as some properties of interpolation functions, we can derive some known and some new identities, relations and formulas for the q-Bernstein basis functions, including the partition of unity property, formulas for representing the monomials, recurrence relations, formulas for derivatives, subdivision identities and integral representations. Furthermore, we give plots of not only our new basis functions, but also their generating functions. Also, we simulate q-Bezier type curves for some selected q values and control points. [ABSTRACT FROM AUTHOR]

In this paper, by using p-adic Volkenborn integral, and generating functions, we give some properties of the Bernstein basis functions, the Apostol-Daehee numbers and polynomials, Apostol-Bernoulli polynomials, some special numbers including the Stirling numbers, the Euler numbers, the Daehee numbers, and the Changhee numbers. By using an integral equation and functional equations of the generating functions and their partial differential equations (PDEs), we give a recurrence relation for the Apostol-Daehee polynomials. We also give some identities, relations, and integral representations for these numbers and polynomials. By using these relations, we compute these numbers and polynomials. We make further remarks and observations for special polynomials and numbers, which are used to study elementary word problems in engineering and in medicine. [ABSTRACT FROM AUTHOR]

We give some alternative forms of the generating functions for the Bernstein basis functions. Using these forms,we derive a collection of functional equations for the generating functions. By applying these equations, we prove some identities for the Bernstein basis functions. Integrating these identities, we derive a variety of identities and formulas, some old and some new, for combinatorial sums involving binomial coefficients, Pascal's rule, Vandermonde's type of convolution, the Bernoulli polynomials, and the Catalan numbers. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In applications of the k-nearest neighbour technique (kNN) with real-valued attributes of interest (Y) the predictions are biased for units with ancillary values of X with poor or no representation in a sample of n units. In this article a model-assisted calibration is proposed that reduces unit-level extrapolation bias. The bias is estimated as the difference in model-based predictions of Y given the X-values of the true k nearest units and the k selected reference units. Calibrated kNN predictions are then obtained by adding this difference to the original kNN prediction. The relationship is modelled between Y and X with decorrelated X-variables, variables scaled to the interval [0,1] and Bernstein basis functions to capture changes in Y as a function of changes in X. Three examples with actual forest inventory data from Italy, the USA and Finland demonstrated that calibrated kNN predictions were, on average, closer to their true values than non-calibrated predictions. Calibrated predictions had a range much closer to the actual range of Y than non-calibrated predictions. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to define and investigate a new class of symmetric beta type distributions with the help of the symmetric Bernstein-type basis functions. We give symmetry property of these distributions and the Bernstein-type basis functions. Using the Bernstein-type basis functions and binomial series, we give some series and integral representations including moment generating function for these distributions. Using generating functions and their functional equations, we also give many new identities related to the moments, the polygamma function, the digamma function, the harmonic numbers, the Stirling numbers, generalized harmonic numbers, the Lah numbers, the Bernstein-type basis functions, the array polynomials, and the Apostol–Bernoulli polynomials. Moreover, some numerical values of the expected values for the logarithm of random variable are given. [ABSTRACT FROM AUTHOR]

The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new identities, relations, derivative formulas, and combinatorial sums with the inclusion of binomials coefficients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson–Charlier polynomials, combinatorial numbers and polynomials, the Bersntein basis functions, and the probability distribution functions. Furthermore, by applying the p-adic integrals and Riemann integral, we obtain some combinatorial sums including the binomial coefficients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind). Finally, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution. [ABSTRACT FROM AUTHOR]