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Öğe Multidimensional Bernstein polynomials and Bezier curves: Analysis of machine learning algorithm for facial expression recognition based on curvature(Elsevier Science Inc, 2019) Küçükoğlu, İrem; Şimşek, Buket; Şimşek, YilmazIn this paper, by using partial derivative formulas of generating functions for the multidimensional unification of the Bernstein basis functions and their functional equations, we derive derivative formulas and identities for these basis functions and their generating functions. We also give a conjecture and some open questions related to not only subdivision property of these basis functions, but also solutions of a higher-order special differential equations. Moreover, we provide an implementation for a real world problem of human facial expression recognition with the help of curvature of Bezier curves whose machine learning supported by statistical evaluations on feature vectors using in the aforementioned machine learning algorithm. (C) 2018 Elsevier Inc. All rights reserved.Öğe New Formulas and Numbers Arising from Analyzing Combinatorial Numbers and Polynomials(MTJPAM Turkey, 2021) Kucukoglu, Irem; Şimşek, YilmazIn this paper, we derive various identities involving the negative higher-order combinatorial numbers and polynomials and other kinds of special numbers and polynomials such as the Stirling numbers, the Lah numbers, the negative higher-order Changhee numbers and polynomials, and the positive higher-order Bernoulli numbers and polynomials. Furthermore, by using the integral formulas of not only the negative higher-order combinatorial numbers and polynomials but also their generating functions, we obtain some identities and combinatorial sums. We give some infinite series, involving the negative higher-order combinatorial numbers, with their values in terms of the falling factorials, the Catalan numbers, the Daehee numbers (linear combination of the Stirling numbers and the Bernoulli numbers) and the Changhee numbers (linear combination of the Stirling numbers and the Euler numbers). As application of these infinite series, we also set two new sequences of special numbers with their generating functions, and investigate their properties. We pose an open question related to one of these number sequences. By using an infinite series arising from the integral of the generating functions for the negative higher-order combinatorial numbers and polynomials, we also introduce a new family of polynomials associated with the Bernstein basis functions. In addition, we derive symmetry property, integral formulas and derivative formula for these newly introduced polynomials. Moreover, by implementing an explicit formula of these newly introduced polynomials in Mathematica with the aid of the Wolfram programming language, we present some plots of these newly introduced polynomial functions for some of their randomly selected special cases. We also give some further results including series representations, combinatorial sums, integral formulas and relations for some of combinatorial numbers and poynomials. Finally, we present some observations and comments on our results. © 2021, MTJPAM Turkey. All rights reserved.Öğe Some Certain Classes of Combinatorial Numbers and Polynomials Attached to Dirichlet Characters: Their Construction by p-Adic Integration and Applications to Probability Distribution Functions(Springer, 2021) Şimşek, Yilmaz; Kucukoglu, IremThe aim of this chapter is to survey on old and new identities for some certain classes of combinatorial numbers and polynomials derived from the non-trivial Dirichlet characters and p-adic integrals. This chapter is especially motivated by the recent papers (Simsek, Turk J Math 42:557–577, 2018; Srivastava et al., J Number Theory 181:117–146, 2017; Kucukoglu et al. Turk J Math 43:2337–2353, 2019; Axioms 8(4):112, 2019) in which the aforementioned combinatorial numbers and polynomials were extensively investigated and studied in order to obtain new results. In this chapter, after recalling the origin of the aforementioned combinatorial numbers and polynomials, which goes back to the paper (Simsek, Turk J Math 42:557–577, 2018), a compilation has been made on what has been done from the paper (Simsek, Turk J Math 42:557–577, 2018) up to present days about the main properties and relations of these combinatorial numbers and polynomials. Moreover, with the aid of some known and new formulas, relations, and identities, which involve some kinds of special numbers and polynomials such as the Apostol-type, the Peters-type, the Boole-type numbers and polynomials the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Genocchi numbers and polynomials, the Stirling numbers, the Cauchy numbers (or the Bernoulli numbers of the second kind), the binomial coefficients, the falling factorial, etc., we give further new formulas and identities regarding these combinatorial numbers and polynomials. Besides, some derivative and integral formulas, involving not only these combinatorial numbers and polynomials, but also their generating functions, are presented in addition to those given for their positive and negative higher-order extensions. By using Wolfram programming language in Mathematica, we present some plots for these combinatorial numbers and polynomials with their generating functions. Finally, in order to do mathematical analysis of the results in an interdisciplinary way, we present some observations on a few applications of the positive and negative higher-order extension of the generating functions for combinatorial numbers and polynomials to the probability theory for researchers to shed light on their future interdisciplinary studies. © 2021, Springer Nature Switzerland AG.Öğe Unified presentations of the generating functions for a comprehensive class of numbers and polynomials(MTJPAM Turkey, 2024) Kucukoglu, Irem; Şimşek, YilmazThe aim of this paper is to construct unified presentations of the ordinary and exponential generating functions for a comprehensive class of numbers and polynomials by p-adic analysis of a uniformly differentiable function. Moreover, we classify these generating functions according to which of the classes of “Appell sequences”, “Sheffer sequences” and accordingly other classes they belong to. In addition, we give some applications of the constructed unified presentations of the generating functions. In particular, we derive some identities and provide some differential equations satisfied by the constructed unified presentations of the generating functions. We also handle some special cases of these generating functions and provide several tables which summarize the special cases discussed throughout this paper. Over and above, by applying the p-adic integrals, we derive some further identities regarding the constructed unified presentations of the generating functions. Lastly, we raise some open questions and conclude our paper by providing final comments and observations on our results. © 2024, MTJPAM Turkey. All rights reserved.
