Yazar "Küçükoğlu, İrem" seçeneğine göre listele

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    An approach to negative hypergeometric distribution by generating function for special numbers and polynomials
    (2019) Küçükoğlu, İrem; Şimşek, Burçin; Şimşek, Yılmaz
    The aim of this paper is to not only provide a definition of a new family of special numbers and polynomials of higher-order with their generating functions, but also to investigate their fundamental properties in the spirit of probabilistic distributions. By applying generating functions methods, we derive miscellaneous novel identities and formulas involving the Chu–Vandermonde-type convolution formulas, combinatorial sums, Bernstein basis functions, and the other well-known special numbers and polynomials. Moreover, we provide a computational algorithm which returns special values of these numbers and polynomials. In addition, we show that our new identities and formulas are connected with the interpolation functions of the Apostol-type numbers and polynomials. Finally, we present some theoretical and applied details on probabilistic distributions arising from the aforementioned Chu–Vandermonde-type convolution formulas.
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    Analysis of higher-order peters-type combinatorial numbers and polynomials by their generating functions and p-adic integration
    (American Institute of Physics Inc., 2020) Küçükoğlu, İrem
    The aim of this paper is to analyze higher-order Peters-type combinatorial numbers and polynomials by means of their generating functions and p-adic integration. By using generating functions we first obtain a combinatorial identity containing not only these numbers and polynomials, but also the Stirling numbers of the first kind, the falling factorial and binomial coefficients. Secondly, by implementation of p-adic integration into the combinatorial sum representation of higher-order Peters-type combinatorial polynomials which includes falling factorial function, we provide both bosonic and fermionic p-adic integral representations of these numbers and polynomials. © 2020 American Institute of Physics Inc.. All rights reserved.
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    Computation of k-ary Lyndon words using generating functions and their differential equations
    (Univ Nis, Fac Sci Math, 2018) Küçükoğlu, İrem; Şimşek, Yılmaz
    By using generating functions technique, we investigate some properties of the k-ary Lyndon words. We give an explicit formula for the generating functions including not only combinatorial sums, but also hypergeometric function. We also derive higher-order differential equations and some formulas related to the k-ary Lyndon words. By applying these equations and formulas, we also derive some novel identities including the Stirling numbers of the second kind, the Apostol-Bernoulli numbers and combinatorial sums. Moreover, in order to compute numerical values of the higher-order derivative for the generating functions enumerating k-ary Lyndon words with prime number length, we construct an efficient algorithm. By applying this algorithm, we give some numerical values for these derivative equations for selected different prime numbers.
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    Derivative formulas related to unification of generating functions for sheffer type sequences
    (Amer Inst Physics, 2019) Küçükoğlu, İrem
    The main aim of this paper is to present partial derivative formulas for an unification, which was introduced by the author in "Unification of the generating functions for Sheffer type sequences and their applications, preprint", of Sheffer type sequences including the Peters polynomials, the Boole polynomials, the Changhee polynomials, the Simsek polynomials and the Korobov polynomials of the first kind. By making use of these derivative formulas, we provide a recurrence relation and a derivative formula for this unification. Furthermore, by using recurrence relation for this unification, we present miscellaneous special cases of this unification. Finally, we give some derivative formulas related to the well-known Sheffer type sequences such us the Peters polynomials and the Simsek polynomials.
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    Generating functions for new families of combinatorial numbers and polynomials: Approach to poisson-charlier polynomials and probability distribution function
    (Mdpi, 2019) Küçükoğlu, İrem; Şimşek, Burçin; Şimşek, Yılmaz
    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.
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    Identities and derivative formulas for the combinatorial and apostol-euler type numbers by their generating functions
    (Univ Nis, Fac Sci Math, 2018) Küçükoğlu, İrem; Şimşek, Yılmaz
    The first aim of this paper is to give identities and relations for a new family of the combinatorial numbers and the Apostol-Euler type numbers of the second kind, the Stirling numbers, the Apostol-Bernoulli type numbers, the Bell numbers and the numbers of the Lyndon words by using some techniques including generating functions, functional equations and inversion formulas. The second aim is to derive some derivative formulas and combinatorial sums by applying derivative operators including the Caputo fractional derivative operators. Moreover, we give a recurrence relation for the Apostol-Euler type numbers of the second kind. By using this recurrence relation, we construct a computation algorithm for these numbers. In addition, we derive some novel formulas including the Stirling numbers and other special numbers. Finally, we also some remarks, comments and observations related to our results.
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    Identities for dirichlet and lambert-type series arising from the numbers of a certain special word
    (Univ Belgrade, Fac Electrical Engineering, 2019) Küçükoğlu, İrem; Şimşek, Yılmaz
    The goal of this paper is to give several new Dirichlet-type series associated with the Riemann zeta function, the polylogarithm function, and also the numbers of necklaces and Lyndon words. By applying Dirichlet convolution formula to number-theoretic functions related to these series, various novel identities and relations are derived. Moreover, some new formulas related to Bernoulli-type numbers and polynomials obtain from generating functions and these Dirichlet-type series. Finally, several relations among the Fourier expansion of Eisenstein series, the Lambert series and the number-theoretic functions are given.
  • Küçük Resim
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    Matrix representations for a certain class of combinatorial numbers associated with bernstein basis functions and cyclic derangements and their probabilistic and asymptotic
    (2021) Küçükoğlu, İrem; Şimşek, Yılmaz
    In this paper, we mainly concerned with an alternate form of the generating functions for a certain class of combinatorial numbers and polynomials. We give matrix representations for these numbers and polynomials with their applications. We also derive various identities such as Rodrigues-type formula, recurrence relation and derivative formula for the aforementioned combinatorial numbers. Besides, we present some plots of the generating functions for these numbers. Furthermore, we give relationships of these combinatorial numbers and polynomials with not only Bernstein basis functions, but the two-variable Hermite polynomials and the number of cyclic derangements. We also present some applications of these relationships. By applying Laplace transform and Mellin transform respectively to the aforementioned functions, we give not only an infinite series representation, but also an interpolation function of these combinatorial numbers. We also provide a contour integral representation of these combinatorial numbers. In addition, we construct exponential generating functions for a new family of numbers arising from the linear combination of the numbers of cyclic derangements in the wreath product of the finite cyclic group and the symmetric group of permutations of a set. Finally, we analyse the aforementioned functions in probabilistic and asymptotic manners, and we give some of their relationships with not only the Laplace distribution, but also the standard normal distribution. Then, we provide an asymptotic power series representation of the aforementioned exponential generating functions.
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    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, Yilmaz
    In 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.
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    New classes of Catalan-type numbers and polynomials with their applications related to p-adic integrals and computational algorithms
    (Scientific Technical Research Council Turkey-Tubitak, 2020) Küçükoğlu, İrem; Şimşek, Burçin; Şimşek, Yılmaz
    The aim of this paper is to construct generating functions for new classes of Catalan-type numbers and polynomials. Using these functions and their functional equations, we give various new identities and relations involving these numbers and polynomials, the Bernoulli numbers and polynomials, the Stirling numbers of the second kind, the Catalan numbers and other classes of special numbers, polynomials and functions. Some infinite series representations, including the Catalan-type numbers and combinatorial numbers, are investigated. Moreover, some recurrence relations and computational algorithms for these numbers and polynomials are provided. By implementing these algorithms in the Python programming language, we illustrate the Catalan-type numbers and polynomials with their plots under the special conditions. We also give some derivative formulas for these polynomials. Applying the Riemann integral, contour integral, Volkenborn (bosonic p-adic) integral and fermionic p-adic integral to these polynomials, we also derive some integral formulas. With the help of these integral formulas, we give some identities and relations associated with some classes of special numbers and also the Cauchy-type numbers.
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    Numerical evaluation of special power series including the numbers of Lyndon words: an approach to interpolation functions for apostol-type numbers and polynomials
    (Kent State University, 2018) Küçükoğlu, İrem; Şimşek, Yılmaz
    Because the Lyndon words and their numbers have practical applications in many different disciplines such as mathematics, probability, statistics, computer programming, algorithms, etc., it is known that not only mathematicians but also statisticians, computer programmers, and other scientists have studied them using different methods. Contrary to other studies, in this paper we use methods associated with zeta-type functions, which interpolate the family of Apostol-type numbers and polynomials of order k. Therefore, the main purpose of this paper is not only to give a special power series including the numbers of Lyndon words and binomial coefficients but also to construct new computational algorithms in order to simulate these series by numerical evaluations and plots. By using these algorithms, we provide novel computational methods to the area of combinatorics on words including Lyndon words. We also define new functions related to these power series, Lyndon words counting numbers, and the Apostoltype numbers and polynomials. Furthermore, we present some illustrations and observations on approximations of functions by rational functions associated with Apostol-type numbers that can provide ideas on the reduction of the algorithmic complexity of these algorithms.
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    On a family of special numbers and polynomials associated with apostol-type numbers and poynomials and combinatorial numbers
    (Univ Belgrade, Fac Electrical Engineering, 2019) Küçükoğlu, İrem; Şimşek, Yılmaz
    In this article, we examine a family of some special numbers and polynomials not only with their generating functions, but also with computation algorithms for these numbers and polynomials. By using these algorithms, we provide several values of these numbers and polynomials. Furthermore, some new identities, formulas and combinatorial sums are obtained by using relations derived from the functional equations of these generating functions. These identities and formulas include the Apostol-type numbers and polynomials, and also the Stirling numbers. Finally, we give further remarks and observations on the generating function including lambda-Apostol-Daehee numbers, special numbers, and finite sums.
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    Remarks on recurrence formulas for the Apostol-type numbers and polynomials
    (Jangjeon Mathematical Society, 2018) Küçükoğlu, İrem; Şimşek, Yılmaz
    In this paper, by differentiating the generating functions for one of the family of the Apostol-type numbers and polynomials with respect to their parameters, we present some partial differential equations including these functions. By making use of these equations, we provide some new formulas, relations and identities including these numbers and polynomials and their derivatives. Furthermore, by using a collection of the generating functions for the aforementioned family and their functional equations, we investigate the numbers and polynomials belonging to this family and their relationships with other well-known special numbers and polynomials including the Apostol-Bernoulli numbers and polynomials of higher order, the Apostol-Euler numbers and polynomials of higher order, the Frobenius-Euler numbers and polynomials of higher order, the ?-array polynomials, the ?-Stirling numbers, and the ?-Bernoulli numbers and polynomials. © 2018 Advanced Studies in Contemporary Mathematics (Kyungshang). All rights reserved.
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    Some new identities and formulas for higher-order combinatorial-type numbers and polynomials
    (Univ Nis, Fac Sci Math, 2020) Küçükoğlu, İrem
    The main purpose of this paper is to provide various identities and formulas for higher-order combinatorial-type numbers and polynomials with the help of generating functions and their both functional equations and derivative formulas. The results of this paper comprise some special numbers and polynomials such as the Stirling numbers of the first kind, the Cauchy numbers, the Changhee numbers, the Simsek numbers, the Peters poynomials, the Boole polynomials, the Simsek polynomials. Finally, remarks and observations on our results are given.