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Öğe Analysis of Generating Functions for Special Words and Numbers and Algorithms for Computation(Springer Basel Ag, 2022) Kucukoglu, Irem; Milovanovic, Gradimir, V; Simsek, YilmazOur aim is to construct and compute efficient generating functions enumerating the k-ary Lyndon words having prime number length which arise in many branches of mathematics and computer science. We prove that these generating functions coincide with the Apostol-Bernoulli numbers and their interpolation functions and obtain other forms of these generating functions including not only the Frobenius-Euler numbers, but also the Fubini type numbers. Moreover, we derive some identities, relations and combinatorial sums including the numbers of the k-ary Lyndon words, the Bernoulli numbers and polynomials, the Stirling numbers and falling factorials. Using these generating functions and recurrence relation for the Apostol-Bernoulli numbers, we give two algorithms to compute these generating functions. Using these algorithms, we compute some infinite series formulas including the number of the k-ary Lyndon words on some special classes of primes with the purpose of providing some numerical evaluations about these generating functions. In addition, we approximate these generating functions by the rational functions of the Apostol-Bernoulli numbers to show that the complexity of the aforementioned algorithms may be decreased by means of approximation method which are illustrated by some numerical evaluations with their plots for varying prime numbers. Finally, using Bell polynomials (i.e., exponential functions) approach to the numbers of Lyndon words, we construct the exponential generating functions for the numbers of Lyndon words. Finally, we define a new family of special numbers related to these special words and investigate some of their fundamental properties.Öğe Computational and implementational analysis of generating functions for higher order combinatorial numbers and polynomials attached to Dirichlet characters(Wiley, 2022) Kucukoglu, IremThe main purpose of this paper is to give computational and implementational analysis of generating functions for some extensions of combinatorial numbers and polynomials attached to Dirichlet characters. By using generating function methods and p-adic q-integral techniques, it is also aimed to derive some computational formulas for these numbers and polynomials. By implementing both the derived computational formulas and the constructed generating functions in Mathematica, we present some tables and plots for these numbers and polynomials, and their generating functions to illustrate the effects of their parameters for some special cases. Moreover, by making an observation on some results of this paper, we derive some novel computational formulas for the finite sums that contain the Dirichlet characters and falling factorials. We also assess some cases of these special finite sums. In the sequel, we pose some open problems regarding these special finite sums. Apart from these findings, we also give not only Fourier series expansion but also asymptotic estimates for the mentioned combinatorial numbers. In addition, we give some Raabe-type multiplication formulas for the mentioned combinatorial polynomials. Finally, we give an observation on decompositions of the generating functions attached to any group homomorphism.Öğe Computational identities for extensions of some families of special numbers and polynomials(2021) Kucukoglu, Irem; Sımsek, YılmazThe main purpose of this paper is to obtain computational identities and formulas for a certain class of\rcombinatorial-type numbers and polynomials. By the aid of the generating function technique, we derive a recurrence\rrelation and an infinite series involving the aforementioned class of combinatorial-type numbers. By applying the\rRiemann integral to the combinatorial-type polynomials with multivariables, we present some integral formulas for these\rpolynomials, including the Bernoulli numbers of the second kind. By the implementation of the p-adic integral approach\rto the combinatorial-type polynomials with multivariables, we also obtain formulas for the Volkenborn integral and the\rfermionic p-adic integral of these polynomials. Furthermore, we provide an approximation for the combinatorial-type\rnumbers with the aid of the Stirling’s approximation for factorials. By coding some of our results in Mathematica using\rthe Wolfram programming language, we also provide some numerical evaluations and illustrations on the combinatorialtype numbers and their Stirling’s approximation with table and figures. We also give some remarks and observations on\rthe combinatorial-type numbers together with their relationships to other well-known special numbers and polynomials.\rAs a result of these observations, we derive some computation formulas containing the Dirichlet series involving the\rMöbius function, the Bernoulli numbers, the Catalan numbers, the Stirling numbers, the Apostol–Bernoulli numbers,\rthe Apostol–Euler numbers, the Apostol–Genocchi numbers and some kinds of combinatorial numbers. Besides, some\rinequalities for the combinatorial-type numbers are presented. Finally, we conclude this paper by briefly overviewing the\rresults with their potential applications.Öğe Computational identities for extensions of some families of special numbers and polynomials(Tubitak Scientific & Technological Research Council Turkey, 2021) Kucukoglu, Irem; Simsek, YilmazThe main purpose of this paper is to obtain computational identities and formulas for a certain class of combinatorial-type numbers and polynomials. By the aid of the generating function technique, we derive a recurrence relation and an infinite series involving the aforementioned class of combinatorial-type numbers. By applying the Riemann integral to the combinatorial-type polynomials with multivariables, we present some integral formulas for these polynomials, including the Bernoulli numbers of the second kind. By the implementation of the p-adic integral approach to the combinatorial-type polynomials with multivariables, we also obtain formulas for the Volkenborn integral and the fermionic p-adic integral of these polynomials. Furthermore, we provide an approximation for the combinatorial-type numbers with the aid of the Stirling's approximation for factorials. By coding some of our results in Mathematica using the Wolfram programming language, we also provide some numerical evaluations and illustrations on the combinatorial type numbers and their Stirling's approximation with table and figures. We also give some remarks and observations on the combinatorial-type numbers together with their relationships to other well-known special numbers and polynomials. As a result of these observations, we derive some computation formulas containing the Dirichlet series involving the Mobius function, the Bernoulli numbers, the Catalan numbers, the Stirling numbers, the Apostol-Bernoulli numbers, the Apostol-Euler numbers, the Apostol-Genocchi numbers and some kinds of combinatorial numbers. Besides, some inequalities for the combinatorial-type numbers are presented. Finally, we conclude this paper by briefly overviewing the results with their potential applications.Öğe Construction and computation of unified Stirling-type numbers emerging from p-adic integrals and symmetric polynomials(Springer-Verlag Italia Srl, 2021) Kucukoglu, Irem; Simsek, YilmazThe aim of this paper is to give construction and computation methods for generalized and unified representations of Stirling-type numbers and Bernoulli-type numbers and polynomials. Firstly, we define generalized and unified representations of the falling factorials. By using these new representations as components of the generating functions, we also construct generalized and unified representations of Stirling-type numbers. By making use of the symmetric polynomials, we give computational formulas and algorithm for these numbers. Applying Riemann integral to the unified falling factorials, we introduce new families of Bernoulli-type numbers and polynomials of the second kind by their computation formulas and plots drawn by the Wolfram programming language in Mathematica. Applying p-adic integrals to the unified falling factorials, we construct two new sequences that involve some well-known special numbers such as the Stirling numbers, the Bernoulli numbers and the Euler numbers. Finally, we give not only further remarks and observations, but also some open questions regarding the potential applications and relations of our results.Öğe Construction of Bernstein-Based Words and Their Patterns(Wiley, 2025) Kucukoglu, Irem; Simsek, YilmazIn this paper, with inspiration of the definition of Bernstein basis functions and their recurrence relation, we give construction of a new word family that we refer Bernstein-based words. By classifying these special words as the first and second kinds, we investigate their some fundamental properties involving periodicity and symmetricity. Providing schematic algorithms based on tree diagrams, we also illustrate the construction of the Bernstein-based words. For their symbolic computation, we also give computational implementations of Bernstein-based words in the Wolfram Language. By executing these implementations, we present some tables of Bernstein-based words and their decimal equivalents. In addition, we present black-white and four-colored patterns arising from the Bernstein-based words with their potential applications in computational science and engineering. We also give some finite sums and generating functions for the lengths of the Bernstein-based words. We show that these functions are of relationships with the Catalan numbers, the centered m$$ m $$-gonal numbers, the Laguerre polynomials, certain finite sums, and hypergeometric functions. We also raise some open questions and provide some comments on our results. Finally, we investigate relationships between the slopes of the Bernstein-based words and the Farey fractions.Öğe Derivation of Some Formulas for Stirling-type Numbers by p-adic Integration(Amer Inst Physics, 2024) Kucukoglu, IremIn this study, by implementing bosonic p-adic integral to some identities regarding the generalized and the non-central factorial coefficients, we derive some formulas that involves the first kind Stirling numbers and Bernoulli numbers, and also the Daehee numbers. Finally, we bring to an end of the paper by providing some concluding remarks with further observations on the findings of the current paper.Öğe FORMULAS FOR Q-COMBINATORIAL SIMSEK NUMBERS AND POLYNOMIALS: ANALYZING WITH COMPUTATIONAL IMPLEMENTATIONS(Univ Belgrade, Fac Electrical Engineering, 2025) Kucukoglu, IremThis paper aims to provide new results and computational implementations for describing and analyzing the q-combinatorial Simsek numbers and polynomials of the first kind. For symbolic computation of these numbers and polynomials, some procedures and illustrations have been provided in the Wolfram programming language. In addition, some computation formulas, derivative formulas, generating functions, interpolation functions, and integral formulas pertaining to these numbers and polynomials have been derived.Öğe Generating functions for divisor sums and totative sums arising from combinatorial Simsek numbers(Univ Nis, Fac Sci Math, 2024) Kucukoglu, IremThe main objective of this paper is to introduce and investigate new number families derived from finite sums running over divisors and totatives and containing higher powers of binomial coefficients. Especially, by making decomposition on the generating functions for a kind of combinatorial number families recently introduced by Simsek [29], we also construct generating functions for the newly introduced number families. For symbolic computation of the newly introduced number families and their generating functions, we also give computational implementations in the Wolfram language. By these implementations, some tables of both these number families and their generating functions have been presented for some arbitrarily chosen special cases. Additionally, we provide some applications regarding the Thacker's (totient) function. In particular, by making summation on all totatives of a positive integer, we investigate some special finite sums containing both the Thacker's (totient) function and higher powers of binomial coefficients. By this investigation, some of the problems regarding these finite sums have been partially answered accompanied by some remarks. Furthermore, we propose an open problem regarding a potential relation between one of these number families and a formula involving the Mo center dot bius function. Finally, the paper have been concluded by providing an overview on the results of this paper and their potential usage areas, and by making suggestions regarding future studies able to be made.Öğe Generating functions for multiparametric Hermite-based Peters-type Simsek numbers and polynomials in several variables(MTJPAM Turkey, 2024) Kucukoglu, IremThe essential target of this study is to introduce multivariable and multiparameter generalization of the Hermite-based Peters-type Simsek numbers and polynomials and to construct their generating functions. Besides, inspired by the generating functions and techniques constructed by Simsek [25], we here introduce and systematically investigate some new families of generating-type functions whose functional equations have been used to reveal generating functions for new families of special numbers and polynomials in this paper. The constructed generating functions unify and generalize, but are not limited to, the generating functions for the Hermite-based Peters-type Simsek numbers and polynomials, the positive and negative higher-order Peters-type Simsek numbers and polynomials, the Peters-type Simsek numbers and polynomials of all kinds, the two-variable Peters-type Simsek polynomials, the two-variable Changhee polynomials. We finalize this paper by putting forward some comments involving open problems and observations on the main results. © 2024, MTJPAM Turkey. All rights reserved.Öğe Identities for the multiparametric higher-order Hermite-based Peters-type Simsek polynomials of the first kind(MTJPAM Turkey, 2023) Kucukoglu, IremThe main aim of this study is to investigate the multiparametric higher-order Hermite-based Peters-type Simsek numbers and polynomials of the first kind, which were introduced by the author in her recent paper [18]. To achieve this aim, we first provide pseudocodes for symbolic computation of these numbers and polynomials. Moreover, we implement these pseudocodes in the Wolfram language. By these implementations, we provide some tables and plots regarding these numbers and polynomials in some arbitrarily chosen special cases. By using their generating functions with their functional equations, we derive some finite sums, identities and derivative formulas concerning these numbers and polynomials. We also investigate the first order multiparametric Hermite-based Peters-type Simsek polynomials and we provide some remarks and observations about their some reductions. Fi-nally, we conclude the paper by providing some remarks and open problems on the potential applications that could emerge from the Sheffer-type sequences, the heat-type equations, the orthogonality and the analytic continuation. © 2023, MTJPAM Turkey. All rights reserved.Öğe Implementation of computation formulas for certain classes of Apostol-type polynomials and some properties associated with these polynomials(2021) Kucukoglu, IremThe 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 $\\lambda$-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 $\\lambda$-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 $\\lambda$-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 $\\lambda$-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 some randomly selected special cases of their parameters. Finally, we conclude the paper with some comments and observations on our results.Öğe IMPLEMENTATION OF COMPUTATION FORMULAS FOR CERTAIN CLASSES OF APOSTOL-TYPE POLYNOMIALS AND SOME PROPERTIES ASSOCIATED WITH THESE POLYNOMIALS(Ankara Univ, Fac Sci, 2021) Kucukoglu, IremThe 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 lambda-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 in finite series representation for the lambda-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 lambda-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 lambda-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.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 Remarks on a Class of Combinatorial Numbers and Polynomials(American Institute of Physics Inc., 2023) Kucukoglu, IremIn this paper, by using the theory of quantum calculus, we introduce a class of combinatorial numbers and polynomials. In particular, the class of q-combinatorial numbers introduced in this work is a q-analogue of the combinatorial numbers recently defined b y Simsek [9]. We also construct a formula f or t he generating f unctions o f these q -combinatorial n umbers i n terms of q-exponential functions. Furthermore, applying q-derivative, we analyze some properties of these q-combinatorial numbers and their generating functions. As a result of this analysis, we give a few remarks related to our findings. Finally, we conclude the paper with a brief observation on our results. © 2023 American Institute of Physics Inc.. 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 Unification of the generating functions for Sheffer type sequences and their applications(MTJPAM Turkey, 2023) Kucukoglu, IremIn this paper, it is aimed to introduce a unification and generalization of the generating functions for Sheffer type sequences such as the Peters polynomials, the Boole polynomials, the Changhee polynomials, the Korobov polynomials of the first kind and the Peters-type Simsek numbers and polynomials. Moreover, by considering a special case of the aforementioned unification, we also introduce and investigate a new family of numbers and polynomials to be referred as the more general kind of the Peters-type Simsek numbers and polynomials. Finally, we give some applications of our findings. © 2023, MTJPAM Turkey. All rights reserved.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.
