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Some array polynomials over special monoid presentations

FIXED POINT THEORY AND APPLICATIONS 2013年第1期2013卷 1-14页

作者： Cevik, Ahmet Sinan Das, Kinkar Chandra Simsek, Yilmaz Cangul, Ismail Naci Selcuk Univ Dept Math Fac Sci TR-42075 Konya Turkey Sungkyunkwan Univ Dept Math Suwon 440746 South Korea Akdeniz Univ Fac Art & Sci Dept Math TR-07058 Antalya Turkey Uludag Univ Fac Arts & Sci Dept Math TR-16059 Bursa Turkey

In a recent joint paper (Cevik et al. in Hacet. J. Math. Stat., acceptted), the authors have investigated the p-Cockcroft property (or, equivalently, efficiency) for a presentation, say , of the semi-direct product of a free abelian monoid rank two by a finite cyclic monoid. Moreover, they have presented sufficient conditions on a special case for to be minimal whilst it is inefficient. In this paper, by considering these results, we first show that the presentations of the form can actually be represented by characteristic polynomials. After that, some connections between representative characteristic polynomials and generating functions in terms of array polynomials over the presentation will be pointed out. Through indicated connections, the existence of an equivalence among each generating function in itself is claimed studied in this paper. MSC: 11B68, 11S40, 12D10, 20M05, 20M50, 26C05, 26C10.

关键词： minimality characteristic polynomials array polynomials

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Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their applications

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FIXED POINT THEORY AND APPLICATIONS 2013年第1期2013卷 1-28页

作者： Simsek, Yilmaz Akdeniz Univ Fac Sci Dept Math TR-07058 Antalya Turkey

The first aim of this paper is to construct new generating functions for the generalized lambda-Stirling type numbers of the second kind, generalized array type polynomials and generalized Eulerian type polynomials and numbers. We derive various functional equations and differential equations using these generating functions. The second aim is to provide a novel approach to derive identities including multiplication formulas and recurrence relations for these numbers and polynomials using these functional equations and differential equations. Furthermore, we derive some new identities for the generalized lambda-Stirling type numbers of the second kind, the generalized array type polynomials and the generalized Eulerian type polynomials. We also give many applications related to the class of these polynomials and numbers.

关键词： Bernoulli polynomials Euler polynomials Apostol Bernoulli polynomials generalized Frobenius Euler polynomials normalized polynomials array polynomials Stirling numbers of the second kind generating function functional equation

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Computation methods for combinatorial sums and Euler-type numbers related to new families of numbers

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2017年第7期40卷 2347-2361页

作者： Simsek, Yilmaz Univ Akdeniz Dept Math Fac Sci TR-07058 Antalya Turkey

The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order. We can show that these numbers are related to the well-known numbers and polynomials such as the Stirling numbers of the second kind and the central factorial numbers, the array polynomials, the rook numbers and polynomials, the Bernstein basis functions and others. In order to derive our new identities and relations for these numbers, we use a technique including the generating functions and functional equations. Finally, we give not only a computational algorithm for these numbers but also some numerical values of these numbers and the Euler numbers of negative order with tables. We also give some combinatorial interpretations of our new numbers. Copyright (c) 2016 John Wiley & Sons, Ltd.

关键词： Euler numbers and polynomials generating functions Stirling numbers functional equations central factorial numbers array polynomials binomial coefficients binomial sum combinatorial sum

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Identities associated with Milne-Thomson type polynomials and special numbers

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2018年第1期2018卷 84-84页

作者： Simsek, Yilmaz Cakic, Nenad Univ Akdeniz Dept Math Fac Sci Antalya Turkey Univ Belgrade Dept Math Fac Elect Engn Belgrade Serbia

The purpose of this paper is to give identities and relations including the Milne-Thomson polynomials, the Hermite polynomials, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the central factorial numbers, and the Cauchy numbers. By using fermionic and bosonic p-adic integrals, we derive some new relations and formulas related to these numbers and polynomials, and also the combinatorial sums.

关键词： Generating function Functional equation Bernoulli numbers and polynomials Euler numbers and polynomials Stirling numbers array polynomials Milne-Thomson polynomials Hermite polynomials Central factorial numbers Cauchy numbers Special functions p-adic integral

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Applications on the Apostol-Daehee numbers and polynomials associated with special numbers, polynomials, and p-adic integrals

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ADVANCES IN DIFFERENCE EQUATIONS 2016年第1期2016卷 1页

作者： Simsek, Yilmaz Yardimci, Ahmet Akdeniz Univ Fac Sci Dept Math TR-07058 Antalya Turkey Akdeniz Univ Dept Biostat & Med Informat Fac Med Antalya Turkey

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.

关键词： Bernoulli numbers and polynomials Apostol-Bernoulli numbers and polynomials Daehee numbers and polynomials Apostol-Daehee numbers array polynomials Stirling numbers of the first kind and the second kind generating function functional equation derivative equation Bernstein basis functions

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Analysis approach to finite monoids

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FIXED POINT THEORY AND APPLICATIONS 2013年第1期2013卷 1-18页

作者： Cevik, A. Sinan Cangul, I. Naci Simsek, Yilmaz Selcuk Univ Dept Math Fac Sci TR-42075 Konya Turkey Uludag Univ Dept Math Fac Arts & Sci TR-16059 Bursa Turkey Akdeniz Univ Dept Math Fac Art & Sci TR-07058 Antalya Turkey

In a previous paper by the authors, a new approach between algebra and analysis has been recently developed. In detail, it has been generally described how one can express some algebraic properties in terms of special generating functions. To continue the study of this approach, in here, we state and prove that the presentation which has the minimal number of generators of the split extension of two finite monogenic monoids has different sets of generating functions (such that the number of these functions is equal to the number of generators) that represent the exponent sums of the generating pictures of this presentation. This study can be thought of as a mixture of pure analysis, topology and geometry within the purposes of this journal. AMS Subject Classification: 11B68, 11S40, 12D10, 20M05, 20M50, 26C05, 26C10.

关键词： efficiency p-Cockcroft property split extension generating functions Stirling numbers array polynomials

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A new approach to connect algebra with analysis: relationships and applications between presentations and generating functions

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BOUNDARY VALUE PROBLEMS 2013年第1期2013卷 1-17页

作者： Cangul, Ismail Naci Cevik, Ahmet Sinan Simsek, Yilmaz Uludag Univ Fac Arts & Sci Dept Math TR-16059 Bursa Turkey Selcuk Univ Fac Sci Dept Math TR-42075 Konya Turkey Akdeniz Univ Fac Art & Sci Dept Math TR-07058 Antalya Turkey

For a minimal group (or monoid) presentation P, let us suppose that P satisfies the algebraic property of either being efficient or inefficient. Then one can investigate whether some generating functions can be applied to it and study what kind of new properties can be obtained by considering special generating functions. To establish that, we will use the presentations of infinite group and monoid examples, namely the split extensions Z(n) X Zand Z(2) X Z, respectively. This study will give an opportunity to make a new classification of infinite groups and monoids by using generating functions.

关键词： efficiency p-Cockcroft property split extension generating functions Stirling numbers array polynomials

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array algebra expansion of matrix and tensor calculus: Part 2

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2002年第2期24卷 509-528页

作者： Rauhala, UA BAE SYST San Diego CA 92150 USA

Part 1 of this paper summarized some extended matrix and tensor operators of the multilinear array algebra and loop inverse estimation. It also introduced the compact shorthand tensor notations of array polynomials for expressing the Taylor expansion of a nonlinear function and its least squares normal equation. These matrix-like operators will be further expanded using the nonlinear tensor transforms of Blaha's Q-surface technique. They enable an analytical derivation of the direct (one hyper iteration) solution of nonlinear systems. New operators are found such as the nonlinear counterpart of the Lm- or general pseudoinverse and neural net operators. The increased pull-in range and rate of convergence of the new technique are applied in some standard examples of nonlinear equations. Some applications of the nonlinear array algebra involving literally billions of parameters are discussed, including the automated stereo mensuration of three-dimensional digital imaging and mapping systems.

关键词： array algebra nonlinear Lm-inverse tensor methods Q-surface hyper iteration array polynomials

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