• Title, Summary, Keyword: a family of polynomials

### REMARKS ON RECURRENCE FORMULAS FOR THE APOSTOL-TYPE NUMBERS AND POLYNOMIALS

• KUCUKOGLU, IREM;SIMSEK, YILMAZ
• Advanced Studies in Contemporary Mathematics
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• v.28 no.4
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• pp.643-657
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• 2018
• 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 ${\lambda}$-array polynomials, the ${\lambda}$-Stirling numbers, and the ${\lambda}$-Bernoulli numbers and polynomials.

### FINDING RESULTS FOR CERTAIN RELATIVES OF THE APPELL POLYNOMIALS

• Ali, Mahvish;Khan, Subuhi
• Bulletin of the Korean Mathematical Society
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• v.56 no.1
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• pp.151-167
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• 2019
• In this article, a hybrid family of polynomials related to the Appell polynomials is introduced. Certain properties including quasimonomiality, differential equation and determinant definition of these polynomials are established. Further, applications of Carlitz's theorem to these polynomials and to certain other related polynomials are considered. Examples providing the corresponding results for some members belonging to this family are also considered.

### A NEW FAMILY OF FUBINI TYPE NUMBERS AND POLYNOMIALS ASSOCIATED WITH APOSTOL-BERNOULLI NUMBERS AND POLYNOMIALS

• Kilar, Neslihan;Simsek, Yilmaz
• Journal of the Korean Mathematical Society
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• v.54 no.5
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• pp.1605-1621
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• 2017
• The purpose of this paper is to construct a new family of the special numbers which are related to the Fubini type numbers and the other well-known special numbers such as the Apostol-Bernoulli numbers, the Frobenius-Euler numbers and the Stirling numbers. We investigate some fundamental properties of these numbers and polynomials. By using generating functions and their functional equations, we derive various formulas and relations related to these numbers and polynomials. In order to compute the values of these numbers and polynomials, we give their recurrence relations. We give combinatorial sums including the Fubini type numbers and the others. Moreover, we give remarks and observation on these numbers and polynomials.

### SIMPLIFYING COEFFICIENTS IN A FAMILY OF ORDINARY DIFFERENTIAL EQUATIONS RELATED TO THE GENERATING FUNCTION OF THE MITTAG-LEFFLER POLYNOMIALS

• Qi, Feng
• Korean Journal of Mathematics
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• v.27 no.2
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• pp.417-423
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• 2019
• In the paper, by virtue of the $Fa{\grave{a}}$ di Bruno formula, properties of the Bell polynomials of the second kind, and the Lah inversion formula, the author simplifies coefficients in a family of ordinary differential equations related to the generating function of the Mittag-Leffler polynomials.

### AN EASILY CHECKING CONDITION FOR THE STAVILITY TEST OF A FAMILY OF POLYNOMIALS WITH NONLIMEARLY PERTURBED COEFFICIENTS

• Kim, Young-Chol;Hong, Woon-Seon
• 제어로봇시스템학회:학술대회논문집
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• pp.5-9
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• 1995
• In many cases of robust stability problems, the characteristic polynomial has real coefficients which or nonlinear functions of uncertain parameters. For this set of polynomials, a new stability easily checking algorithm for reducing the conservatism of the stability bound are given. It is the new stability theorem to determine the stability region just in parameter space. Illustrative example show that the presented method has larger stability bound in uncertain parameter space than others.

### Korobov Polynomials of the Fifth Kind and of the Sixth Kind

• Kim, Dae San;Kim, Taekyun;Kwon, Hyuck In;Mansour, Toufik
• Kyungpook Mathematical Journal
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• v.56 no.2
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• pp.329-342
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• 2016
• Recently, Korobov polynomials have been received a lot of attention, which are discrete analogs of Bernoulli polynomials. In particular, these polynomials are used to derive some interpolation formulas of many variables and a discrete analog of the Euler summation formula. In this paper, we extend these family of polynomials to consider the Korobov polynomials of the fifth kind and of the sixth kind. We present several explicit formulas and recurrence relations for these polynomials. Also, we establish a connection between our polynomials and several known families of polynomials.

### q-SOBOLEV ORTHOGONALITY OF THE q-LAGUERRE POLYNOMIALS {Ln(-N)(·q)}n=0∞ FOR POSITIVE INTEGERS N

• Moreno, Samuel G.;Garcia-Caballe, Esther M.
• Journal of the Korean Mathematical Society
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• v.48 no.5
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• pp.913-926
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• 2011
• The family of q-Laguerre polynomials $\{L_n^{(\alpha)}({\cdot};q)\}_{n=0}^{\infty}$ is usually defined for 0 < q < 1 and ${\alpha}$ > -1. We extend this family to a new one in which arbitrary complex values of the parameter ${\alpha}$ are allowed. These so-called generalized q-Laguerre polynomials fulfil the same three term recurrence relation as the original ones, but when the parameter ${\alpha}$ is a negative integer, no orthogonality property can be deduced from Favard's theorem. In this work we introduce non-standard inner products involving q-derivatives with respect to which the generalized q-Laguerre polynomials $\{L_n^{(-N)}({\cdot};q)\}_{n=0}^{\infty}$, for positive integers N, become orthogonal.

### NEW FAMILY OF ITERATIVE METHODS FOR SOLVING NON-LINEAR EQUATIONS USING NEW ADOMIAN POLYNOMIALS

• RAFIQ, ARIF;PASHA, AYESHA INAM;LEE, BYUNG-SOO
• The Pure and Applied Mathematics
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• v.22 no.3
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• pp.231-243
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• 2015
• We suggest and analyze a family of multi-step iterative methods for solving nonlinear equations using the decomposition technique mainly due to Rafiq et al. [13].

### STRUCTURE RELATIONS OF CLASSICAL MULTIPLE ORTHOGONAL POLYNOMIALS BY A GENERATING FUNCTION

• Lee, Dong Won
• Journal of the Korean Mathematical Society
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• v.50 no.5
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• pp.1067-1082
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• 2013
• In this paper, we will find some recurrence relations of classical multiple OPS between the same family with different parameters using the generating functions, which are useful to find structure relations and their connection coefficients. In particular, the differential-difference equations of Jacobi-Pineiro polynomials and multiple Bessel polynomials are given.

### Some Identities Involving Euler Polynomials Arising from a Non-linear Differential Equation

• Rim, Seog-Hoon;Jeong, Joohee;Park, Jin-Woo
• Kyungpook Mathematical Journal
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• v.53 no.4
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• pp.553-563
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• 2013
• We derive a family of non-linear differential equations from the generating functions of the Euler polynomials and study the solutions of these differential equations. Then we give some new and interesting identities and formulas for the Euler polynomials of higher order by using our non-linear differential equations.