Research Article Received 12 September 2012 Published online 28 May 2013 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/mma.2825 MOS subject classification: 08A40; 11B83; 14F10; 26C05; 26C10; 26C15; 44A05 A new class of polynomials associated with Bernstein and beta polynomials Yilmaz Simsek * † Communicated by K. Guerlebeck The purpose of this paper is to define a new class polynomials. Special cases of these polynomials give many famous family of the Bernstein type polynomials and beta polynomials. We also construct generating functions for these polynomials. We investigate some fundamental properties of these functions and polynomials. Using functional equations and generating functions, we derive various identities related to theses polynomials. We also construct interpolation function that inter- polates these polynomials at negative integers. Finally, we give a matrix representations of these polynomials. Copyright © 2013 John Wiley & Sons, Ltd. Keywords: polynomials; Bernstein polynomials; beta polynomials; generating functions; functional equations; interpolation function; matrix representation; Mellin transform 1. Introduction Polynomials played an extremely important role in the development of several branches of Mathematics, Physics, and Engineering. They have many algebraic operations. Because of their finite evaluation schemes, closure under addition, multiplication, differentia- tion, integration, and composition, they are richly utilized in computational models of scientific and engineering problems (cf. [1]). Recently, many authors have studied on many kind of the polynomials. About 100 years ago, the Bernstein polynomials were defined. The Bernstein type polynomials have been defined by many different ways, which are given by q-series, by complex functions, and by many algorithms. The Bernstein polynomials are used in approximations of functions, in the other fields such as smoothing in statistics, in numerical analysis, in the solution of the differential equations, and constructing Bezier curves and in computer-aided geometric design. In this paper, we define a new class polynomials with their generating functions. Special cases of our polynomials give the Bernstein polynomials, the unification of the Bernstein type polynomials, and beta polynomials. In recent years, generating functions play an important role in the investigation of many fundamental properties of the polynomials and sequences. By using these functions, one can find many identities and formulas for the polynomials and sequences. In Section 2, we construct generating functions for a new class polynomials. We investigate many properties of these generating functions. It is well known that functional equations are one of the main topics in many branches of Mathematics. By using functional equations, which contain generating functions, we give many identities and relations related to the new class polynomials. The remainder of this paper is organized as follows: In Section 2, we give definition and properties of the new class polynomials, which are denote by Y n k .x; b/. In Section 3, we construct generating function for the polynomials Y n k .x; b/. In Section 4, we derive some functional equations of the generating functions for the polynomials Y n k .x; b/. By using these equations, we give relations between the polynomials Y n k .x; b/, the Bernstein polynomials, and the beta polynomials. In Section 5, we give integral representations for the polynomials Y n k .x; b/. In Sections 6 and 7, we find partial differential equations (PDEs) for the generating functions. By using these PDEs, we give differentiating and recurrence relation of the polynomials Y n k .x; b/. In Sections 8 and 9, we investigate some properties of the the polynomials Y n k .x; b/ that are partition of unity and alternating sum. In Section 10, by applying Mellin transform to the generating functions, we construct interpolation function of the polynomials Y n k .x; b/. Finally, in Section 11, we give a matrix representation for the polynomials Y n k .x; b/. Department of Mathematics, Faculty of Science, University of Akdeniz, 07058 Antalya, Turkey *Correspondence to: Yilmaz Simsek, Department of Mathematics, Faculty of Science, University of Akdeniz, 07058 Antalya, Turkey. † E-mail: ysimsek@akdeniz.edu.tr 676 Copyright © 2013 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2014, 37 676–685