Browsing by Author "Asfaw, Nasir"

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    Asymptotic Zeros of Hypergeometric Bernoulli Polynomials
    (Addis Ababa University, 2017-05-17) Asfaw, Nasir; Hassen, Abdulkadir(Professor)
    Bernoulli polynomials are named after the Swiss mathematician Jacob Bernoulli (1654 -1705). These are the class of polynomials fBn(x)g de_ned by zexz ez 1=1x=no 0BN(X) Bn(x) for jzj < 2_: With Bn = Bn(x), the rational numbers Bn are called Bernoulli numbers. In 1999, A. P. Veselov and J. P. Ward[28] established an asymptotic representation for Bn(x) and described several properties of real zeros of Bn(x) for large values of n. Later in 2008, John Mangual[24] considered another method and discussed the asymptotic real and complex zeros of Bn(x). He precisely explained the asymptotic complex zeros of Bn(nx) by introducing a curve to which the complex zeros are attracted as n goes to in_nity. In 2008, Abdulkadir Hassen and Hieu D. Nguyen [15] considered a generalization of Bn(x) called Hypergeometric Bernoulli polynomials of order N, Bn(N; x), de_ned by zNexz=N! ez TN1(z) 1Xn=0 Bn(N,X) Zn nl where TN(z) = PN k=0 zk k! . When N = 2, we obtain the class of polynomials fBn(2; x)g _rst considered by F. T. Howard[21] (with another notation). In this thesis, we introduce some properties of Bn(N ; x) which are analogous to that of Bernoulli polynomials. We establish an asymptotic formula for Bn(2 ; x) and determine their asymptotic zeros. We briey explain the behavior of the real and complex zeros of Bn(2; x) for su_ciently large positive integers n. We prove that the complex zeros of Bn(2 ; nz) asymptotically lie on a curve whose equation is given by r1ejzj = ey1=(z) : =(z) > 0 ey1=(z) : =(z) < 0 where z1 = x1 + iy1 and _z1 = x1 iy1 are roots of '(z) = ez 1 z of the minimum modulus r1 = jz1j = j_z1j