Asymptotic Zeros Of Hypergeometric Bernoulli Polynomials

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Bernoulli polynomials are named after the Swiss mathematician Jacob Bernoulli (1654 -1705). Thesernare the class of polynomials fBn(x)g de_ned byrnzexzrnez 1=1x=norn0BN(X)rnBn(x)rnfor jzj < 2_:rnWith Bn = Bn(x), the rational numbers Bn are called Bernoulli numbers.rnIn 1999, A. P. Veselov and J. P. Ward[28] established an asymptotic representation for Bn(x) andrndescribed several properties of real zeros of Bn(x) for large values of n. Later in 2008, John Mangual[24]rnconsidered another method and discussed the asymptotic real and complex zeros of Bn(x). He preciselyrnexplained the asymptotic complex zeros of Bn(nx) by introducing a curve to which the complex zerosrnare attracted as n goes to in_nity.rnIn 2008, Abdulkadir Hassen and Hieu D. Nguyen [15] considered a generalization of Bn(x) calledrnHypergeometric Bernoulli polynomials of order N, Bn(N; x), de_ned byrnzNexz=N!rnez TN1(z)rn1Xn=0rnBn(N,X)rnZn nlrnwhere TN(z) =rnPNrnk=0rnzkrnk! . When N = 2, we obtain the class of polynomials fBn(2; x)g _rst consideredrnby F. T. Howard[21] (with another notation).rnIn this thesis, we introduce some properties of Bn(N ; x) which are analogous to that of Bernoullirnpolynomials. We establish an asymptotic formula for Bn(2 ; x) and determine their asymptotic zeros.rnWe briey explain the behavior of the real and complex zeros of Bn(2; x) for su_ciently large positivernintegers n. We prove that the complex zeros of Bn(2 ; nz) asymptotically lie on a curve whose equationrnis given byrnr1ejzj =rney1=(z) : =(z) > 0rney1=(z) : =(z) < 0rnwhere z1 = x1 + iy1 and _z1 = x1 iy1 are roots of '(z) = ez 1 z of the minimum modulusrnr1 = jz1j = j_z1j

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Asymptotic Zeros Of Hypergeometric Bernoulli Polynomials

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