### Abstract:

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