Dynamics of the Zeros of Analytic Continued (h, q)-Euler Polynomials

and Applied Analysis 3 0 1 2 3 4 5 6 2 4 6 8


Introduction
By using software, many mathematicians can explore concepts much more easily than in the past.The ability to create and manipulate figures on the computer screen enables mathematicians to quickly visualize and produce many problems, examine properties of the figures, look for patterns, and make conjectures.This capability is especially exciting because these steps are essential for most mathematicians to truly understand even basic concept.Recently, the computing environment would make more and more rapid progress and there has been increasing interest in solving mathematical problems with the aid of computers.Mathematicians have studied different kinds of the Euler, Bernoulli, Tangent, and Genocchi numbers and polynomials.Numerical experiments of Bernoulli polynomials, Euler polynomials, Genocchi polynomials, and Tangent polynomials have been the subject of extensive study in recent year and much progress has been made both mathematically and computationally (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]).Throughout this paper, we always make use of the following notations: N denotes the set of natural numbers, N 0 denotes the set of nonnegative integers, Z denotes the set of integers, R denotes the set of real numbers, and C denotes the set of complex numbers.Let  be a complex number with || < 1 and ℎ ∈ Z. Bernoulli equation is one of the well known nonlinear differential equations of the first order.It is written as   +  ()  =  ()   ( any real number) , (1) where () and () are continuous functions.For  = 0 and  = 1 the equation is linear, and otherwise it is nonlinear.When  = 2, the Bernoulli equation has the solution which is the function of exponential generating function of the Euler numbers.Simsek [18] introduced the (ℎ, )-Euler numbers  (ℎ) , and polynomials  (ℎ) , ().He gave recurrence identities (ℎ, )-Euler polynomials and the alternating sums of powers of consecutive (ℎ, )-integers.In [13], we described the beautiful zeros of the (ℎ, )-Euler polynomials  (ℎ)  , () using a numerical investigation.Also we investigated distribution and structure of the zeros of the (ℎ, )-Euler polynomials  (ℎ)  , () by using computer.
Thus (ℎ, )-Euler numbers  (ℎ)  , are defined by means of the generating function As is well known, when  = 2 a special Bernoulli equation has the solution That is, the Bernoulli equation has the solution which is the function of exponential generating function of the (ℎ, )-Euler numbers.Thus, a realistic study for the analytic continued polynomials  (ℎ)  (, ) is very interesting by using computer.It is the aim of this paper to observe an interesting phenomenon of "scattering" of the zeros of the analytic continued polynomials  (ℎ)   (, ) in complex plane.By using computer, the (ℎ, )-Euler numbers  (ℎ)  ,, can be determined explicitly.A few of them are Theorem 1.For  ∈ N 0 , we have By Theorem 1, after some elementary calculations, we have Since  (ℎ) ,, (0) =  (ℎ) , , by (9), we obtain Then, it is easy to deduce that  (ℎ) , () are polynomials of degree .Here is the list of the first (ℎ, )-Euler's polynomials: 2. Analytic Continuation of (ℎ, )-Euler Numbers  (ℎ)
From (4), we note that By using the above equation, we are now ready to define (ℎ, )-Euler zeta functions.
By using (3), we note that By ( 16), we are now ready to define the Hurwitz-type (ℎ, )-Euler zeta functions.
In Figure 6(b), we draw  and  axes but no  axis in three dimensions.In Figure 6(c), we draw  and  axes but no  axis in three dimensions.In Figure 6(d), we draw  and  axes but no  axis in three dimensions.
Our numerical results for approximate solutions of real zeros of  (ℎ)   (, ),  = −1/2, ℎ = 3 are displayed.We observe a remarkably regular structure of the complex roots of (ℎ, )-Euler polynomials.We hope to verify a remarkably regular structure of the complex roots of (ℎ, )-Euler polynomials (Table 1).
This introduces the exciting concept of the dynamics of the zeros of analytic continued Euler polynomials, the idea of looking at how the zeros move about in the  complex plane as we vary the parameter .To have a physical picture of the motion of the zeros in the complex  plane, imagine that each time as  increases gradually and continuously by one, an additional real zero flies in from positive infinity along the real positive axis, gradually slowing down as if "it is flying through a viscous medium." For more studies and results in this subject, you may see [6,[11][12][13][14][15].