Some Identities for Euler and Bernoulli Polynomials and Their Zeros

: In this paper, we study some special polynomials which are related to Euler and Bernoulli polynomials. In addition, we give some identities for these polynomials. Finally, we investigate the zeros of these polynomials by using the computer

The Bernoulli polynomials B (r) n (x) of order r are defined by the following generating function The Frobenius-Euler polynomials of order r, denoted by H ( The values at x = 0 are called Frobenius-Euler numbers of order r; when r = 1, the polynomials or numbers are called ordinary Frobenius-Euler polynomials or numbers.
In this paper, we study some special polynomials which are related to Euler and Bernoulli polynomials.In addition, we give some identities for these polynomials.Finally, we investigate the zeros of these polynomials by using the computer.

Cosine-Bernoulli, Sine-Bernoulli, Cosine-Euler and Sine-Euler Polynomials
In this section, we define the cosine-Bernoulli, sine-Bernoulli, cosine-Euler and sine-Euler polynomials.Now, we consider the Euler polynomials that are given by the generating function to be On the other hand, we observe that e (x+iy)t = e xt e iyt = e xt (cos yt + i sin yt).
From Equations ( 6) and ( 7), we have and Thus, by ( 8) and ( 9), we can derive It follows that we define the following cosine-Euler polynomials and sine-Euler polynomials. and respectively.

and E
(S) By (12), we get Therefore, by comparing the coefficients on the both sides, we obtain the following theorem: and Taking r = 1 in Theorem 5, we obtain the following corollary: Corollary 2. For n ≥ 0, we have and From Corollary 2, we note that and By (12), we get Comparing the coefficients on the both sides of (27), we have Similarly, for n ≥ 1, we have Now, we consider the Bernoulli polynomials that are given by the generating function to be We also have Thus, by ( 28) and (29), we can derive and It follows that we define the following cosine-Bernoulli and sine-Bernoulli polynomials.

Now, we observe that
Thus, by (36), we get Therefore, by (37), we obtain the following theorem: and B (S) Now, we define the new type polynomials that are given by the generating functions to be n (y) Note that n (y), (n ≥ 0).The new type polynomials can be determined explicitly.A few of them are and 6 (x, y) = −3y − 5y 3 − 3y 5 .From (38) and (39), we derive the following equations: and By ( 38)-(41), we get and From ( 12), ( 13), ( 38) and (39), we derive the following theorem: For n ≥ 0, we have k (y).
Now, we define the new type polynomials that are given by the generating functions to be and t e t − 1 sin yt = From ( 44) and (45), we derive the following equations: and By ( 44)-( 47), we get and From ( 32), ( 33), ( 44) and (45), we derive the following theorem: Theorem 10.For n ≥ 0, we have k (y).
We remember that the classical Stirling numbers of the first kind S 1 (n, k) and S 2 (n, k) are defined by the relations (see [12]) respectively.Here, (x denotes the falling factorial polynomial of order n. The numbers S 2 (n, m) also admit a representation in terms of a generating function By ( 12), (51) and by using Cauchy product, we get where By comparing the coefficients on both sides of (52), we have the following theorem: Theorem 11.For n ∈ Z + , we have By ( 12), ( 38), ( 50), (51) and by using Cauchy product, we have ((e t − 1) + 1) x cos(yt) By comparing the coefficients on both sides of (53), we have the following theorem: Theorem 12.For n ∈ Z + , we have n−i (y).
By ( 4), ( 12), (38), (50), (51) and by using Cauchy product, we have By comparing the coefficients on both sides, we have the following theorem: Theorem 13.For n ∈ Z + and r ∈ N, we have By ( 5), ( 12), ( 38), ( 50), (51) and by using the Cauchy product, we get By comparing the coefficients on both sides, we have the following theorem: Theorem 14.For n ∈ Z + and r ∈ N, we have n−l (i, y).
By Theorems 12-14, we have the following corollary.
Corollary 3.For n ∈ Z + and r ∈ N, we have i (x).

Distribution of Zeros of the Cosine-Euler and Sine-Euler Polynomials
This section aims to demonstrate the benefit of using numerical investigation to support theoretical prediction and to discover a new interesting pattern of the zeros of the cosine-Euler and sine-Euler polynomials.Using a computer, a realistic study for the cosine-Euler polynomials E n (x, y) in a complex plane.We investigate the beautiful zeros of the cosine-Euler and sine-Euler polynomials by using a computer.We plot the zeros of the cosine-Euler polynomials E (C) n (x, y) (Figure 1).
We observe that E (C) n (x, a), x ∈ C has Re(x) = 1 2 reflection symmetry in addition to the usual Im(x) = 0 reflection symmetry analytic complex functions, where a ∈ R( Figures 1 and 2).Since Hence, we have the following theorem: n (1/2, y) = 0, for n ∈ N.
Our numerical results for numbers of real and complex zeros of the cosine-Euler polynomials n (x, y) = 0 are displayed (Table 1).Our numerical results for numbers of real and complex zeros of the sine-Euler polynomials n (x, y) = 0 are displayed (Table 2).In Figure 5 (left), we choose x = −3.In Figure 3 (right), we choose x = 1.The plot of real zeros of the sine-Euler polynomials E (S) n (x, y) for 1 ≤ n ≤ 40 structure are presented (Figure 6).In Figure 6 (left), we choose x = −3.In Figure 6 (right), we choose x = 1.We observe a remarkable regular structure of the complex roots of the cosine-Euler polynomials E (C) n (x, y).We also hope to verify a remarkable regular structure of the complex roots of the cosine-Euler polynomials E (C) n (x, y).Next, we calculated an approximate solution satisfying E (C) n (x, y) = 0, x ∈ R. The results are given in Table 3.
(C) n (x, y) and sine-Euler polynomials E (S) n (x, y) is very interesting.It is the aim of this paper to observe an interesting phenomenon of "scattering" of the zeros of the the cosine-Euler polynomials E (C) n (x, y) and sine-Euler polynomials E (S)

Table 1 .
Numbers of real and complex zeros of E n (x, y).
n n (x, y) = 0, y ∈ R. The results are given in Table4.