Poly-Cauchy and Peters mixed-type polynomials

The Peters polynomials are a generalization of Boole polynomials. In this paper, we consider Peters and poly-Cauchy mixed type polynomials and investigate the properties of those polynomials which are derived from umbral calculus. Finally, we give various identities of those polynomials associated with special polynomials.


Introduction
The Peters polynomials are defined by the generating function to be x , (see [14]) .
The higher-order Cauchy polynomials of the first kind are defined by the generating function to be and the higher-order Cauchy polynomials of the second kind are given by The Stirling number of the first kind is given by Thus, by (10), we get [14]) .
It is easy to show that Let C be the complex number field and let F be the algebra of all formal power series in the variable t over C as follows: Let P = C [x] and let P * be the vector space of all linear functionals on P. L| p (x) denotes the action of linear functional L on the polynomial p (x), and we recall that the vector space operations on P * are defined by For f (t) ∈ F , let us define the linear functional on P by setting (14) f (t)| x n = a n , (n ≥ 0) .
The map L −→ f L (t) is a vector space isomorphism from P * onto F . Henceforth, F denotes both the algebra of formal power series in t and vector space of all linear functionals on P, and so an element f (t) of F will be thought of as both a formal power series and a linear functional. We call F the umbral algebra and the umbral calculus is the study of umbral algebra. The order O (f ) of the power series f (t) ( = 0) is the smallest integer for which the coefficient of t k does not vanish. If with O (f (t)) = 1 and O (g (t)) = 0, there exists a unique sequence s n (x) of polynomials such that g (t) f (t) k s n (x) = n!δ n,k , (n, k ≥ 0). The sequence s n (x) is called the Sheffer sequence for (g (t) , f (t)) which is denoted by s n (x) ∼ (g (t) , f (t)) .
For f (t) , g (t) ∈ F and p (x) ∈ P,we have
In this paper, we consider Peters and poly-Cauchy mixed type polynomials with umbral calculus viewpoint and investigate the properties of those polynomials which are derived from umbral calculus. Finally, we give some interesting identities of those polynomials associated with speical polynomials.
It is not difficult to show that From (14), we have Therefore, by (32), we obtain the following theorem. Alternatively, Therefore, by (33), we obtain the following theorem.
Remark. By the same method, we get From (20) and (28), we have From (31), we note that Therefore, by (36) and (37), we obtain the following theorem.
Theorem 3. For n ≥ 0, we have Remark. By the same method as Theorem 3, we get From (28), we note that By (24), (39) and (40), we get Thus, by (41), we see that Therefore, by (42), we obtain the following theorem.
Theorem 4. For n ≥ 0, we have Remark. By the same method as Theorem 4, we get From (12), we note that Thus, by (44), we see that Therefore, by (48), we obtain the following theorem.
It is easy to see that By the same method as Theorem 5, we get From (20) and (28), we have Now, we observe that Therefore, by (52) and (53), we obtain the following theorem.
Theorem 6. For n ≥ 0, we have Remark. By the same method as Theorem 6, we get n−j (x; λ, µ) (y) j .
By (23), we get By the same method as (65), we get Now, we compute the following equation in two different ways: On the one hand, On the other hand, Then, by (26), we get C n,m (72) Therefore, by (71) and (72), we obtain the following the theorem. Remark. By the same method as Theorem 10, we havê  Therefore, by (75) and (76), we obtain the following theorem.  Therefore, by (77) and (78), we obtain the following theorem.