Some Identities Involving the High-Order Cauchy Polynomials

In this paper, we consider the Cauchy numbers and polynomials of order k and give some relation between Cauchy polynomials of order k and special polynomials by using generating functions and the Riordan matrix methods. In addition, we establish some new equalities and relations involving high-order Cauchy numbers and polynomials, high-order Daehee numbers and polynomials, the generalized Bell polynomials, the Bernoulli numbers and polynomials, high-order Changhee polynomials, high-order Changhee-Genocchi polynomials, the combinatorial numbers, Lah numbers and Stirling numbers, etc.


Introduction
Combinatorial constants are widely used in many disciplines such as probabilistic calculations, theoretical physics problem solving, computer algorithm analysis, etc. Cauchy numbers are special sequences that are widely used in number theory, numerical analysis, etc. In recent years, many papers in [1] [2] [3] [4] have been devoted to the study of Cauchy numbers and polynomials identities by various methods. High-order Cauchy numbers and polynomials are introduced by Taekyun Kim, Dae San Kim, Hyuck In Kwon and Jongjin Seo in [1]. Higher-order Cauchy of the first kind and poly-Cauchy of the first kind mixed type polynomials are introduced by D. S. Kim, T in [2]. About Cauchy numbers and polynomials and other polynomials, more combinatorial identities are de-rived. In this paper, we establish some new identities and properties by using High-order Cauchy polynomials.
The High-Order Cauchy polynomials of the first kind and the second are defined by the following generating function in [1] [2].
The High-Order Cauchy polynomials of the first kind are defined by the following generating function When 0 x = , The generating functions of the relevant special combinatorial sequences involved in this paper are as follows [3]- [17]: The α-Cauchy numbers of the first kind are given by the generating function The generalized Bell polynomials of the first kind are given by the generating function The generalized Bernoulli polynomials are given by the generating function to be For α + ∈  , the high-order degenerate Bernoulli numbers of the second are given by the generating function The high-order Changhee polynomials are defined by When are called the Changhee-Genocchi numbers. When 0 k = , we get the following generating function The negative order Changhee polynomials are defined by The high-order Changhee-Genocchi polynomials are defined by When are called the high-order Changhee-Genocchi numbers.
The Lah numbers are given by the generating function The generalized Lah numbers are given by the generating function When 0 The classical Harmonic numbers are given by the generating function

Properties about Cauchy Polynomials
In this section, we establish some identities and give some properties of high-order Cauchy polynomials by using generating functions.
Comparing the coefficients of ! n t n in both sides, we get the identities.
Similarly, we can obtain Theorem 2.2. For nonnegative integer n, we obtain Comparing the coefficients of ! n t n in both sides, we can easily get the identities.
Similarly, we can obtain Theorem 2.4. For integer Theorem 2.5. For nonnegative integer n, we obtain Proof From generating function (1), (6), we get Journal of Applied Mathematics and Physics , , Comparing the coefficients of ! n t n in both sides, we can easily get the identities.
Similarly, we can obtain Theorem 2.6. For nonnegative integer n, we have Proof From generating function (1), (7), we get Comparing the coefficients of ! n t n in both sides, we can easily get the identities.
Theorem 2.8. For nonnegative integer n, we have Comparing the coefficients of ! n t n in both sides, we can easily get the identities.
The proof of the Theorem 2.9 is similar to the proof of the Theorem 2.8, we can obtain Theorem 2.9. For nonnegative integer n, we have Proof From generating function (1), (7), (17), we get x C x L n m r i j k n x Proof From generating function (1), (14), we get

Identities about High-Order Cauchy Polynomials
In this section, by means of the Riordan matrix, we derive some new equalities between High-order Cauchy polynomials and Striling numbers, Bell numbers, Bernoulli numbers, Lah numbers, Changhee numbers and so on.