The Properties Of Polynomials Related To Eulerian Polynomials

The Eulerian polynomial has been an active topic in enumerative combinatorics.Many properties of the Eulerian polynomial have been further studied since the Eulerian number and the generating function are put forward.For example,the ?-positivity,the unimodality,the symmetry,the asymptotic normality of their coefficients;the recurrence relation,the reality of roots of them;the exponential generating function,the strongly q-log-convexity(the strongly q-log-concavity)and the q-Stieltjes moment(q-SM for short)of their sequence.Soon afterwards,polynomials are associated with the Eulerian polynomials,which have also been extensively studied,such as the Eulerian polynomials of the types B and D,the affine Eulerian polynomial,the derangement polynomial and so on.Therefore,some properties of them play important roles in enumerative combinatorics.We have known three types of polynomials are related to the Eulerian polynomial.The first type is the cyclic derangement polynomial.It is the generalization of the derangement polynomial,which is respectively equivalent to the derangement polynomial of types A and B under certain conditions.Steingr??msson has given the definition of the cyclic Eulerian polynomial and the recursion of the cyclic Eulerian numbers.We can know the cyclic derangement polynomial by the cyclic Eulerian polynomial.The second type is the r-derangement polynomial.It is the generalization of the derangement polynomial.It is equivalent to the derangement polynomials under certain conditions.Wang has given the recurrence relation and the exponential generating function of the r-derangement number.On this basis,we define the r-derangement polynomial.We can know more properties on the derangement polynomial by researching the r-derangement polynomial.The third type is the binomial-Eulerian polynomial.The Eulerian polynomial is converted into it by the binomial transformation.Shareshian has given the definition,the ?-positivity and the unimodality of it.We do further research based on researches of scholars.The details are as follows:In the first part,we give the recurrence relation of the cyclic Eulerian polynomial by using the recurrence relation of the cyclic Eulerian number and the definition of the cyclic Eulerian polynomial.Later,we find the explicit relation between them by comparing the definition of the cyclic Eulerian polynomial to the definition of the cyclic derangement polynomial.Therefore,we obtain the recurrence relation and the reality of roots of the cyclic derangement polynomial.Further more,we get the asymptotic normality and spiral property of their coefficients by applying the reality of its roots and Lyapunov theorem.In the second part,the r-derangement polynomial is defined by the theoretical knowledge of the r-derangement number and the weak excedance statistics.Then,we gain the recurrence relation of the r-derangement polynomial by combinatorial methods and recurrence relation of the r-derangement number.Finally,the exponential generating function of the r-derangement polynomial is obtained by the exponential generating function of the derangement polynomial.In the third part,we give the recurrence relation and the exponential generating function of the binomial-Eulerian polynomial by the recurrence relation and exponential generating function of the Eulerian polynomial.Due to the Eulerian polynomial has the strongly q-log-convexity and the q-SM,we gain the binomial-Eulerian polynomial has the strongly q-log-convexity and the q-SM.