General Eulerian Numbers and Eulerian Polynomials

In this paper, we will define general Eulerian numbers and Eulerian polynomials based on general arithmetic progressions. Under the new definitions, we have been successful in extending several well-known properties of traditional Eulerian numbers and polynomials to the general Eulerian polynomials and numbers.


Journal of Mathematics
Each Eulerian polynomial can be presented as a generating function of Eulerian numbers [4], also introduced by Euler, as (6) Furthermore, the corresponding exponential generating function [3, (3 e following combinatorial de�nition of Eulerian numbers was discovered by Riordan in the 1950s.
It is well known that Eulerian numbers have the following symmetric property: Proposition 3. Given a positive integer , and ≤ ≤ , = .
Proof. It is obvious because if a permutation has ascents then its reverse has ascents.
Furthermore, the values of can be expressed in a form of a triangular array as shown in Table 1.
Besides the recursive formula (9), can be calculated directly by the following analytic formula [3, (3.5), page 264]: where en the -Eulerian polynomials ( ) are de�ned as Like the traditional Eulerian numbers, the -Eulerian numbers ( ) have the following recursive formula: Similarly to the traditional Eulerian numbers, we can also construct a triangular array for -Eulerian numbers as in Table 2.
In 1974, Carlitz [6] completed his study of his -Eulerian numbers by giving a combinatorial meaning to his -Eulerian numbers: where functions ( ) are as de�ned in �e�nition 4. Interested readers can �nd more details about the history of Eulerian numbers, Eulerian polynomials, and the corresponding concepts in -environment from [3].
In this paper, instead of studying -sequences, we will generalize Euler's work on Eulerian numbers and Eulerian polynomials to any general arithmetic progression: In Section 2, we will give a new de�nition of general Eulerian numbers based on a given arithmetic progression as de�ned in (17). �nder the new de�nition, some well-known combinatorial properties of traditional Eulerian numbers become special cases of our more general results. In Section 3, we will de�ne general Eulerian polynomials. en, (2), (3), (4), (5), (6), (7), (9), and (10) become special cases of our more general results.

General Eulerian Numbers e traditional Eulerian numbers
, play an important role in the well-known Worpitzky's identity [7]: �efore we give a general de�nition of Eulerian numbers based on a given arithmetic progression (17), we shall mention a property associated with the traditional Eulerian numbers .
Given an arithmetic progression (17), we want to de�ne general Eulerian numbers ( so that the important properties of traditional Eulerian numbers such as the recursive formula (9), the triangular array (Table 1) Like the traditional Eulerian numbers and -Eulerian numbers, the �rst general Eulerian numbers can be presented in the form of a triangular array as in Table 3.
We intentionally choose values to start with , because by doing so Table 1 becomes the special case of Table 3 when = = in the arithmetic progression: In other words, Table 1 corresponds to the sequence of natural numbers: erefore when = = , the entries in the �rst column of Table 3 become zeroes e�cept initially de�ned ( = . With the new �e�nition 6, we are able to prove the following two properties. Again note that if = = , then the following identity is just the conventional Worpitzky's identity. (17),

Lemma 7 (General Worpitzky's Identity). Given an arithmetic progression as in
Proof. We will prove Lemma 7 by induction on .
(i) When = , using the values in Table 3, By (20) With Lemma 7, we can prove the following Lemma which is a generalization of Proposition 5. (17),

Lemma 8. Given an arithmetic progression as in
Proof. We will prove Lemma 8 by induction on .

General Eulerian Polynomials
�e�nition ��. We de�ne the general Eulerian polynomials associated to an arithmetic progression as in (17) as �e�nition 10 is a generalization of the traditional Eulerian polynomials as in (6). e following lemma gives the relation between the general Eulerian polynomials and the traditional Eulerian polynomials.
Using the results from Lemma 12, we can derive the following lemma, which is a general version of (5). (17)

Proposition 13. Given an arithmetic progression as in
Proof.
On the other hand, by Lemma 12, By comparing the coefficients of / , we have For the �nite summation ∑ =1 [ ( − 1) , we have the following property which is a generalization of (2) and (3).

Lemma 14.
Let be the general Eulerian polynomials as �e�ne� in �e�nition ��: Proof. We will use (2) and (3) by Lemma 11. us, we have obtained the second part of (50).
which gives the �rst term of (50).