#### Abstract

We will generalize the definitions of Eulerian numbers and Eulerian polynomials to 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.

#### 1. Introduction

Bernoulli [1, pages 95–97] had introduced his famous Bernoulli numbers, denoted by ( for ) to evaluate the sum of the th power of the first integers. He then proved the following summation formula: when .

Two decades later, Euler  studied the alternating sum . He ended up with giving the following general result [3, (2.8), page 259]: Another simplified form of is the following [3, (3.3), page 263]: where , are called Eulerian polynomials and are recursively defined by [3, (2.7), page 264]:

Note that for , if we put , then becomes the alternating sum of the th power of the first positive integers. Furthermore, as , we have [3, (3.2), page 263]:

Each Eulerian polynomial can be presented as a generating function of Eulerian numbers , also introduced by Euler, as

Furthermore, the corresponding exponential generating function [3, (3.1), page 262] is

The following combinatorial definition of Eulerian numbers was discovered by Riordan in the 1950s.

Definition 1. A Eulerian number is the number of permutations of the first numbers that have ascents (or descents), that is, places where (or ), .

Example 2. When , and , because there are four permutations with only one ascent: then satisfies the recurrence: ,    and

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]:

Since the 1950s, Carlitz ([5, 6]) and his successors have generalized Euler’s results to -sequences . Under Carlitz’s definition, the -Eulerian numbers are given by where Then the -Eulerian polynomials are defined 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.

Definition 4. Given a permutation of the first numbers , define functions

In 1974, Carlitz  completed his study of his -Eulerian numbers by giving a combinatorial meaning to his -Eulerian numbers: where functions are as defined in Definition 4. Interested readers can find more details about the history of Eulerian numbers, Eulerian polynomials, and the corresponding concepts in -environment from .

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 definition of general Eulerian numbers based on a given arithmetic progression as defined in (17). Under the new definition, some well-known combinatorial properties of traditional Eulerian numbers become special cases of our more general results. In Section 3, we will define general Eulerian polynomials. Then, (2), (3), (4), (5), (6), (7), (9), and (10) become special cases of our more general results.

#### 2. General Eulerian Numbers

The traditional Eulerian numbers , play an important role in the well-known Worpitzky’s identity :

Before we give a general definition of Eulerian numbers based on a given arithmetic progression (17), we shall mention a property associated with the traditional Eulerian numbers .

Proposition 5. Let be the traditional Eulerian numbers as defined in Definition 1, then

Proof. See [8, (4), page 348].

Given an arithmetic progression (17), we want to define general Eulerian numbers so that the important properties of traditional Eulerian numbers such as the recursive formula (9), the triangular array (Table 1), Worpitzky’s Identity (18), and Proposition 5, and so forth, become special cases of more general results under the new definition.

Definition 6. General Eulerian numbers associated with an arithmetic progression as in (17) are defined as and

Like the traditional Eulerian numbers and -Eulerian numbers, the first 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: Therefore when , the entries in the first column of Table 3 become zeroes except initially defined .

With the new Definition 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.

Lemma 7 (General Worpitzky’s Identity). Given an arithmetic progression as in (17),

Proof. We will prove Lemma 7 by induction on .
(i) When , using the values in Table 3,
(ii) Now suppose

By (20) With Lemma 7, we can prove the following Lemma which is a generalization of Proposition 5.

Lemma 8. Given an arithmetic progression as in (17),

Proof. We will prove Lemma 8 by induction on .(i) When , by Lemma 7, since , for . Therefore, which finishes the base case .(ii) Now suppose (iii) Then from Table 3, Definition 6, and Lemma 7,

The general Eulerian numbers can be calculated directly from the following formula, which is a generalization of (10).

Lemma 9. For a given arithmetic progression , the general Eulerian numbers satisfy

Proof. The following proof is given by inductions on both and .
For , .
For , .
For , .
Now suppose . Then from the recursive formula (20), Note that Combine the results above, we have

#### 3. General Eulerian Polynomials

Definition 10. We define the general Eulerian polynomials associated to an arithmetic progression as in (17) as

Definition 10 is a generalization of the traditional Eulerian polynomials as in (6). The following lemma gives the relation between the general Eulerian polynomials and the traditional Eulerian polynomials.

Lemma 11. Let be the general Eulerian polynomials as in Definition 10, . Then where are traditional Eulerian polynomials as defined in (4) and (6).

Proof. (i) When , .
(ii) Now suppose
Note that by definition, . So , which implies On the other hand, from [3, (3.4), page 263], we have With results as in (39) and (40), we have Therefore, expression (37) becomes

The following result is a generalization of (7).

Lemma 12. Let be as defined in Lemma 11. Then

Proof. From Lemma 11

Using the results from Lemma 12, we can derive the following lemma, which is a general version of (5).

Proposition 13. Given an arithmetic progression as in (17), let be the general Eulerian Polynomials associated to the arithmetic progression . Then

Proof.
On the other hand, by Lemma 12, By comparing the coefficients of , we have

For the finite summation , we have the following property which is a generalization of (2) and (3).

Lemma 14. Let be the general Eulerian polynomials as defined in Definition 10:

Proof. We will use (2) and (3) to prove expressions (49) and (50), respectively. For (49), if we use (2) to evaluate , we have This gives the second term in (49). which gives the first term of (49). So we have proved (49) by using expression (2).
For (50), if we use (3) to evaluate , we have by Lemma 11. Thus, we have obtained the second part of (50). which gives the first term of (50).

#### Acknowledgments

The authors want to express their deep appreciation to Professor Dominique Foata and the referee for their kind help.