Research Article  Open Access
General Eulerian Numbers and Eulerian Polynomials
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 wellknown 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 [2] 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 [4], 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 [6] 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 [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 definition of general Eulerian numbers based on a given arithmetic progression as defined in (17). Under the new definition, some wellknown 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 wellknown Worpitzky’s identity [7]:
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.
References
 J. Bernoulli, Ars conjectandi, opus posthumum. Accedit tractatus de seriebus infinitis, et epistola gallicè scripta de ludo pilae reticularis, Impensis Thurnisiorum, fratrum, Basel, Switzerland, 1713.
 L. Euler, Institutiones calculi differentialis cum eius usu in analysi finitorum ac Doctrina serierumm, chapter VII, Academiae Imperialis Scientiarum Petropolitanae, Petersburg, Russia, 1755.
 D. Foata, “Eulerian polynomials: from Euler's time to the present,” in The Legacy of Alladi Ramakrishnan in the Mathematical Sciences, pp. 253–273, Springer, New York, NY, USA, 2010. View at: Publisher Site  Google Scholar  MathSciNet
 L. Euler, “Methodus universalis series summandi ulterius promota,” in Commentarii academia scientiarum imperialis Petropolitana 8, pp. 147–158, Academiae scientiarum imperialis Petropolitanae, 1736. View at: Google Scholar
 L. Carlitz, “qBernoulli and Eulerian numbers,” Transactions of the American Mathematical Society, vol. 76, pp. 332–350, 1954. View at: Google Scholar  Zentralblatt MATH  MathSciNet
 L. Carlitz, “A combinatorial property of qEulerian numbers,” The American Mathematical Monthly, vol. 82, pp. 51–54, 1975. View at: Publisher Site  Google Scholar  MathSciNet
 J. Worpitzky, “Studien uber die Bernoullischen und Eulerischen Zahlen,” Journal für die reine und angewandte Mathematik, vol. 94, pp. 203–232, 1883. View at: Google Scholar
 H. Tsao, “Sums of powers and Eulerian numbers,” The Mathematical Gazette, vol. 95, pp. 347–349, 2011. View at: Google Scholar
Copyright
Copyright © 2013 Tingyao Xiong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.