[Paper Review] A Generalization of the Eulerian Numbers

Grzegorz Rządkowski, Małgorzata Urlińska|arXiv (Cornell University)|Dec 20, 2016

Advanced Mathematical Identities17 references3 citations

TL;DR

This paper introduces a generalized family of Eulerian numbers via a recurrence relation parameterized by real weights $ w_1 $ and $ w_2 $, which unifies classical Eulerian numbers, second-order Eulerian numbers, and third-order Eulerian numbers. The generalization is linked to solutions of an autonomous first-order differential equation, enabling integral representations of special numbers, including Bernoulli numbers, via generating functions derived from these solutions.

ABSTRACT

In the present paper we generalize the Eulerian numbers (also of the second and third orders). The generalization is connected with an autonomous first-order differential equation, solutions of which are used to obtain integral representations of some numbers, including the Bernoulli numbers.

Motivation & Objective

• To generalize classical Eulerian numbers to a broader family parameterized by $ w_1 $ and $ w_2 $, encompassing known variants like second- and third-order Eulerian numbers.
• To establish a connection between the generalized Eulerian numbers and solutions of an autonomous first-order differential equation.
• To derive integral representations of special number sequences, including Bernoulli numbers, using the generating functions derived from these differential equations.
• To provide a unified framework for understanding the combinatorial and analytic properties of Eulerian-type numbers through differential equations and generating functions.

Proposed method

• Define a generalized sequence $ G(n,k) $ via a linear recurrence: $ G(n+1,k) = (n w_1 - n + k + 1)G(n,k) + (n w_2 - k + 1)G(n,k-1) $, with $ G(0,0)=1 $.
• Show that the row sums of $ G(n,k) $ form a polynomial in $ w_1 + w_2 $, given by $ \prod_{m=0}^{n-1} (m(w_1 + w_2) - m + 1) $.
• Associate the recurrence with an autonomous differential equation $ u'(z) = f(u) $, where $ f(u) $ depends on $ w_1 $ and $ w_2 $, and solve it explicitly for specific parameter choices.
• Construct the generating function $ g(u,w) $ from the solution of the differential equation, and derive its integral $ f(w) = \int_0^1 g(u,w) du $, which yields coefficients related to generalized Eulerian numbers.
• Use the Taylor expansion of $ f(w) $ to extract coefficients $ f_n $, which are shown to match the generalized Eulerian numbers $ G_n(u) $, and relate them to Bernoulli numbers via integral identities.
• Verify numerically that $ \int_0^1 G_n(u) du = \frac{B_{n+1}}{n+1} $ for second-order Eulerian numbers, suggesting a deep link to Bernoulli number theory.

Experimental results

Research questions

• RQ1 Can Eulerian numbers be generalized beyond the classical and second-order cases using a unified recurrence with real parameters $ w_1 $ and $ w_2 $?
• RQ2 How are solutions of an autonomous first-order differential equation related to the generating functions of generalized Eulerian numbers?
• RQ3 Can integral representations of special number sequences—especially Bernoulli numbers—be derived from such differential equations?
• RQ4 Is there a functional relationship between the generating function of generalized Eulerian numbers and the exponential generating function of Bernoulli numbers?
• RQ5 Does the integral $ \int_0^1 G_n(u) du $ yield $ \frac{B_{n+1}}{n+1} $ for second-order Eulerian numbers, as numerically observed for $ n \leq 9 $?

Key findings

• The generalized Eulerian numbers $ G(n,k) $ defined by the recurrence with parameters $ w_1 $ and $ w_2 $ reduce to classical Eulerian numbers when $ w_1 = w_2 = 1 $, second-order Eulerian numbers when $ w_1 = 1, w_2 = 2 $, and third-order Eulerian numbers when $ w_1 = 1, w_2 = 3 $.
• The row sums of $ G(n,k) $ are given by the polynomial $ \prod_{m=0}^{n-1} (m(w_1 + w_2) - m + 1) $, which generalizes the known row sum formulas for Eulerian numbers.
• For $ w_1 = \frac{1}{2}, w_2 = 1 $, the differential equation $ u'(z) = \sqrt{u}(u - 1) $ is solved explicitly, and its generating function leads to a closed-form integral representation of the generalized Eulerian numbers.
• For $ w_1 = 1, w_2 = 2 $, the differential equation $ u'(z) = u(u-1)^2 $ is solved implicitly, and the resulting generating function integral matches the exponential generating function of Bernoulli numbers.
• Numerical evidence supports the identity $ \int_0^1 G_n(u) du = \frac{B_{n+1}}{n+1} $ for second-order Eulerian numbers, suggesting a novel integral representation of Bernoulli numbers.
• The identity $ \sum_{k=0}^{n-1} (-1)^k \frac{\left<\!\genfrac{<}{>}{0pt}{}{n}{k}\!\right>}{\binom{2n+1}{k+1}} = 2B_{n+1} $ is derived as a consequence of the integral identity, linking combinatorics and special functions.