A note on Eulerian numbers and Toeplitz matrices

Abstract This note presents a new formula of Eulerian numbers derived from Toeplitz matrices via Riordan array approach.


Introduction
In 1958, Riordan [14] had been proposed the general problem of inverting combinatorial sums: an = n k= r n,k b k ⇐⇒ bn = n k= r * n,k a k .
A matrix representation a = Rb where a = (a , a , ...) T , b = (b , b , ...) T and R = (r n,k ) n,k≥ of these combinatorial sums is useful for inverting the sums. In particular, the combinatorial sums involving binomial coe cients can be represented by Riordan arrays. In general, Sprugnoli [16] showed that the sums involving the rows of a Riordan array can be performed by operating a suitable transformation on a generating function and then by extracting a coe cient from the resulting function.
In this paper, we consider combinatorial sums and inverting the sums involving Eulerian numbers together with the binomial coe cients of the form n + m − k. This matrix representation follows Appel matrix which is a Riordan array of a Toeplitz form. By using this approach, the paper demonstrates among others how the Riordan arrays can be applied to nd closed forms for combinatorial sums and inverting the sums. The inverse relationship shown in (1) in terms of Riordan arrays is studied in [6].
It is well known that Eulerian numbers de ned by (see, for example, Hsu and Shiue [8]) where A( , ) = and A(m, ) = for m > . For t = n ∈ N, n+m−k m ≠ implies that k ≤ n. Hence, (2) yields for n = , , . . ., which can be presented as a matrix form: where the Toeplitz matrix (see [1]) on the right-had side of the last equation is an Appel Riordan array generated by (d, t) with its rst column generating function as Riordan arrays are in nite, lower triangular matrices de ned by two generating functions. They form a group, called the Riordan group (see Shapiro et al. [15]). More formally, consider the set of formal power series , is the minimal number r ∈ N such that fr ≠ . The set of formal power series of order r is denoted by Fr. It is known that F is the set of invertible f.p.s. and F is the set of compositionally invertible f.p.s., that is, the f.p.s. f (t) for which the compositional inversef (t) or, in other words, d(t)h(t) k is the generating function for the entries of column k.
Let [f , f , f , . . .] T be a column vector with f (t) = n≥ fn t n . It is convenient to switch freely between a sequence, a sequence written as a column vector, and the ordinary generating function for that sequence. We then have the fundamental theorem of Riordan arrays (see, for example, [7,15]) It follows quickly that the usual row-by-column product of two Riordan arrays is also a Riordan array: The Riordan array I = ( , t) is everywhere 0 except for all 's on the main diagonal; it can be easily proved that I acts as an identity for this product, that is, ) be a Riordan array. Then its inverse is whereh(t) is the compositional inverse of h(t). In this way, the set R of proper Riordan arrays forms a group (see [15]). From (8), the inverse of (d, t) is Thus, from (4), we obtain which give a new proof of the following formula (other proofs can be seen, for example, [3,12,13]) for evaluating Eulerian numbers. (2). Then they can be calculated by the formula

Proposition 1.1. (New proof for a well known result) Let A(m, n), m > , be the Eulerian numbers de ned by
Proof. The rst equation of (10) follows from (9). The second equation of (10) is obvious because m > .
Formula (10) might be called the decomposition of Eulerain numbers A(m, n) in terms of integer powers. Obviously, (3) and (10) are a pair of inverse formulas. In this short note, we shall use the same approach to construct the formulas of the decompositions of generalized Eulerian numbers shown in [8], pn-associated Eulerian numbers de ned in [12], and Carlitz numbers and non-central Carlitz numbers discussed in [2].

Main results
For any real or complex number θ we de ne Recall that Howard's degenerate weighted Stirling numbers (see Howard [10]) S(p, j, λ|θ) ( ≤ j ≤ p) may be de ned by with S(p, j, λ) = S(p, j, λ| ). Following the relationship between Eulerian numbers and Stirling numbers, [8] de ne generalized Eulerian numbers Ap(x, λ|θ) as with the deduced case A particular kind of generalized Stirling numbers called Dickson-Stirling numbers is de ned by In Proposition of [8] the following two formulas are given Theorem 2.1. Let (x + λ|θ)p, A(p, j, λ|θ), A(p, j, α), and Dp(x, α) be de ned by (11), (12), (13), and (14), respectively. Then we have the following decomposition formulas for the generalized Eulerian numbers A(p, n, λ|θ) and theirs special forms A(p, n, α).
Formulas (15) and (17) and (16) and (18) where Let {pn(t), n = , , . . .} be a class of polynomials with degree of pn(t) being n and p (t) = . The coe cients A n,k of the factorial series expansion of pn(t) namely are called the pn−associated Eulerian numbers in [12]. Particularly, if pn(t) = (t + r) n , where r ∈ R, then the expansion de nes the Non-central Eulerian numbers A(n, k, r). By using the same approach shown in Theorem 2.1, we obtain the following decomposition formula for A n,k and A(n, k, r), which provides their expressions in terms of pn(t) and (t + r) n , respectively. (20) and (21), respectively. Then

Theorem 2.2. (see also Proposition 2.1 of [12] and Section 14.7 of [2]) Let A(n, k) and A(n, k, r) be de ned by
where B(n, k; s, r) are called non-central Carlitz numbers. The inverse formulas of (25) and (26) are presented below. It worth noticing that the approach we made use above is not limited to the expansion in terms of t+n−k n . For instance, replacing t by −t and m to n in (2) and noting we obtain In (27) Therefore, the Eulerian numbers A(n, k) can be presented in terms of t n , t = , , . . ., as It worth pointing out, the matrix algorithm in the computation of number sequences can be extended to non-Toeplitz matrix. For instance, Kelly [11] present the following identity for integer power sums S j = j + j + · · · + n j + k j= k + j S j (n) = (n + ) k+ , and use use an elementary method to prove it. If we consider (28) as an equation to de ne S j (n), then a recessive process can be applied to evaluate S j (n) (see [11]). Following the matrix approach presented before, by substituting k = , , . . ., we may evaluate S j (n) from the lower triangular system Here, the matrix is no longer a Toeplitz matrix, which inverse will be studied in another paper. We now consider the problem related to (28): FindŜ j (n) de ned by for k = , , . . .. For instance, if E(n + , k + ) = (n + ) k+ , thenŜ k (n) = n k . Riordan array approach can be applied to evaluate other famous numbers. Note that in [5] one of the author presented two formulas for the computation of generalized Stirling numbers by using two sequence characterizations (see [4,9]) of Riordan arrays.