Context-Free Grammars for Several Triangular Arrays

: In this paper, we present a uniﬁed grammatical interpretation of the numbers that satisfy a kind of four-term recurrence relation, including the Bell triangle, the coefﬁcients of modiﬁed Hermite polynomials, and the Bessel polynomials. Additionally, as an application, a criterion for real zeros of row-generating polynomials is also presented.


Introduction
Let A denote an alphabet, the letters of which are considered as independent commutative indeterminates.Then, the context-free grammar G over A is defined as a set of replacement rules that substitute the letters in A with formal functions on A. The formal derivative D is a linear operator, which is defined relative to a context-free grammar G (see [1]).For example, for A = {u, v} and G = {u → uv, v → v}, then D(u) = uv, D 2 (u) = u(v + v 2 ), D n (u) = u ∑ n k=1 S(n, k)v k , where S(n, k) is the Stirling number of the second kind, i.e., the number of ways to partition [n] into k blocks.
In [2], Hao, Wang, and Yang presented a grammatical interpretation of the numbers T(n, k) that satisfy the following three-term recurrence relation: Very recently, there is a large literature devoted to the numbers t(n, k) that satisfy the following four-term recurrence relation (see [3][4][5][6][7]): with t 0,0 = 1 and t n,k = 0, unless 0 ≤ k ≤ n.For example, Ma [8] showed that if G = {x → xy, y → yz, z → y 2 }, then D n (x 2 ) = x 2 ∑ n k=0 R(n + 1, k)y k z n−k , where R(n, k) is the number of permutations in S n with k alternating runs, and it satisfies the recurrence relation Clearly, a(n, k) is the number of set partitions of {1, 2, . . ., n} in which exactly k of the blocks have been distinguished.The numbers a(n, k) satisfy the recurrence relation with a(0, 0) = 1, a(0, k) = 0 for k = 0 (see [9,10]).The triangular array {a(n, k)} n,k is known as the classical Bell triangle and is given as follows: , which implies that the first column of the triangle array is made up of the Bell numbers B n .A natural question is whether there exists a grammatical interpretation of the numbers a(n, k).
This paper is motivated by exploring the grammatical interpretation of the triangular array {B(n, k)} 0≤k≤n that satisfies the following four-term recurrence relation where a i , b i , and c are integers for 1 In Section 2, we present grammatical interpretations of the triangular array {B(n, k)}.
In Section 3, we present grammatical interpretations of several combinatorial sequences, including the Bell triangle, the modified Hermite polynomials, the Bessel polynomials, and so on.In Section 4, we show the result of the real-rootedness of row-generating functions for {B(n, k)}, and apply the proposed criteria to the Bell triangular array as an example.

Grammatical Interpretations of the Triangular Array B(n, k)
We now present the first main result of this paper.
Theorem 1. Suppose that a i , b i , and c are integers for Then, we have where the coefficients B(n, k) satisfy the recurrence relation (3).
Proof.Note that D(I) = (a 2 + a 3 )IX + b 3 IY.Suppose that (4) holds for n.Then, by induction, we obtain Applying the rules of G, we can derive Collate and merge similar items Extracting the coefficient of IX k Y n+1−k , we obtain (3).This completes the proof.
Along the same lines of the proof of Theorem 1, one can easily derive the following result.
Then, we have where M(n, k) satisfy the following five-term recursive relation: where a i , b i , c, and d are integers for 1 ≤ i ≤ 3.

The Bell Triangle
The Bell triangle was proposed by Aigner [9] to provide a characterization of the sequence of Bell numbers by means of the determinants of Hankel matrices.As a special case of Theorem 1, we now present a grammatical interpretations of the Bell triangle.
From Leibniz's formula, we obtain the following corollary:

On the Coefficients of Modified Hermite Polynomials
The modified Hermite polynomials have the following form: It should be noted that the numbers T(n, k) are the coefficients of the modified Hermite polynomials (see A099174 [11]) and Using Theorem 1, we obtain the following proposition. ).

The Bessel Polynomials
As a well-known orthogonal sequence of polynomials, the Bessel polynomials y n (x) were introduced by Krall and Frink in [12], which can be defined as the polynomial solutions of the second-order differential equation After that, the Bessel polynomials have been extensively studied and applied (see [13][14][15]).Moreover, the polynomials y n (x) can be generated by using the Rodrigues formula (see [11] [A001498]): Explicitly, we can obtain It is easy to verify that The polynomials y n (x) satisfy the recurrence relation The first three Bessel Polynomials are expressed as We present here a grammatical characterization of the Bessel polynomials y n (x).

The Exponential Riordan Array
Definition 1 (see [16]).The exponential Riordan group G is a set of infinite lower-triangular integer matrices, and each matrix in G is defined by a pair of generating function g(x) = g 0 + The associated matrix is the matrix whose i-th column has exponential generating function g(x) f (x) i /i! (columns marked from 0).The matrix corresponding to the pair f , g is defined by [g, f ].
. From Leibniz's formula, we obtain the following corollary: Corollary 5.For n ≥ 0, we have In Table 1, we list some combinatorial sequences that satisfy (3).More examples can be found in similar tables in [17][18][19].By using Theorem 1, we give the grammatical interpretation of the corresponding sequences, so that we can obtain more convolution formulas.

Real Rootedness
In this section, as an application, we will pay attention to the property of real roots of the row-generating functions in the array {B(n, k)} 0≤k≤n in (3).For the sake of proving our results, some known results should be introduced beforehand.
Let {P n (x)} denote a Sturm sequence, which is a sequence of standard polynomials meeting the condition of deg P n = n and P n−1 (r)P n+1 (r) < 0 whenever P n (r) = 0 and n ≥ 1.Let RZ represent the set of polynomials with only real roots.{P n (x)} is known as a generalized Sturm sequence (GSS) if P n ∈ RZ and zeros of P n (x) are separated by those of P n−1 (x) for n ≥ 1.As a special case of Corollary 2.4 in Liu and Wang [20] (also see Zhu, Yeh, and Lu [7]), the following result provides a unified method to many polynomials with only real zeros.
For nonnegative array B(n, k), which satisfies the recurrence relation (3), it is sufficient to assume that, for n ≥ 1, as the row-generating functions of B(n, k).Thus, B 0 (x) = 1 and Moreover, it turns out that B n (x) follows from the recurrence relation (3) as Theorem 2. Let {B(n, k)} n,k≥0 be the array defined in (3).Assume that b 2 = a 2 + c.Then, we have the following results: (i) There exist polynomials A n (x) for n ≥ 0 such that where A n (x)satisfies the recurrence relation It can be proven that (i) holds by induction on n as follows.
As n = 1, we can obtain Thus, we have By the induction hypothesis, it now turns out that ).
It follows from that recurrence relation (7) that, for n ≥ 2, Thus, for n ≥ 1, we can prove that ).
(ii) Evidently, in light of (i), B n (x) forms a generalized Sturm sequence if and only if (iff) A n (x) forms a generalized Sturm sequence.The nonnegativity of the coefficients for A n (x) needs to be considered firstly.Let A n (x) = ∑ n k=0 A(n, k)x k for n ≥ 0.Then, according to the recurrence relation (7), we obtain for n ≥ 1.Following from the nonnegativity of {B(n, k)} n,k≥0 , it holds a 1 + a 2 ≥ 0, a 1 ≥ 0, a 2 + a 3 ≥ 0 Furthermore, by the hypothesis condition, we obtain Thus, {B(n, k)} n,k≥0 is a nonnegative array.According to the recurrence relation ( 7) and Lemma 1, we can conclude that the polynomials A n (x) form a generalized Sturm sequence if a 2 ≤ 0. For the same reason, the polynomials B n (x) form a generalized Sturm sequence.
For example, the row-generating function of the Bell triangle a(n, k) in Section 3 is a n (x) = ∑ n k=0 a(n, k)x k .Then, the polynomials satisfy a n (x) = (1 + x)a n−1 (x) + (1 + x)a n−1 (x), with a 0 (x) = 1.Using Theorem 2 (i), there exists an array A(n, k) such that where A n (x) for n ≥ 1 satisfies the recurrence relation where A 0 (x) = 1 and A 1 (x) = 1.Obviously, A(n, k) = S(n, n − k) for n ≥ 1. Applying Theorem 2 (ii), it can be proven that {a n (x)} for n ≥ 0 is a generalized Sturm sequence.