1 Introduction↩︎

Orthogonal polynomials are classical objects that play a central role in a wide range of areas of mathematics. Since the pioneering work of Flajolet [1] and Viennot [2], [3] in the 1980s, researchers have found them to possess many interesting combinatorial properties. See [3]–[5] and references therein for more details.

Orthogonal polynomials can be defined as a sequence \(\{p_n(x)\}_{n\ge0}\) of polynomials with \(\deg p_n(x) =n\) such that there is a linear functional \(\mathcal{L}\) satisfying \(\mathcal{L}(p_n(x)p_m(x)) = K_n \delta_{n,m}\), where \(K_n\ne 0\). Their moments \(\{\sigma_n\}_{n\ge0}\) are defined by \[\sigma_n = \frac{\mathcal{L}(x^n)}{\mathcal{L}(1)}.\] More generally, the mixed moments \(\{\sigma_{n,k}\}_{n,k\ge 0}\) are defined by \[\sigma_{n,k} = \frac{\mathcal{L}(x^np_k(x))}{\mathcal{L}(p_k(x)^2)}.\] By the orthogonality, we have \[x^n = \sum_{k=0}^{n} \sigma_{n,k} p_k(x),\] which can be taken as the definition of the mixed moments. We also define the coefficients (or dual mixed moments) \(\{\nu_{n,k}\}_{n,k\ge 0}\) by \[p_n(x) = \sum_{k=0}^{n} \nu_{n,k} x^k.\]

Viennot [2], [3] found a combinatorial interpretation for \(\sigma_{n,k}\) using Motzkin paths. To illustrate, suppose that \(\{ p_n(x) \}_{n\ge 0}\) is a sequence of monic orthogonal polynomials satisfying the 3-term recurrence \[p_{n+1}(x) = (x-b_n)p_n(x)-\lambda_{n}p_{n-1}(x).\] A Motzkin path is a path on \(\mathbb{Z}^2\) from \((a, b)\) to \((c,d)\) consisting of up steps \((1,1)\), horizontal steps \((1,0)\), and down steps \((1,-1)\) that never go below the line \(y = 0\). The weight \(w(p)\) of a Motzkin path \(p\) is the product of the weights of the steps in \(p\), where the weight of an up step is always \(1\), the weight of a horizontal step at height \(k\) is \(b_k\), and the weight of a down step starting at height \(k\) is \(\lambda_k\). Viennot showed that \(\sigma_{n,k}\) is the sum of \(w(p)\) for all Motzkin paths \(p\) from \((0,0)\) to \((n,k)\). For example, \[\sigma_{3,1} = b_0^2+b_0b_1+b_1^2+\lambda_1+\lambda_2\] is the generating function for all Motzkin paths from \((0,0)\) to \((3,1)\) as shown in 1. Using this combinatorial interpretation for \(\sigma_{n,k}\) one can show that the moments of Hermite, Charlier, and Laguerre polynomials are generating functions for perfect matchings, set partitions, and permutations, respectively. Viennot also found a combinatorial model for \(\nu_{n,k}\) using lattice paths called Favard paths.

There is a natural way to extend univariate orthogonal polynomials to multivariate orthogonal polynomials using bialternant formulas. In the expansion of the multivariate orthogonal polynomials (resp. Schur polynomials) in terms of Schur polynomials (resp. multivariate orthogonal polynomials), the coefficients are given as determinants of \(\nu_{n,k}\) (resp. \(\sigma_{n,k}\)). Therefore one may ask whether the Lindström–Gessel–Viennot (LGV) lemma [6] (see also [7]) can be used to find an interpretation for the Schur coefficients using nonintersecting paths. However, neither Motzkin paths nor Favard paths give such a model, since their underlying graphs are not planar. One of the motivations of this paper is to find a planar graph that provides a lattice path interpretation for the mixed moments, so that it can be extended naturally to multivariate orthogonal polynomials.

Corteel and Kim [8] introduced the lecture hall graph in which paths are in natural bijection with lecture hall partitions and anti-lecture hall compositions. Lecture hall partitions with \(n\) nonnegative parts were originally defined in the enumeration of certain affine permutations coming from the affine Coxeter group \(\tilde{C}_n\) [9]. They also have many connections with other combinatorial objects; see [10] for example.

Corteel and Kim [8] found a path model in the lecture hall graph for both mixed moments and coefficients of the little \(q\)-Jacobi polynomials. A nice feature of their combinatorial model is that it can be extended to give a tableau model for multivariate little \(q\)-Jacobi polynomials. Their discovery of the combinatorial model for little \(q\)-Jacobi polynomials, however, was somewhat by accident, because the coefficient \(\nu_{n,k}\) of the little \(q\)-Jacobi polynomial happens to be equal to a formula, previously established in [11], for a generating function for truncated lecture hall partitions.

The goal of this paper is to develop a general interpretation of the mixed moments \(\sigma_{n,k}\) and the coefficients \(\nu_{n,k}\) of orthogonal polynomials as generating functions for paths in the lecture hall graph. We show that it is not a coincidence that paths on the lecture hall graph and little \(q\)-Jacobi polynomials are related. In fact, we prove that the relation is much stronger and that the orthogonal polynomials in the whole \(q\)-Askey scheme [12] can be studied this way.

We give a simple algorithm to construct a possible candidate giving a lecture hall path interpretation for both coefficients and mixed moments of orthogonal polynomials. This algorithm allows us to rediscover the lecture hall path interpretations for the little \(q\)-Jacobi polynomials without any prior knowledge of the generating function for truncated lecture hall partitions.

Our main result may be summarized in the following “meta” theorem. This is deliberately quite vague for now. The purpose of our manuscript is to develop the tools and results necessary to make it precise.

For example, 45 gives a combinatorial model for the mixed moments of the Askey–Wilson polynomials, and 9 together with this provides a model for their coefficients.

For certain polynomials in the \(q\)-Askey scheme, we can give different models: some will be just on the lecture hall graph of finite height; others will be on the graph of infinite height. For the polynomials in the \(q\)-Askey scheme, the models with finite height have the benefit that we can take the limit \(q\rightarrow 1\) and those with infinite rows give a combinatorial model for the formal power series expansion of the mixed moments and the coefficients.

Here are some advantages of our lecture hall graph models. The first advantage of this method is that it gives a combinatorial model for the mixed moments and the coefficients with the same graph model, where southeast paths are used for mixed moments and northeast paths without consecutive east steps are used for coefficients. Note that Viennot’s combinatorial models do not have this uniform property because Motzkin paths and Favard paths have different underlying graphs.

The second advantage is that it has a nice relationship with the \(q\)-Askey scheme. We start with the Stieltjes–Wigert polynomials and we establish simple lemmas that let us go up in the scheme. The only family that needs to be treated as a special case is that of the continuous \(q\)-Hermite polynomials.

The third advantage is that, since the lecture hall graph is planar, our combinatorial models can be naturally extended to study multivariate orthogonal polynomials using the Lindström–Gessel–Viennot lemma, which is not possible for Viennot’s model. This was done in [8] for little \(q\)-Jacobi polynomials and we can now generalize this to the whole \(q\)-Askey scheme. In a forthcoming paper, we will give applications of our combinatorial models to multivariate versions of orthogonal polynomials, total positivity, and random matrix theory.

The last but not least advantage is that it gives a combinatorial model for the mixed moments of Askey–Wilson polynomials. Using this model we give the first combinatorial proof of the symmetry of the parameters \(a,b,c,d\) in the Askey–Wilson polynomials, which until now has been understood only analytically. We note that there is a combinatorial model, called staircase tableaux, for moments of Askey–Wilson polynomials with some reparametrization of \(a,b,c,d\) to \(\alpha,\beta,\gamma,\delta\); see [13] and [14]. It is known that the new parameters \(\alpha,\beta,\gamma,\delta\) are also symmetric, but finding a combinatorial proof is still open.

The paper is organized as follows. In 2 we define the lecture hall graph and we introduce mixed moments relative to different bases. In 3 we prove useful properties of the lecture hall graph, which will be used throughout the paper. In 4 we find weight systems for the mixed moments of Stieltjes–Wigert polynomials, \(q\)-Bessel polynomials, little \(q\)-Jacobi polynomials, big \(q\)-Jacobi polynomials, and Askey–Wilson polynomials, using a bootstrapping method. In 5 we give another bootstrapping method to find a lecture hall graph model for mixed moments of Askey–Wilson polynomials relative to continuous \(q\)-Hermite polynomials. In 6 we give another combinatorial model for mixed moments of Askey–Wilson polynomials. Using this model we give the first combinatorial proof of the symmetry of \(a,b,c,d\) in Askey–Wilson polynomials.

Finally, we remark that our method can also be applied to all orthogonal polynomials in the \(q\)-Askey scheme and in the Askey scheme. The remaining orthogonal polynomials will be covered in a forthcoming paper.

2 Preliminaries↩︎

In this section we give basic definitions and lemmas on the lecture hall graph. We then define mixed moments and coefficients of orthogonal polynomials relative to other bases. We will follow the standard notation of basic hypergeometric series; see for example [15].

2.1 Lecture hall graphs↩︎

The lecture hall graph was first introduced by Corteel and Kim [8] in their study of little \(q\)-Jacobi polynomials. It has further been studied in [16] and [17].

For nonnegative integers \(t,i,j\) with \(j\le i\), we denote \[v^t_{i,j} = \left(i,t+ \frac{j}{i+1}\right) \in \mathbb{Z}\times\mathbb{Q}.\]

We consider three kinds of steps:

• an east step is a directed edge of the form \((v^t_{i,j}, v^t_{i+1,j})\) in \(\mathcal{G}\),

• a north step is a directed edge of the form \((v^t_{i,j}, v^t_{i,j+1})\) or \((v^t_{i,i}, v^{t+1}_{i,0})\),

• a south step is a directed edge of the form \((v^t_{i,j}, v^t_{i,j-1})\) or \((v^t_{i,0}, v^{t-1}_{i,i})\).

For two vertices \(u\) and \(v\) in \(\mathcal{G}\), a path from \(u\) to \(v\) is a sequence \((v_0,v_1,\dots,v_n)\) of vertices of \(\mathcal{G}\) such that \(v_0=u\), \(v_n=v\), and each \((v_{i}, v_{i+1})\) is an east, north, or south step. We also consider a path from \(u\in \mathcal{G}\) to \(v=(a,\infty)\), which is a sequence \((v_0,v_1,v_2,\ldots)\) of vertices of \(\mathcal{G}\) such that \(v_0=u\), \(\lim_{n\to \infty}v_n=v\), and each \((v_{i}, v_{i+1})\) is an east, north, or south step. Similarly, a path from \(u=(a,\infty)\) to \(v\in \mathcal{G}\) is a sequence \((\ldots, v_{2},v_{1},v_0)\) of vertices of \(\mathcal{G}\) such that \(v_0=v\), \(\lim_{n\to \infty}v_{n}=u\), and each \((v_{i+1}, v_{i})\) is an east, north, or south step.

Let \(\operatorname{SE}(u\to v)\) denote the set of paths from \(u\) to \(v\) in \(\mathcal{G}\) consisting of south and east steps. We also define \(\operatorname{NE}^*(u\to v)\) to be the set of paths from \(u\) to \(v\) in \(\mathcal{G}\) consisting of north and east steps with no consecutive east steps. Here, consecutive east steps means a pair \((s_1,s_2)\) of steps of the form \(s_1=(v^t_{i,j},v^t_{i+1,j})\) and \(s_2=(v^t_{i+1,j},v^t_{i+2,j})\). For example, in 3, the path \(p_1\in \operatorname{SE}((1,\infty)\to (5,0))\) has consecutive east steps \((v^2_{2,2}, v^2_{3,2})\) and \((v^2_{3,2}, v^2_{4,2})\) but the path \(p_2\in \operatorname{NE}^*((1,0)\to (5,\infty))\) does not have any consecutive east steps.

Paths in \(\operatorname{SE}(u\to v)\) or \(\operatorname{NE}^*(u\to v)\) are closely related to lecture hall partitions. To describe their connection we need some definitions.

A lecture hall partition is a nonnegative integer sequence \(\lambda=(\lambda_{1},\dots,\lambda_n)\) such that \[\frac{\lambda_{1}}{n} \ge \frac{\lambda_{2}}{n-1} \ge \cdots \ge \frac{\lambda_{n}}{1}.\] An anti-lecture hall composition is a nonnegative integer sequence \(\alpha=(\alpha_{1},\dots,\alpha_n)\) such that \[\frac{\alpha_{1}}{1} \ge \frac{\alpha_{2}}{2} \ge \cdots \ge \frac{\alpha_{n}}{n}.\] As variations of the above objects, we consider the set \(AL_{n,k}\) of truncated anti-lecture hall compositions and the set \(RL^*_{n,k}\) of truncated strict reverse lecture hall partitions defined by \[\begin{align} AL_{n,k} &= \left\{ (\alpha_{k+1},\dots,\alpha_n) : \frac{\alpha_{k+1}}{k+1} \ge \frac{\alpha_{k+2}}{k+2} \ge \cdots \ge \frac{\alpha_{n}}{n} \right\}, \\ RL^*_{n,k} &= \left\{ (\rho_{k+1},\dots,\rho_n): \frac{\rho_{k+1}}{k+1} < \frac{\rho_{k+2}}{k+2} < \cdots < \frac{\rho_{n}}{n} \right\}. \end{align}\]

Now we define a map \(\phi\), which sends a path to a sequence of integers. Suppose that \(p\) is a path from \((k,a)\) to \((n,b)\) for some \(k,n\in \mathbb{Z}_{\ge0}\) and \(a,b\in \mathbb{Z}_{\ge0}\cup\{\infty\}\). For \(i=k+1,\dots,n\), suppose that the east step of \(p\) between \(x=i-1\) and \(x=i\) is the \(\alpha_i\)th east step among all east steps between \(x=i-1\) and \(x=i\) from the bottom. In other words, if the east step of \(p\) between \(x=i-1\) and \(x=i\) is \((v^t_{i-1,j},v^t_{i,j})\) in \(\mathcal{G}\), then \(\alpha_i = ti + j\). We define \(\phi(p) = (\alpha_{k+1},\dots,\alpha_n)\). For example, if \(p_1\) and \(p_2\) are the paths in 3, then \(\phi(p_1) = (4,5,6,3)\) and \(\phi(p_2) = (0,1,7,12)\). This map gives natural bijections from paths in a lecture hall graph to truncated anti-lecture hall compositions and truncated reverse lecture hall partitions.

Our main goal is to find combinatorial models for mixed moments \(\sigma_{n,k}\) and coefficients \(\nu_{n,k}\) of orthogonal polynomials. We will see later in 9 that a combinatorial model for \(\sigma_{n,k}\) using \(\operatorname{SE}(u\to v)\) immediately gives a combinatorial model for \(\nu_{n,k}\) using \(\operatorname{NE}^*(u\to v)\), and vice versa. Hence, in this paper we will mostly consider the set \(\operatorname{SE}(u\to v)\) rather than \(\operatorname{NE}^*(u\to v)\). For brevity, we will write \(p:u\to v\) to mean \(p\in \operatorname{SE}(u\to v)\).

A weight system is a function \(w\) that assigns a weight \(w(s)\) to each east step \(s=(v^t_{i,j}, v^t_{i+1,j})\) in \(\mathcal{G}\). We denote \[w(t;i,j) := w(v^t_{i,j},v^t_{i+1,j}).\] In other words, \(w(t;i,j)\) is the weight of the \(j\)th east step from the bottom, where the bottommost one is the \(0\)th step, among the east steps in the region \(\{(x,y): i\le x\le i+1, t\le y<t+1 \}\); see 4. Given a weight system \(w\), the weight \(w(p)\) of a path \(p\) is defined to be the product of the weights of all east steps in \(p\).

The following definition will be used throughout this paper.

By definition, we have \(h^w_{n,k}=e^w_{n,k}=0\) if \(n<k\) and \(h^w_{n,n}=e^w_{n,n}=1\). Note that \(h^w_{n,k}\) and \(e^w_{n,k}\) generalize the homogeneous symmetric function \[h_n(x_0,x_1,\ldots) = \sum_{0\le i_0\le i_1 \le \cdots \le i_{n-1}} x_{i_0} x_{i_1} \cdots x_{i_{n-1}}\] and the elementary symmetric function \[e_n(x_0,x_1,\ldots) = \sum_{0\le i_0< i_1 < \cdots < i_{n-1}} x_{i_0} x_{i_1} \cdots x_{i_{n-1}}\] in the following sense: if \(w\) is the weight system defined by \[w(t;i,j) = \begin{cases} x_t & if \( j=0 \),\\ 0 & otherwise, \end{cases}\] then we have \[h_n(x_0,x_1,\ldots) = h^w_{n,0}, \qquad e_n(x_0,x_1,\ldots) = e^w_{n,0}.\]

The quantities \(h^w_{n,k}\) and \(e^w_{n,k}\) are dual to each other in the following sense.

The following simple lemma will be used frequently in this paper.

2.2 Mixed moments relative to other bases↩︎

Recall that the mixed moments \(\{\sigma_{n,k}\}_{n,k\ge 0}\) of orthogonal polynomials \(\{p_n(x)\}_{n\ge0}\) with respect to a linear functional \(\mathcal{L}\) are defined by \[\sigma_{n,k} = \frac{\mathcal{L}(x^np_k(x))}{\mathcal{L}(p_k(x)^2)},\] or equivalently, \[x^n = \sum_{k=0}^{n} \sigma_{n,k} p_k(x).\] By abuse of terminology, we extend the definition of the mixed moments \(\sigma_{n,k}\) to polynomials that are not necessarily orthogonal polynomials.

More generally, we also define mixed moments and coefficients by choosing a basis other than \(\{x^n\}_{n\ge0}\).

Note that the original mixed moments \(\{\sigma_{n,k}\}_{n,k\ge 0}\) of \(\{p_n(x)\}_{n\ge0}\) are the mixed moments of \(\{p_n(x)\}_{n\ge0}\) relative to the standard basis polynomials \(\{x^n\}_{n\ge0}\). Note also that, by definition, the mixed moments \(\{\sigma_{n,k}\}_{n,k\ge 0}\) and the coefficients \(\{\nu_{n,k}\}_{n,k\ge 0}\) of \(\{p_n(x)\}_{n\ge0}\) relative to \(\{q_n(x)\}_{n\ge0}\) are the entries of the change-of-basis matrices between the two bases \(\{p_n(x)\}_{n\ge0}\) and \(\{q_n(x)\}_{n\ge0}\) of the space of polynomials. Therefore the two matrices \((\sigma_{n,k})_{n,k=0}^\infty\) and \((\nu_{n,k})_{n,k=0}^\infty\) are inverses of each other. By this fact and 5, we obtain the following proposition.

9 implies that if we have a lecture hall graph model for \(\sigma_{n,k}\) then the same weight system also gives a lecture hall graph model for \((-1)^{n-k}\nu_{n,k}\) and vice versa. Therefore we will henceforth only focus on finding a lecture hall graph model for the mixed moments \(\sigma_{n,k}\).

By the following lemma with \(b_n(x) = x^n\), we can use the mixed moments \(\{\tilde{\sigma}_{n,k}\}_{n,k\ge 0}\) relative to \(\{q_n(x)\}_{n\ge0}\) as an intermediate step to study the original mixed moments \(\{\sigma_{n,k}\}_{n,k\ge 0}\).

We note that an equivalent statement of 10, for the case that \(\{p_n(x)\}_{n\ge0}\) and \(\{q_n(x)\}_{n\ge0}\) are orthogonal polynomials and \(b_n(x) = x^n\), has been proved in [18]. 10 allows us to find a lecture hall graph model for \(\sigma_{n,k}\) using those for \(\tau_{n,k}\) and \(\tilde{\sigma}_{n,k}\). To give a precise statement we introduce some definitions.

For a sequence \(d=(d_0,d_1,d_2,\ldots)\), define \[(x|d)^n = (x-d_0)(x-d_1) \cdots (x-d_{n-1}).\] We will call the mixed moments \(\tilde{\sigma}_{n,k}\) relative to \(\{(x|d)^n\}_{n\ge0}\) the factorial mixed moments. In most cases we will consider the original mixed moments \(\sigma_{n,k}\), the factorial mixed moments \(\tilde{\sigma}_{n,k}\) for some \((x|d)_n\), and the mixed moments relative to the continuous \(q\)-Hermite polynomials.

3 Properties of weight systems↩︎

In this section we prove useful properties of weight systems of height \(1\) and weight systems of infinite height. Firstly, we consider weight systems of height \(1\). In this case our lecture hall graph model is equivalent to a special case of the planar network of Fomin and Zelevinsky [19]. Using their result [19], given a lower unitriangular matrix \((\sigma_{n,k})_{n,k=0}^\infty\), one can deduce that there is a unique weight system \(w\) of height \(1\) satisfying \(h^w_{n,k} = \sigma_{n,k}\). We give an explicit formula for the weight system \(w\) using minors of the matrix \((\sigma_{n,k})_{n,k=0}^\infty\).

In the case of weight systems of infinite height, there can be many weight systems \(w\) satisfying \(h^w_{n,k} = \sigma_{n,k}\). We provide a simple algorithm to discover a weight system of infinite height. Although our algorithm does not always produce a correct weight system, it does provide a valid one for all orthogonal polynomials in the \(q\)-Askey scheme except for the case of the continuous \(q\)-Hermite polynomials. In particular, this algorithm allows us to discover systematically the weight system for the mixed moments of little \(q\)-Jacobi polynomials in [8], which was first discovered “by accident” when they happened to come across the results in lecture hall partitions [11]. Furthermore, this algorithm can also be used to discover weight systems for big \(q\)-Jacobi polynomials and Askey–Wilson polynomials in this paper. We also provide various recurrences for weight systems which can be used to prove that a given weight system \(w\) indeed satisfies \(h^w_{n,k} = \sigma_{n,k}\).

3.1 Weight systems of height 1↩︎

In this subsection we will show that weight systems of height 1 are particularly useful because their weights are uniquely determined by \(h^w_{n,k}\) and they can be used to build other weight systems. Observe that the lecture hall graph of height \(1\) can be identified with a staircase grid as shown in 5. This implies that the total number of paths \(p:(k,1) \to (n,0)\) is \(\binom{n}{k}\).

In order to show the uniqueness for a weight system of height \(1\), we introduce some notation. Given a matrix \(A=(a_{i,j})_{i,j=0}^\infty\), let \(A_{r}(i,j)\) denote the minor of \(A\) with row indices \(i,i+1,\ldots,i+r-1\) and column indices \(j,j+1,\ldots,j+r-1\).

The following proposition shows that given an array \(\{a_{n,k}\}\), finding a height 1 weight system \(w\) with \(h^w_{n,k} = a_{n,k}\) is equivalent to the same problem with \(e^w_{n,k} = a_{n,k}\). Hence, by 13, there is also a unique weight system \(w\) of height \(1\) such that \(e^w_{n,k} = a_{n,k}\).

The following two lemmas give recurrences for \(h^w_{n,k}\), which can be used to prove an identity of the form \(\sigma_{n,k} = h^w_{n,k}\).

Using 18 we can prove the following proposition, which states that a weight system for mixed moments can be obtained from a weight system for factorial mixed moments by adding a weight system of height \(1\) at the bottom.

3.2 Weight systems of infinite height↩︎

In this subsection we consider weight systems of infinite height. To see the difference between weight systems of infinite height and those of height \(1\), suppose that given a triangular array \(\{a_{n,k}\}_{n\ge k\ge0}\) we want to find a weight system \(w\) such that \(h_{n,k}^w = a_{n,k}\). If \(w\) is a weight system of height \(1\), then it is uniquely determined and we can use the formula in 13 to get the weight system. If, on the other hand, \(w\) is of infinite height, the weight \(w\) is not uniquely determined in general. How can we find such a weight system then? We propose the following algorithm to construct a possible candidate \(w\).

It may seem too optimistic to hope that such a simple algorithm provides a correct weight system. However, this algorithm gives a valid weight system of infinite height for all orthogonal polynomials in the \(q\)-Askey scheme except the continuous \(q\)-Hermite polynomials. Once we have a candidate weight system \(w\) explicitly constructed via this algorithm, we can prove the identity \(h_{n,k}^w = a_{n,k}\) using recursions such as the one in the following lemma.

If there is a weight system \(w_1\) of height \(1\) and a weight system \(w\) of infinite height with the “shifting property”, which is given in ?? below, we can construct a weight system of any height.

4 A bootstrapping method from Stieltjes–Wigert to Askey–Wilson↩︎

In this section, we construct lecture hall graph models for the mixed moments of Stieltjes–Wigert polynomials, \(q\)-Bessel polynomials, little \(q\)-Jacobi polynomials, big \(q\)-Jacobi polynomials, and Askey–Wilson polynomials. We first find a lecture hall graph model of height \(1\) for the mixed moments of Stieltjes–Wigert polynomials, and models of height \(1\) and infinite height for \(q\)-Bessel polynomials. We then prove “expanding lemmas” (31 and 32), which turn a weight system of height \(1\) to a weight system of height \(2\). Using the expanding lemmas, we obtain lecture hall graph models of height \(1\) and infinite height for little \(q\)-Jacobi polynomials, big \(q\)-Jacobi polynomials, and Askey–Wilson polynomials.

4.1 Stieltjes–Wigert↩︎

The monic Stieltjes–Wigert polynomials are defined by \[S_n(x;q) = (-1)^{n} q^{-n^2} {}_{1}\phi_{1} \left( \begin{matrix} q^{-n}\\ 0\\ \end{matrix} ; q,-q^{n+1}x \right) .\] The mixed moments \(\sigma_{n,k}\) and the coefficients \(\nu_{n,k}\) of Stieltjes–Wigert polynomials are given by \[\begin{align} \sigma_{n,k} &= q^{k^2-n^2+\binom{n-k}{2}} \genfrac{[}{]}{0pt}{}{n}{k}_{q}, \\ \nu_{n,k} &= (-1)^{n-k} q^{k^2-n^2} \genfrac{[}{]}{0pt}{}{n}{k}_{q}. \end{align}\] Hence the mixed moments of Stieltjes–Wigert polynomials are \(q\)-binomial coefficients with some factors. The \(q\)-binomial coefficients satisfy the following recurrences: \[\label{eq:rec-q-bin} \genfrac{[}{]}{0pt}{}{n}{k}_{q} = q^k\genfrac{[}{]}{0pt}{}{n-1}{k}_{q} + \genfrac{[}{]}{0pt}{}{n-1}{k-1}_{q} = \genfrac{[}{]}{0pt}{}{n-1}{k}_{q} + q^{n-k}\genfrac{[}{]}{0pt}{}{n-1}{k-1}_{q}.\tag{7}\] We begin with two simple weight systems giving \(q\)-binomial coefficients with some possible factors.

We now give a weight system for the mixed moments of Stieltjes–Wigert polynomials.

Since \(\sigma_{n,k}\) is a polynomial in \(q\), we do not consider its weight system of infinite height.

4.2 \(q\)-Bessel polynomials↩︎

The (monic) \(q\)-Bessel polynomials (also known as alternative \(q\)-Charlier polynomials) are defined by \[\label{eq:alt-q-charlier} p_n(x;a;q) = \frac{1}{(-1)^{n}q^{-\binom{n}{2}}(-aq^n;q)_{n}} {}_{2}\phi_{1} \left( \begin{matrix} q^{-n},-aq^n\\ 0\\ \end{matrix} ; q,qx \right) .\tag{8}\] Let \(\sigma^b_{n,k}(a;q)\) and \(\nu^b_{n,k}(a;q)\) denote the mixed moments and the dual mixed moments of the \(q\)-Bessel polynomials: \[x^n = \sum_{k=0}^{n} \sigma^b_{n,k}(a;q) p_k(x;a;q), \qquad p_n(x;a;q) = \sum_{k=0}^{n} \nu^b_{n,k}(a;q) x^k.\] These quantities have simple product formulas.

We will find several weight systems for \(\sigma^b_{n,k}(a;q)\). We first give a weight system of height \(1\).

To find a weight system of infinite height for \(\sigma^b_{n,k}(a;q)\), we need some lemmas. The following lemma allows us to find a weight system of infinite height for \(\{\sigma_{n,k}\}_{n,k\ge0}\) if a certain recurrence is satisfied.

The mixed moments \(\sigma^b_{n,k}(a;q)\) satisfy a recurrence of the form in 27.

We are now ready to give a weight system of infinite height for \(\sigma^b_{n,k}(a;q)\). Note that this weight system can be constructed using 1 with the monomial ordering defined by \(q<a\). See 9 for the weight system in the following proposition.

Since we have both a weight system \(w_1\) of height \(1\) and a weight system \(w\) of infinite height for \(\sigma^b_{n,k}\), we can construct a weight system of any height using 21.

4.3 Expanding weight systems↩︎

In this subsection we prove some lemmas which allow us to expand one row of a weight system to two rows. Here, a row means the part of a weight system between \(y=\ell-1\) and \(y=\ell\) for some \(\ell\ge1\).

See 10 for the weight systems in the following lemma.

For an integer \(k\), let \(\chi_o(k)=1\) if \(k\) is odd and \(\chi_o(k)=0\) otherwise. Similarly, let \(\chi_e(k)=1\) if \(k\) is even and \(\chi_e(k)=0\) otherwise.

4.4 Little \(q\)-Jacobi↩︎

Let \(p_n(x;a,b;q)\) be the monic little \(q\)-Jacobi polynomial: \[\label{eq:def-lqj} p_n(x;a,b;q) = \frac{(aq;q)_n}{(-1)^n q^{-\binom{n}{2}}(abq^{n+1};q)_n} {}_{2}\phi_{1} \left( \begin{matrix} q^{-n},abq^{n+1}\\ aq\\ \end{matrix} ; q,qx \right) .\tag{12}\] We denote by \(\sigma^L_{n,k}(a,b;q)\) and \(\nu^L_{n,k}(a,b;q)\) the mixed moments and the dual mixed moments of the little \(q\)-Jacobi polynomials, respectively: \[x^n = \sum_{k=0}^{n} \sigma^L_{n,k}(a,b;q) p_k(x;a,b;q), \qquad p_n(x;a,b;q) = \sum_{k=0}^{n} \nu^L_{n,k}(a,b;q) x^k.\]

The following lemma will be used to find formulas for \(\sigma^L_{n,k}(a,b;q)\) and \(\nu^L_{n,k}(a,b;q)\).

The following result has been proved in [8] using the \(q\)-Saalschütz summation formula. We provide another proof using 33 because the same technique will be used later for big \(q\)-Jacobi polynomials and Askey–Wilson polynomials.

Now we give a weight system of height \(1\) for \(\sigma^L_{n,k}(a,b;q)\).

Using the expanding lemma (31) we obtain a weight system of infinite height for \(\sigma^L_{n,k}(a,b;q)\). We note that this is also discoverable by 1 using the monomial ordering \(q<a<b\). See 11 for the weight system in the following proposition.

4.5 Big \(q\)-Jacobi↩︎

Let \(p_n(x;a,b,c;q)\) be the monic big \(q\)-Jacobi polynomial: \[\label{eq:big-q-jacobi-def} p_n(x;a,b,c;q) = \frac{(aq,cq;q)_n}{(abq^{n+1};q)_n} {}_{3}\phi_{2} \left( \begin{matrix} q^{-n},abq^{n+1},x\\ aq,cq\\ \end{matrix} ; q,q \right) .\tag{13}\] We denote by \(\sigma^B_{n,k}(a,b,c;q)\) and \(\nu^B_{n,k}(a,b,c;q)\) the mixed moments and the dual mixed moments of the big \(q\)-Jacobi polynomials, respectively: \[x^n = \sum_{k=0}^{n} \sigma^B_{n,k}(a,b,c;q) p_k(x;a,b,c;q), \qquad p_n(x;a,b,c;q) = \sum_{k=0}^{n} \nu^B_{n,k}(a,b,c;q) x^k.\] We will first consider the factorial mixed moments \(\tilde{\sigma}^B_{n,k}(a,b,c;q)\) and factorial dual mixed moments \(\tilde{\nu}^B_{n,k}(a,b,c;q)\) defined by \[(x|\mathbf{q})^n = \sum_{k=0}^{n} \tilde{\sigma}^B_{n,k}(a,b,c;q) p_k(x;a,b,c;q), \qquad p_n(x;a,b,c;q) = \sum_{k=0}^{n} \tilde{\nu}^B_{n,k}(a,b,c;q) (x|\mathbf{q})^k,\] where \(\mathbf{q}=(1,q^{-1},q^{-2},\dots)\) so that \[(x|\mathbf{q})^k=(x-1)(x-q^{-1})\cdots(x-q^{-k+1}).\]

We give a weight system of height \(1\) and a weight system of infinite height for \(\tilde{\sigma}^B_{n,k}(a,b,c;q)\) in the next two propositions.

See 12 for the weight system in the following proposition.

By 19, we can construct a weight system for \(\sigma^B_{n,k}(a,b,c;q)\) from the weight system \(\widetilde{w}\) in 40 by adding one row coming from the factorial basis \(\{(x|\mathbf{q})^n\}_{n\ge0}\) at the bottom. Miraculously, the new row and the bottom row of \(\widetilde{w}\) cancel each other and the resulting weight system is \(\widetilde{w}\) with bottom row removed as described in the next proposition. We note that this weight system is also discoverable by 1 using the monomial ordering \(q<c<a<b\). See 13 for the weight system in the following proposition.

4.6 Askey–Wilson polynomials↩︎

The monic Askey–Wilson polynomials \(p_n(x;a,b,c,d|q)\) are defined by \[\label{eq:AW-def} p_n(x;a,b,c,d|q) = \frac{(ab,ac,ad;q)_n}{2^na^n (abcdq^{n-1};q)_n} {}_{4}\phi_{3} \left( \begin{matrix} q^{-n},abcdq^{n-1},ae^{i\theta}, ae^{-i\theta}\\ ab,ac,ad\\ \end{matrix} ; q,q \right) ,\tag{14}\] where \(x=\cos \theta = (e^{i\theta}+e^{-i\theta})/2\). Let \(f=(f_0,f_1, \dots)\) be the sequence given by \(f_j = (aq^{j}+a^{-1}q^{-j})/2\). Then it is easy to check that \[(ae^{i\theta};q)_k (ae^{-i\theta};q)_k = (-2a)^k q^{\binom{k}{2}} (x|f)^k.\]

We denote by \(\sigma^{AW}_{n,k}(a,b,c,d;q)\) and \(\nu^{AW}_{n,k}(a,b,c,d;q)\) the mixed moments and the dual mixed moments of the Askey–Wilson polynomials, respectively: \[x^n = \sum_{k=0}^{n} \sigma^{AW}_{n,k}(a,b,c,d;q) p_k(x;a,b,c,d|q), \qquad p_n(x;a,b,c,d|q) = \sum_{k=0}^{n} \nu^{AW}_{n,k}(a,b,c,d;q) x^k.\] As in the case of big \(q\)-Jacobi polynomials, we will first consider the factorial mixed moments \(\tilde{\sigma}^{AW}_{n,k}(a,b,c,d;q)\) and factorial dual mixed moments \(\tilde{\nu}^{AW}_{n,k}(a,b,c,d;q)\) defined by \[(x|f)^n = \sum_{k=0}^{n} \tilde{\sigma}^{AW}_{n,k}(a,b,c,d;q) p_k(x;a,b,c,d|q), \qquad p_n(x;a,b,c,d|q) = \sum_{k=0}^{n} \tilde{\nu}^{AW}_{n,k}(a,b,c,d;q) (x|f)^k.\]

As in the previous subsections, we give a weight system of height \(1\) for \(\tilde{\sigma}^{AW}_{n,k}(a,b,c,d;q)\).

Now we give a weight system of infinite height for \(\tilde{\sigma}^{AW}_{n,k}(a,b,c,d;q)\). This is also discoverable by 1 using the monomial ordering \(q<d<c<a<b\). See 15 for the weight system in the following proposition.

Finally, by 19 and 44, we obtain a weight system for the original mixed moments of Askey–Wilson polynomials. See 16 for the weight system in the following theorem.

5 Another bootstrapping method from continuous \(q\)-Hermite polynomials↩︎

In this section, we provide another bootstrapping method to find a combinatorial model for mixed moments of Askey–Wilson polynomials relative to the continuous \(q\)-Hermite polynomials, which are one of the most well-studied families of orthogonal polynomials in the \(q\)-Askey scheme. At the end of this section, we give a lecture hall graph model for the mixed moments of the continuous \(q\)-Hermite polynomials. As applications of the results in this section, we prove 54, which establishes the total positivity of the matrix of mixed moments of Askey–Wilson polynomials relative to continuous \(q\)-Hermite polynomials, and we also give the first combinatorial proof of the symmetry of \(a,b,c,d\) in the Askey–Wilson polynomials in 6.

Our results build on those of Kim and Stanton [18], who used a bootstrapping method to compute the moments of Askey–Wilson polynomials from continuous \(q\)-Hermite polynomials.

5.1 Definitions↩︎

The (monic) continuous \(q\)-Hermite polynomials \(H_n(x|q)\), continuous big \(q\)-Hermite polynomials \(H_n(x;a|q)\), Al–Salam–Chihara polynomials \(Q_n(x;a,b|q)\), and continuous dual \(q\)-Hahn polynomials \(p^{dH}_n(x; a, b, c|q)\) are defined by \[\begin{align} H_n(x|q) &= \frac{e^{in\theta}}{2^n} {}_{2}\phi_{0} \left( \begin{matrix} q^{-n},0\\ -\\ \end{matrix} ; q,q^n e^{-2i\theta} \right) ,\\ H_n(x;a|q) &= \frac{1}{(2a)^n} {}_{3}\phi_{2} \left( \begin{matrix} q^{-n},ae^{i\theta},ae^{-i\theta}\\ 0,0\\ \end{matrix} ; q,q \right) = \frac{e^{in\theta}}{2^n} {}_{2}\phi_{0} \left( \begin{matrix} q^{-n},ae^{i\theta}\\ -\\ \end{matrix} ; q,q^n e^{-2i\theta} \right) , \\ Q_n(x;a,b|q) &= \frac{(ab;q)_{n}}{(2a)^n} {}_{3}\phi_{2} \left( \begin{matrix} q^{-n},ae^{i\theta},ae^{-i\theta}\\ ab,0\\ \end{matrix} ; q,q \right) ,\\ p^{dH}_n(x; a, b, c|q) &= \frac{(ab, ac; q)_n}{(2a)^n} {}_{3}\phi_{2} \left( \begin{matrix} q^{-n},a e^{i\theta}, a e^{-i\theta}\\ ab, ac\\ \end{matrix} ; q,q \right) . \end{align}\] Here, \(x=\cos\theta\). Note that these polynomials are special cases of the Askey–Wilson polynomials \(p_n(x;a,b,c,d|q)\) defined in 14 , namely, \[\begin{align} H_n(x|q) &= p_n(x;0,0,0,0|q),\\ H_n(x;a|q) &= p_n(x;a,0,0,0|q),\\ Q_n(x;a,b|q) &= p_n(x;a,b,0,0|q),\\ p^{dH}_n(x; a, b, c|q) &= p_n(x;a,b,c,0|q). \end{align}\]

Let \(\sigma^{H}_{n,k}(q)\) be the mixed moments of the continuous \(q\)-Hermite polynomials \(\{H_n(x|q)\}_{n\ge0}\): \[x^n = \sum_{k=0}^n \sigma^{H}_{n,k}(q) H_k(x|q).\] The Touchard–Riordan-like formula due to Josuat-Vergès [22] and Cigler and Zeng [23] (see also [18]) can be stated as \[\sigma^H_{n,k}(q)= \frac{1}{2^{n-k}} \sum_{r=k}^n \left( \binom{n}{\frac{n-r}{2}} - \binom{n}{\frac{n-r}{2}-1}\right) (-1)^{(r-k)/2} q^{\binom{(r-k)/2+1}2} \genfrac{[}{]}{0pt}{}{\frac{r+k}{2}}{\frac{r-k}{2}}_{q},\] where we define \(\binom{i}{j}=\genfrac{[}{]}{0pt}{}{i}{j}_{q}=0\) unless both \(i\) and \(j\) are integers.

We denote by \(\sigma^{bH,H}_{n,k}(a;q)\), \(\sigma^{Q,H}_{n,k}(a,b;q)\), \(\sigma^{dH,H}_{n,k}(a,b,c;q)\), and \(\sigma^{AW,H}_{n,k}(a,b,c,d;q)\) the mixed moments of continuous big \(q\)-Hermite polynomials, Al–Salam–Chihara polynomials, continuous dual \(q\)-Hahn polynomials, and Askey–Wilson polynomials relative to the continuous \(q\)-Hermite polynomials: \[\begin{align} \label{eq:8} H_n(x|q) &= \sum_{k=0}^n \sigma^{bH,H}_{n,k}(a;q) H_k(x;a|q),\\ \notag H_n(x|q) &= \sum_{k=0}^n \sigma^{Q,H}_{n,k}(a,b;q) Q_k(x;a,b|q),\\ \notag H_n(x|q) &= \sum_{k=0}^n \sigma^{dH,H}_{n,k}(a,b,c;q) p^{dH}_k(x;a,b,c|q),\\ \notag H_n(x|q) &= \sum_{k=0}^n \sigma^{AW,H}_{n,k}(a,b,c,d;q) p_k(x;a,b,c,d|q). \end{align}\tag{15}\]

5.2 Another bootstrapping method↩︎

By re-normalizing the results in [18], we have \[\begin{align} \label{eq:cc0} H_n(x|q) &= \sum_{k=0}^n \genfrac{[}{]}{0pt}{}{n}{k}_{q} \left( \frac{a}{2} \right)^{n-k} H_k(x;a|q) ,\\ \notag H_n(x;a|q) &=\sum_{k=0}^n \genfrac{[}{]}{0pt}{}{n}{k}_{q} \left( \frac{b}{2} \right)^{n-k} Q_k(x;a,b|q) ,\\ \notag Q_n(x;a,b|q) &= \sum_{k=0}^n \genfrac{[}{]}{0pt}{}{n}{k}_{q} \left( \frac{c}{2} \right)^{n-k} (abq^k;q)_{n-k} p^{dH}_k(x;a,b,c|q) ,\\ \notag p^{dH}_n(x;a,b,c|q) &=\sum_{k=0}^n \genfrac{[}{]}{0pt}{}{n}{k}_{q} \left( \frac{d}{2} \right)^{n-k} \frac{(abq^k,acq^k,bcq^k;q)_{n-k}}{(abcdq^{2k};q)_{n-k}} p_k(x;a,b,c,d|q). \end{align}\tag{16}\] We will first find weight systems for the coefficients in the sums of the above equations. Then, using 12, we can build a weight system for \(\sigma^{AW,H}_{n,k}(a,b,c,d;q)\) by successively stacking these weight systems.

See Figures 17, 18, and 19 for the weight systems \(w^{(1)}\), \(w^{(2)}\), and \(w^{(3)}\), respectively in the following lemma.

By 12 and 46, we obtain the following lemma.

Rewriting \(h^{w^{(1)} \sqcup w^{(2)} \sqcup w^{(3)} \sqcup w^{(4)}}_{n,k}\) in 47 we obtain the following combinatorial model for \(\sigma^{AW,H}_{n,k}(a,b,c,d;q)\).

5.3 A weight system for continuous \(q\)-Hermite polynomials↩︎

In order to obtain a weight system for the original mixed moment \(\sigma^{AW}_{n,k}(a,b,c,d;q)\) of Askey–Wilson polynomials from 48, we can simply add a weight system for the mixed moment \(\sigma^H_{n,k}(q)\) of continuous \(q\)-Hermite polynomials below the weight system \(w^{AW,H}\). In this subsection we find a weight system for \(\sigma^H_{n,k}(q)\). To do this we first consider continuous big \(q\)-Hermite polynomials \(H_n(x;a|q)\).

Let \(\sigma_{n,k}^{bH}(a;q)\) be the mixed moment of continuous big \(q\)-Hermite polynomials: \[x^n = \sum_{k=0}^n \sigma^{bH}_{n,k}(a;q) H_k(x;a|q).\] Since \(H_n(x;a|q)= p_n(x;a,0,0,0|q)\), setting \(b=c=d=0\) in 45 gives the following proposition; see 20 for the weight system.

To find a weight system for \(\sigma^H_{n,k}(q)\), we compare two polynomials \(H_n(x|q)\) and \(H_n(x;a|q)\). See 21 for the weight system in the following lemma.

By 12, a weight system for \(\sigma_{n,k}^{H}(q)\) can be obtained by adding the weight system \(w'\) in 50 at the top of a weight system of \(\sigma_{n,k}^{bH}(a;q)\). Therefore, by 49, \(w^{bH}\sqcup w'\) is a weight system for \(\sigma_{n,k}^{H}(q)\). However, since \(\sigma_{n,k}^{H}(q)\) is independent of \(a\), we can replace \(a\) by any number. In particular, if we set \(a=1\) in the weight system \(w^{bH}\sqcup w'\), we obtain the following proposition; see 22 for the weight system.

Using 48 and 51, we obtain another weight system for the original mixed moments \(\sigma^{AW}_{n,k}(a,b,c,d;q)\) of Askey–Wilson polynomials.

6 Combinatorial properties of Askey–Wilson polynomials↩︎

In this section, we study in more detail the weight system \(w^{AW,H}\) in 48 for the mixed moment \(\sigma^{AW,H}_{n,k}\) of Askey–Wilson polynomials relative to continuous \(q\)-Hermite polynomials. We then find a more efficient combinatorial model for \(\sigma^{AW,H}_{n,k}\). Using our new combinatorial model we give the first combinatorial proof of the well-known fact that the mixed moments (and hence the coefficients as well) of Askey–Wilson polynomials are symmetric in \(a,b,c,d\). As applications we also find some interesting properties of Askey–Wilson polynomials.

6.1 Another combinatorial model for mixed moments↩︎

Let \(\tilde{\sigma}^{AW,H}_{n,k}(a,b,c,d;q)\) be the following rescaled mixed moments of Askey–Wilson polynomials relative to continuous \(q\)-Hermite polynomials: \[\tilde{\sigma}^{AW,H}_{n,k}(a,b,c,d;q) := \frac{2^{n-k}}{\mathbf{i}^{n-k}} \sigma^{AW,H}_{n,k}(\mathbf{i}a,\mathbf{i}b,\mathbf{i}c,\mathbf{i}d;q),\] where \(\mathbf{i}\) is the imaginary number \(\sqrt{-1}\). Using the substitution \((a,b,c,d)\mapsto (\mathbf{i}a,\mathbf{i}b,\mathbf{i}c,\mathbf{i}d)\), we can restate 48 as follows; see 23 for the weight system.

Observe that \(\widetilde{w}^{AW,H}(t;i,j)\) in ?? is a monomial in \(a,b,c,d\), and \(q\). Thus 53 implies that \(\tilde{\sigma}^{AW,H}_{n,k}(a,b,c,d;q)\) is a formal power series in \(a,b,c,d,q\) with nonnegative integer coefficients. More generally, we have the following corollary.

We will find another combinatorial model for \(\tilde{\sigma}^{AW,H}_{n,k}(a,b,c,d;q)\) derived from 53. To do this, we need some definitions. First, observe that every weight \(\widetilde{w}^{AW,H}(t;i,j)\) in 53 is of the form \(a^{i_1}b^{i_2}c^{i_3}d^{i_4}q^j\) such that \(i_1+i_2+i_3+i_4\) is odd and \(|i_r-i_s|\le 1\) for all \(1\le r,s\le 4\).

For example, \((3,3,4,3) \le (3,3,3,4)\) and \((4,3,3,3) \le (2,3,3,3)\). The motivation of the above definition is the following lemma, which says that \((\mathcal{T},\le)\) is essentially the totally ordered set on the tuples \((t_1,t_2,t_3,t_4)\) of the powers of \(a,b,c,d\) appearing in the weights \(\widetilde{w}^{AW,H}(t;i,j)\) ordered by their heights \(t\).

From now on, we will use the parameters \(a_1,a_2,a_3,a_4\) in place of \(a,b,c,d\), respectively. For any quadruple \(T=(t_1,t_2,t_3,t_4)\) of nonnegative integers, we define \[\begin{align} \mathbf{a}^T &:= a_1^{t_1}a_2^{t_2}a_3^{t_3}a_4^{t_4},\\ |T| &:= t_1 +t_2+t_3+t_4. \end{align}\] For integers \(n\ge k\ge 0\), let \[\mathcal{T}_{n,k} = \{ (T_{k},T_{k+1},\dots,T_{n-1})\in \mathcal{T}^{n-k}: T_{k} \ge T_{k+1} \ge \cdots \ge T_{n-1} \}.\] Note that the set \(\mathcal{T}_{n,k}\) depends only on the difference \(n-k\). However, we keep both \(n\) and \(k\) in the notation \(\mathcal{T}_{n,k}\) to emphasize that we index the elements of a tuple \((T_{k},T_{k+1},\dots,T_{n-1})\in \mathcal{T}_{n,k}\) from \(k\) to \(n-1\).

For \(\mathbf{T}=(T_{k},\dots,T_{n-1}) \in \mathcal{T}_{n,k}\), the sum \(s(\mathbf{T})\) and the multiplicity \(m(\mathbf{T})\) of \(\mathbf{T}\) are defined by \[\begin{align} s(\mathbf{T}) &:= T_k + \cdots + T_{n-1},\\ m(\mathbf{T}) &:= (m_1,\dots,m_r), \end{align}\] where \(T_{k} + \cdots + T_{n-1}\) means the component-wise addition and \(m_1,\dots,m_r\) are the positive integers such that \(m_1 + \cdots + m_r = n-k\) and \[\label{eq:15} T_{k} = \cdots = T_{k+m_1-1} > T_{k+m_1} = \cdots = T_{k+m_1+m_2-1} > \cdots > T_{k+m_1 + \cdots + m_{r-1}} = \cdots = T_{n-1}.\tag{18}\] We will use the following notation: \[\begin{align} \lVert \mathbf{T}\rVert_{n,k} &:= \sum_{i=k}^{n-1} \frac{i(|T_i|-1)}{2}, \\ \genfrac{[}{]}{0pt}{}{n}{k,m(\mathbf{T})}_{q} &:= \genfrac{[}{]}{0pt}{}{n}{k,m_1,\dots,m_r}_{q}= \frac{[n]_q!}{[k]_q! [m_1]_q! \cdots [m_r]_q!}, \end{align}\] where \(m(\mathbf{T}) = (m_1,\dots,m_r)\).

We are now ready to state another combinatorial formula for \(\tilde{\sigma}^{AW,H}_{n,k}(a_1,a_2,a_3,a_4;q)\).

Using 57, we obtain the following “shifting” property of \(\tilde{\sigma}^{AW,H}_{n,k}(a_1,a_2,a_3,a_4;q)\). Here, \([\mathbf{a}^{\mathbf{s}}] f(a_1,a_2,a_3,a_4)\) means the coefficient of \(\mathbf{a}^{\mathbf{s}}\) in \(f(a_1,a_2,a_3,a_4)\).

6.2 The symmetry of \(a,b,c,d\) in mixed moments↩︎

As another application of 57, we give a combinatorial proof of the symmetry of the parameters of \(a_1,a_2,a_3,a_4\) in \(\tilde{\sigma}^{AW,H}_{n,k}(a_1,a_2,a_3,a_4;q)\). To this end we introduce some definitions.

For a function \(f(a_1,a_2,a_3,a_4)\) in the variables \(a_1,a_2,a_3,a_4\) and a permutation \(\tau \in \mathfrak{S}_4\), let \[\tau \cdot f(a_1,a_2,a_3,a_4) = f(a_{\tau(1)},a_{\tau(2)},a_{\tau(3)},a_{\tau(4)}).\] Our goal is to show that \(\tilde{\sigma}^{AW,H}_{n,k}(a_1,a_2,a_3,a_4;q)\) is symmetric in \(a_1,a_2,a_3,a_4\), that is, for any permutation \(\tau \in \mathfrak{S}_4\), \[\label{eq:21} \tau \cdot \tilde{\sigma}^{AW,H}_{n,k}(a_1,a_2,a_3,a_4;q) = \tilde{\sigma}^{AW,H}_{n,k}(a_1,a_2,a_3,a_4;q) .\tag{21}\] Since the simple transpositions generate \(\mathfrak{S}_4\), it suffices to prove 21 for the cases \(\tau = (1,2)\), \(\tau= (2,3)\), and \(\tau = (3,4)\).

An interval of \(\mathcal{T}\) is a subset \(I\) of \(\mathcal{T}\) such that if \(T_1,T_2\in I\) and \(T_1 \le T \le T_2\), then \(T\in I\). For an interval \(I\) of \(\mathcal{T}\), we define \[\mathcal{T}_{n,k}(I) := \mathcal{T}_{n,k} \cap I^{n-k} = \{ (T_{k},T_{k+1},\dots,T_{n-1})\in I^{n-k}: T_{k} \ge T_{k+1} \ge \cdots \ge T_{n-1} \}\] and \[\label{eq:13} \sigma_{n,k}(I) = \sum_{\mathbf{T}\in \mathcal{T}_{n,k}(I)} \mathbf{a}^{s(\mathbf{T})} q^{\lVert \mathbf{T}\rVert_{n,k}} \genfrac{[}{]}{0pt}{}{n}{k,m(\mathbf{T})}_{q} .\tag{22}\] For two intervals \(I_1\) and \(I_2\) of \(\mathcal{T}\) we write \(I_1 < I_2\) if \(T_1<T_2\) for all \(T_1\in I_1\) and \(T_2\in I_2\). An increasing interval partition of \(\mathcal{T}\) is an infinite sequence \((I_0,I_1,\dots)\) of nonempty intervals \(I_0 <I_1 < \cdots\) in \(\mathcal{T}\) such that \(I_0 \cup I_1 \cup \cdots = \mathcal{T}\). Note that in this case the intervals \(I_j\)’s are pairwise disjoint and indeed form a partition of \(\mathcal{T}\).

By 59, in order to prove that \(\tilde{\sigma}^{AW,H}_{n,k}(a_1,a_2,a_3,a_4;q)\) is invariant under a permutation \(\tau\in \mathfrak{S}_4\), it suffices to show that there is an increasing interval partition \((I_0, I_1,\dots)\) of \(\mathcal{T}\) such that \(\tau \cdot \sigma_{r,s}(I_j) = \sigma_{r,s}(I_j)\) for any integers \(r\ge s\ge 0\) and \(j\ge0\). We will find such an increasing interval partition of \(\mathcal{T}\) for the three cases that \(\tau=(2,3)\), \(\tau=(3,4)\), and \(\tau=(1,2)\) in this order.

First, we consider the simplest case \(\tau=(2,3)\).

Before finding an increasing interval partition of \(\mathcal{T}\) for \(\tau=(3,4)\), we need some definitions. Let \(\mathfrak{S}(0^r,1^s)\) be the set of words \(\pi = \pi_1 \cdots \pi_{r+s}\) consisting of \(r\) \(0\)’s and \(s\) \(1\)’s. For such a word \(\pi = \pi_1 \cdots \pi_{r+s}\), an inversion is a pair \((i,j)\) of integers \(1\le i<j\le r+s\) such that \(\pi_i>\pi_j\). Let \(\operatorname{inv}(\pi)\) denote the number of inversions of \(\pi\). It is well known [24] that \[\label{eq:23} \sum_{\pi\in \mathfrak{S}(0^r, 1^s)} q^{\operatorname{inv}(\pi)} = \genfrac{[}{]}{0pt}{}{r+s}{r}_{q}.\tag{25}\]

Now we are ready to consider the case \(\tau=(3,4)\).

Finally we consider the case \(\tau=(1,2)\), which is similar to the case \(\tau=(3,4)\).

By Lemmas 60, 61, and 62, we establish the symmetry of \(a,b,c,d\) in the mixed moments of Askey–Wilson polynomials. Let \(\nu^{AW,H}_{n,k}(a,b,c,d;q)\) be the coefficient in the expansion \[p_n(x;a,b,c,d|q) = \sum_{k=0}^n \nu^{AW,H}_{n,k}(a,b,c,d;q)\, H_k(x|q).\]

Acknowledgments↩︎

The authors are grateful to Donghyun Kim for fruitful discussions. They also thank the anonymous referees for their careful reading of the manuscript and for helpful comments. SC is partially supported by NSF grant DMS-2054482 and ANR grants ANR-19-CE48-0011 and ANR-18-CE40-0033. BJ and JPK are pleased to acknowledge support from ERC Advanced Grant 740900 (LogCorRM). JSK is supported by NRF grant RS-2025-00557835.

References↩︎