On linearization coefficients of $q$-Laguerre polynomials

The linearization coefficient $\mathcal{L}(L_{n_1}(x)\dots L_{n_k}(x))$ of classical Laguerre polynomials $L_n(x)$ is known to be equal to the number of $(n_1,\dots,n_k)$-derangements, which are permutations with a certain condition. Kasraoui, Zeng and Stanton found a $q$-analog of this result using $q$-Laguerre polynomials with two parameters $q$ and $y$. Their formula expresses the linearization coefficient of $q$-Laguerre polynomials as the generating function for $(n_1,\dots,n_k)$-derangements with two statistics counting weak excedances and crossings. In this paper their result is proved by constructing a sign-reversing involution on marked perfect matchings.


Introduction
A family of polynomials P n (x) are called orthogonal polynomials with respect to a linear functional L if deg P n (x) = n for n ≥ 0 and L(P m (x)P n (x)) = 0 if and only if m = n. The nth moment µ n of the orthogonal polynomials is defined by µ n = L(x n ). It is well known that monic orthogonal polynomials P n (x) satisfy a three-term recurrence of the form P n+1 (x) = (x − b n )P n (x) − λ n P n−1 (x). (1) Viennot [11] developed a combinatorial theory to study orthogonal polynomials. In particular, he showed that orthogonal polynomials P n (x) and the moments µ n are expressed as weighted sums of certain lattice paths. There are several classical orthogonal polynomials whose moments have simple combinatorial meanings. For example, the nth moment of the Hermite (respectively, Charlier and Laguerre) polynomials is the number of perfect matchings (respectively, set partitions and permutations) on [n] := {1, 2, . . . , n}.
By definition of orthogonal polynomials, it is easily seen that P m (x)P n (x) = ℓ c ℓ m,n P ℓ (x), c ℓ m,n = L(P ℓ (x)P m (x)P n (x))/L(P ℓ (x) 2 ).
Thus the coefficients c ℓ m,n can be computed using the quantities L(P n1 (x) . . . P n k (x)). We call L(P n1 (x) . . . P n k (x)) a linearization coefficient.
For the above mentioned classical orthogonal polynomials, the linearization coefficients also have nice combinatorial interpretations. If P n (x) are the Hermite (respectively, Charlier and Laguerre) polynomials, then L(P n1 (x) . . . P n k (x)) is the number of inhomogeneous perfect matchings (respectively, set partitions and permutations) on [n 1 ] ⊔ · · · ⊔ [n k ], see [2,4,5,6,12] and references therein. Here, [n 1 ] ⊔ · · · ⊔ [n k ] is the disjoint union of [n i ]'s and a perfect matching m (respectively, set partition π and permutation σ) is inhomogenous if there are no edges (respectively, two elements in the same block and two elements j and σ(j)) that are contained in the same set [n i ].
There are q-analogs of the above combinatorial formulas for linearization coefficients of Hermite, Charlier and Laguerre polynomials due to Ismail, Stanton and Viennot [7], Anshelevich [1] and Kasraoui, Stanton and Zeng [8], respectively. We refer the reader to the survey [2] for more details on these linearization coefficients.
Suppose that P n (x) are orthogonal polynomials whose moments L(x n ) have a combinatorial model as in the case of Hermite, Charlier or Laguerre polynomials. Since P n (x) satisfy a simple recurrence (1), one may also give a combinatorial model for P n (x) with possibly negative signs involved. These combinatorial models for P n (x) and L(x n ) naturally yield a combinatorial meaning to L(P n1 (x) . . . P n k (x)), which may have negative signs. Therefore, if there is a combinatorial formula for L(P n1 (x) . . . P n k (x)) with only positive terms, the most satisfying combinatorial Date: January 8, 2020.
proof of this formula would be finding a sign-reversing involution on the combinatorial models for L(P n1 (x) . . . P n k (x)) whose fixed points give the positive terms in the formula.
Indeed, the formulas for linearization coefficients of q-Hermite [7] and q-Charlier polynomials [1] have been proved in this way by Ismail, Stanton and Viennot [7] and Kim, Stanton and Zeng [9]. However, such a proof is missing in the case of q-Laguerre polynomials. In this paper, we prove the formula for linearization coefficients of q-Laguerre polynomials due to Kasraoui, Stanton and Zeng [8] by finding a sign-reversing involution. We now describe their result below.
The q-Laguerre polynomials L n (x; q, y) are defined by the three-term recurrence relation with L 0 (x; q, y) = 1 and L 1 (x; q, y) = x − y. Here, we use the notation [n] q = 1 + q + · · · + q n−1 . From now on L denotes the linear functional with respect to which the q-Laguerre polynomials are orthogonal.
Kasraoui, Stanton and Zeng [8] showed that the nth moment is given by They also proved the following formula for the linearization coefficients of q-Laguerre polynomials.
In [8] they proved Theorem 1.1 using a recurrence relation for L(L n1 (x; q, y) · · · L n k (x; q, y)) and induction. The purpose of this paper is to give a proof of Theorem 1.1 by constructing a sign-reversing involution. Our fundamental combinatorial objects are matchings instead of permutations.
The remainder of this paper is organized as follows. In Section 2 we give basic definitions and combinatorial interpretations for L n (x; q, y) and µ n (q, y) using matchings and perfect matchings. In Section 3 we give a combinatorial model for the linearization coefficient in terms of marked perfect matchings. We then construct a sign-reversing involution on marked perfect matchings. Section 4 is devoted to showing that our map in Section 3 is indeed a sign-reversing involution that preserves the desired weights on marked perfect matchings. In the final section we discuss future work.

q-Laguerre polynomials and their moments
In this section we give combinatorial interpretations for the q-Laguerre polynomials L n (x; q, y) and their moments µ n (q, y) using matchings and perfect matchings. The results in this section generalize the combinatorial models for Laguerre polynomials and their moments due to Viennot [11,Ch. 6]. We start with basic definitions.
Definition 2.1. Let K n,n be the complete bipartite graph with 2n vertices, i.e., the graph with vertex set {1, 2, . . . , n, 1, 2, . . . , n} and edge set {(i, j) : 1 ≤ i, j ≤ n}. A matching of degree n is a subgraph π of K n,n such that π contains every vertex of K n,n and no two distinct edges of π have common vertices. A matching π of degree n is called a perfect matching if π has exactly n edges. Denote the set of all matchings (respectively, perfect matchings) of degree n by M n (respectively, PM n ). For π ∈ M n , we denote by E(π) the set of edges in π and let e(π) = |E(π)|.
We visualize a matching π of degree n by placing the vertices 1, 2, . . . , n in the upper row and the vertices 1, 2, . . . , n in the lower row as shown in Figure 1. We call 1, 2, . . . , n the upper vertices and 1, 2, . . . , n the lower vertices of π. If there is no possible confusion, we will simply write j instead of j. For example, since every edge of a matching is of the form (i, j), we will also write this edge as (i, j). For π ∈ M n , if (i, j) ∈ π, we denote π(i) = j and e i = (i, π(i)). For example, if π is the matching in Figure 1, then π(1) = 4, π(3) = 2 and e 1 = (1, 4), e 3 = (3, 2). An upper vertex i of π is said to be unmatched if there is no edge of the form (i, j). Similarly, a lower vertex j of π is unmatched if there is no edge of the form (i, j). Note that if π ∈ PM n , there are no unmatched vertices and we can identify π with the permutation σ ∈ S n given by σ(i) = π(i) for all i ∈ [n]. We will often use this identification in this paper.
By the identification of PM n and S n we can rewrite (3) as follows: For the remainder of this section we will find a combinatorial model for L n (x; q, y) in Theorem 2.4 and give yet another expression for µ n (q, y) in (7). To do this, we define some statistics for matchings. Given a matching π ∈ M n , let P = (B 1 , . . . , B l ) be the unique ordered set partition of the upper vertices of π satisfying the following conditions: • Each block B r consists of consecutive elements. In other words, B r is of the form B r = {i, i + 1, . . . , j}. • For each i ∈ [n], i is the largest element in some block B r if and only if i is an unmatched vertex or i = n. We define the upper block index bindex U π (i) of a vertex i to be the integer r such that i ∈ B r . Note that bindex U π (i) is equal to one more than the number of unmatched vertices appearing before i in the upper row. The lower block index bindex L π (i) is defined similarly by considering the ordered set partition of the lower vertices of π.
A crossing of π is a pair (e, e ′ ) of edges e = (i, π(i)) and e ′ = (j, π(j)) in π such that i < j and π(i) > π(j). The number of crossings of π is denoted by cr(π).
Example 2.5. There are 7 matchings of degree 2 as shown in Figure 3. Then by Theorem 2.4, we have Proof of Theorem 2.4. The proof is by induction on n. The cases for n = 0, 1 are easy to check.
For n ≥ 2 we will show that the right hand side of (5) satisfies the three-term recurrence (2), which we recall here: For each matching π ∈ M n+1 there are three cases as follows. Case 1: Two vertices n+1 and n + 1 are both unmatched. Let π ′ ∈ M n be the matching obtained from π by deleting the last vertex in each row. Clearly all statistics but the number of unmatched vertices of π and π ′ are equal. Then this case contributes xL n (x; q, y) to the right-hand side of (6).

Case 2:
The vertex n + 1 is matched to some vertex i, i.e., there is an edge e i = (i, n + 1) ∈ E(π). Let π ′ be the matching obtained from π by deleting e i and its end vertices and we regard π ′ as a matching in M n . Since the deleted vertex i is matched in π, the block indices of vertices of π and π ′ are equal, so are the block differences. That is, bdiff π (e) = bdiff π ′ (e) for e ∈ E(π) \ {e i }. Since the number of block weak excedances and the block weight of π depend only on the block differences, we only need to consider the contribution of e i to bwex(π) and bwt(π). The lower block index bindex L π (n + 1) is one more than the number of unmatched vertices in the lower row, so bindex L π (n + 1) = n + 2 − e(π). Then e i is automatically a block weak excedance, so bwex(π) = bwex(π ′ ) + 1. To consider the block weight, let m be the number of matched upper vertices j such that i < j. It is clear that an edge e j crosses e i if and only if i < j, and hence cr(π) = cr(π ′ ) + m. It is easy to check that bindex U π (i) = i + 1 − e(π) + m, so bdiff π (e i ) = n + 1 − i − m and bwt(π) = bwt(π ′ )+n+1−i−m. Thus Case 2 corresponds to the term Case 3: The vertex n + 1 is unmatched and the vertex n + 1 is matched to some vertex i where i ≤ n. This case is similar to Case 2, except that the edge (n + 1, i) is not a block weak excedance. Letting M n be the set of matchings in M n such that n is unmatched and we obtain that Case 3 contributes −[n] q L n (x; q, y).
From Cases 1, 2 and 3, we have Comparing this with (6), it is enough to show that L n (x; q, y) = L n (x; q, y) − y[n] q L n−1 (x; q, y).
By the same argument in Case 2, the second term (including the negative sign) in the right-hand side of the above equation is equal to π∈Mn\ Mn then the proof follows.
Now we modify the combinatorial expression (4) for the moment µ n (q, y) so that the new expression is more suitable for our approach. For π ∈ PM n , the weight wt(π) of π is defined by In fact, this definition is obtained from the definition of the block weight by replacing block differences bdiff π (e) by π(i) − i. The following lemma gives a relation between ov(π), wt(π) and cr(π). Lemma 2.6. For π ∈ PM n , ov(π) = wt(π) − cr(π).
Proof. We prove that wt(π) = ov(π) + cr(π). By the definition of weight, On the right-hand side of the last equation, it is clear that the sum of the first and third summands is equal to ov(π).
One can see that each crossing of π is counted twice in the right-hand side of the above equation.
To be precise, a pair (i, j) of integers such that (e i , e j ) is a crossing of π is counted once either in the first or last summand depending on the sign of i − π(j), and counted once again either in the second or third summand depending on the sign of j − π(i). This completes the proof.
In the next section we will use Theorem 2.4 and (7) to give a combinatorial meaning to the linearization coefficients of q-Laguerre polynomials.

Linearization coefficients and a sign-reversing involution
3.1. A combinatorial interpretation of linearization coefficients. In this section we give a combinatorial interpretation of the linearization coefficient C(n 1 , . . . , n k ) := L(L n1 · · · L n k ) of the q-Laguerre polynomials L n = L n (x; q, y). First we recall the expression of L n in terms of matchings in Theorem 2.4: L n = π∈Mn (−1) e(π) y bwex(π) q bwt(π)+cr(π) x n−e(π) .
Let π ∈ M N . We say that an edge (i, π(i)) of π is homogeneous with respect to (n 1 , . . . , n k ) if n 1 + · · · + n r−1 + 1 ≤ i, π(i) ≤ n 1 + · · · + n r , for some 1 ≤ r ≤ k, and inhomogeneous otherwise. For simplicity, we omit the expression 'with respect to (n 1 , . . . , n k )' when there is no confusion. Note that M n1,...,n k is the set of matchings in M N such that every edge is homogeneous. We will write E H (π) for the set of homogeneous edges of π.
Here we recall the formula of the nth moment in (7): Note that N − e(π), the power of x in (9), represents the number of unmatched vertices in the upper (or lower) row, or equivalently, the number of edges we need to add to make it a perfect matching. Thus, applying the functional L to x N −e(π) is interpreted as summing up all possible ways to complete π into a perfect matching, by adding edges on the unmatched vertices, allowing inhomogeneous edges.  Example 3.1. Figure 4 describes an example of the application of L. The matching on the left side represents a term x 3 y 3 q 2 in L 2 L 3 L 2 , which is the product of three terms −xyq, xyq and −xy in L 2 , L 3 and L 2 , respectively. Applying L gives an equation π∈PM3 y wex(π) q wt(π)−cr(π) y 3 q 2 , where each summand corresponds to a way to add edges to remaining vertices, represented in dashed lines.
In order to describe the expansion of L(L n1 · · · L n k ), we introduce a perfect matching model containing the information of which edges are newly added by applying L. Let PM * n1,...,n k be the set of pairs m = (π, S) such that • S is a subset of edges in π, which contains all inhomogeneous edges of π, i.e., E(π) \ E H (π) ⊆ S.
We call an element m = (π, S) of PM * n1,...,n k a marked perfect matching. An edge e of π is said to be marked if e ∈ S. In other words, S is the set of marked edges. With marks on edges, we can distinguish new edges added by applying L from the original edges from L n1 · · · L n k . The condition E(π)\ E H (π) ⊆ S is needed since inhomogeneous edges cannot be present in the original matching coming from L n1 · · · L n k . Now we give a bijective correspondence between the terms in the expansion of L(L n1 · · · L n k ) and PM * n1,...,n k . To do this, we extend our former definitions of statistics on M n and PM n to marked perfect matchings. In detail, we consider the decomposition of m into unmarked and marked portions. For m = (π, S) ∈ PM * n1,...,n k , define π \ S and π| S as follows: • π \ S (unmarked portion of m) is the matching in M n1,...,n k with n 1 + · · · + n k − |S| edges obtained from π by deleting the |S| marked edges but leaving their incident vertices not deleted. • π| S (marked portion of m) is the perfect matching in PM |S| obtained from π by deleting all unmarked edges and their adjacent vertices.  More precisely, for a marked perfect matching m = (π, S) ∈ PM * n1,...,n k , let P = (B 1 , . . . , B l ) be the unique ordered set partition of upper vertices satisfying the following conditions: • Each B r consists of consecutive elements. In other words, B r is of the form B r = {i, i + 1, . . . , j}. • For i ∈ {1, . . . , n 1 + · · · + n k }, i is the largest element in some block B r if and only if i is incident to a marked edge or i = n 1 + · · · + n k .
The upper block index bindex U m (i) of a vertex i is defined to be the integer r such that i ∈ B r . Note that bindex U m (i) is equal to one more than the number of vertices incident to marked edges appearing before i. The lower block index bindex L m (i) is defined similarly. The block difference bdiff m (e) of an edge e = (i, π(i)) is defined by bdiff m (e) = bindex L m (π(i)) − bindex U m (i). The definitions of bwex(m) and wt(m) in Definition 3.2 are indeed equivalent to those in Definition 2.2 with bdiff π replaced by bdiff m .

3.2.
Construction of a sign-reversing involution. In order to prove Theorem 3.5, we give a sign-reversing involution Φ on PM * n1,...,n k that preserves the statistics bwex and wt +cr. Indeed, Φ will be a map that marks or unmarks a single homogeneous edge, or does not change anything. First we introduce some facts and definitions that we need to describe the map Φ.
For m = (π, S) ∈ PM * n1,...,n k , let us observe a change in the block difference of an edge e j while marking or unmarking a homogeneous edge e i . If we mark e i that was unmarked before, the upper (respectively, lower) index bindex U m (j) (respectively, bindex L m (π(j))) increases by 1 if and only if j > i (respectively, π(j) > π(i)). Therefore the block difference bdiff m (e j ) = bindex L m (π(j)) − if e j = e i , or e j and e i do not cross each other, bdiff m (e j ) + 1 if j < i and π(j) > π(i), bdiff m (e j ) − 1 if j > i and π(j) < π(i).

(11)
Conversely, if we unmark a marked edge e i ∈ E H (π) so that m = (π, S) turns into m ′ = (π, S\{e i }), then we have if e j = e i , or e j and e i do not cross each other, bdiff m (e j ) − 1 if j < i and π(j) > π(i), bdiff m (e j ) + 1 if j > i and π(j) < π(i).
From now on, let us adopt an expression e j crosses e i from the left, or equivalently e i crosses e j from the right for the relation j < i and π(j) > π(i). With this observation, we define the convertibility of a homogeneous edge, which is a key ingredient of the map Φ.
Definition 3.6. Let m = (π, S) ∈ PM * n1,...,n k . An edge e ∈ E H (π) is said to be convertible (in m) if it satisfies the following conditions.
(1) If e is unmarked, i.e., e / ∈ S, then for every edge e ′ that crosses e, either • e ′ crosses e from the left and bdiff m (e ′ ) ≥ 0, or • e ′ crosses e from the right and bdiff m (e ′ ) ≤ −1.
(2) If e is marked, i.e., e ∈ S, then for every edge e ′ that crosses e, either • e ′ crosses e from the left and bdiff m (e ′ ) > 0, or • e ′ crosses e from the right and bdiff m (e ′ ) < −1.
Note that if an edge e ∈ E H (π) is convertible, then the status of other edges being block weak excedances does not change under the map m = (π, S) → m ′ = (π, S△{e}), where X△Y denotes the symmetric difference (X ∪ Y ) \ (X ∩ Y ). In particular, marking or unmarking a convertible edge preserves the statistic bwex. Note also that an edge e is convertible in m = (π, S) if and only if it is convertible in m ′ = (π, S△{e}).

Remark 3.7.
Suppose that e ′ = (i, π(i)) is an inhomogeneous edge of m = (π, S) ∈ M n1,...,n k . Then n 1 + · · · + n r−1 + 1 ≤ i ≤ n 1 + · · · + n r and n 1 + · · · + n s−1 + 1 ≤ π(i) ≤ n 1 + · · · + n s for some r = s. It is easy to check that the block difference bdiff m (e ′ ) is nonzero, and its sign is determined by r and s. Thus, marking or unmarking a homogeneous edge e ∈ E H (m) does not change the status of whether e ′ is a block weak excedance or not. Therefore, it is sufficient to consider the changes of block differences of homogeneous edges when we toggle e.
We are now ready to define the involution Φ. In other words, we mark or unmark the homogeneous edge whose lower endpoint is the leftmost one among the homogeneous edges. Case 2: Suppose m has homogeneous edges and bdiff m (e) < 0 for some e ∈ E H (π). Let i = min j : e j ∈ E H (π), bdiff m (e j ) < 0 . Depending on the convertibility of the edge e i , we consider two subcases. For the well-definedness of Φ, the only part that is not clear is the existence of the number i ′ in Subcase 2-(b), or equivalently, j < i : e j ∈ E H (π), bdiff m (e j ) = 0, e j crosses e i = ∅.
We will prove this in Lemma 4.3.
Note that except for Case 0, Φ toggles only one edge's marking status. Hence Φ is sign-reversing. In the following section, we will prove that Φ is indeed a well-defined involution that preserves the statistics bwex and wt +cr.

Proof of Theorem 3.5
We start with a simple fact which will be used frequently throughout this section. Moreover, the inequality is strict if e i ∈ S or e j ∈ S.
Proof. By the assumption, we have i < j and π(i) > π(j) and the first statement follows from the relations Since the inequality (13) (respectively, (14)) holds strictly if e i ∈ S (respectively, e j ∈ S), we obtain the second statement.
Before proving Theorem 3.5, we verify the well-definedness of Φ in the following two lemmas. Recall that D(n 1 , . . . , n k ) is identified with the set of marked perfect matchings in PM * n1,...,n k such that all edges are inhomogeneous. Proof. Suppose that e is not marked, i.e., e / ∈ S. By the assumption that e is not convertible, there are two possibilities: • There is an edge e ′ ∈ E H (π) such that e ′ crosses e from the left and bdiff m (e ′ ) < 0, or • There is an edge e ′ ∈ E H (π) such that e ′ crosses e from the right and bdiff m (e ′ ) > −1. By the minimality of i, the first case cannot occur. For the second case, since e ′ crosses e from the right and bdiff m (e) < 0, we have bdiff m (e ′ ) ≤ bdiff m (e) < 0, which contradicts the fact that bdiff m (e ′ ) > −1. Therefore, e is a marked edge. Proof. By Lemma 4.2, e = e i is marked. By the assumption that e is not convertible, we have two possible cases: • There is an edge e ′ ∈ E H (π) such that e ′ crosses e from the left and bdiff m (e ′ ) ≤ 0, or • There is an edge e ′ ∈ E H (π) such that e ′ crosses e from the right and bdiff m (e ′ ) ≥ −1. Suppose an edge e ′ ∈ E H (π) crosses e from the right. Then we have bdiff m (e ′ ) < bdiff m (e) < 0, where the first inequality follows from the fact that e is marked. Thus the latter case cannot happen. Therefore there exists an edge e ′ corresponding to the first case. By the minimality of i, we have bdiff m (e ′ ) = 0 and, therefore, the given set is not empty.
We now give a proof of Theorem 3.5 by a sequence of lemmas. The first objective is to prove that the edge chosen by Φ to be toggled is convertible. Proof. We consider each case in Definition 3.8 except for Case 0, which does not occur since m ∈ D(n 1 , . . . , n k ). Case 1: Suppose m has homogeneous edges and bdiff m (e) ≥ 0 for all e ∈ E H (π). Then Φ(m) = (π, S△ {e i }) for the integer i satisfying π(i) = min π(j) : e j ∈ E H (π) . Suppose an edge e ∈ E H (π) crosses e i . By the minimality of π(i), e must cross e i from the left. By Proposition 4.1, we have bdiff m (e i ) ≤ bdiff m (e) if e i is unmarked and bdiff m (e i ) < bdiff m (e) if e i is marked. Since bdiff m (e i ) ≥ 0 we conclude that e i is convertible. Case 2: Suppose m has homogeneous edges and bdiff m (e) < 0 for some e ∈ E H (π). Let i = min{j : e j ∈ E H (π), bdiff m (e j ) < 0}.
There are two subcases. Let e = e i and e ′ = e i ′ . To check the convertibility of e ′ , let e ′′ = e i ′′ ∈ E H (π) be an edge that crosses e ′ .
Case 1: Suppose m has homogeneous edges and bdiff m (e) ≥ 0 for all e ∈ E H (π). Then m ′ = Φ(m) = (π, S△ {e i }) for the integer i satisfying π(i) = min π(j) : e j ∈ E H (π) . Since e i is convertible by Lemma 4.4, we also have bdiff m ′ (e) ≥ 0 for all e ∈ E H (π). That is, when we apply Φ to m ′ we are still in Case 1. Since the edge e i toggled by Φ depends only on π, the map Φ toggles the same edge e i in m ′ and we have Φ(m ′ ) = m. Case 2: Suppose m has homogeneous edges and bdiff m (e) < 0 for some e ∈ E H (π). Let i = min{j : e j ∈ E H (π), bdiff m (e j ) < 0}.
Note that in this case, the set of edges having negative block difference is invariant under Φ by the convertibility of the edge toggled by Φ. Hence when we apply Φ to m ′ we are Lemmas 4.5, 4.6 and 4.7 imply that Φ is a sign-reversing and weight-preserving involution on PM * n1,...,n k with fixed point set D(n 1 , . . . , n k ). Thus C(n 1 , . . . , n k ) = m∈D(n1,...,n k ) (−1) e(m) y bwex(m) q wt(m)+cr(m) .
It is an open problem to find a combinatorial interpretation for (16) generalizing both (17) and (18). In this paper we found a sign-reversing involution proving (17). Unfortunately, our approach does not apply directly to (18). Finding a combinatorial proof of (18) is still open. If y = q = 1, then a simple combinatorial proof of (18) was found by Foata and Zeilberger [5].
Finally, we note that there is a combinatorial interpretation of the linearization coefficient L(P n1 (x) . . . P n k (x)) for any orthogonal polynomials in terms of weighted Motzkin paths due to de Médicis and Stanton [3]. Their result immediately implies the nonnegativity of (16). It might be interesting to study the linearization coefficients of the q-Laguerre polynomials considered here using their combinatorial model.