Evaluations of certain Catalan-Hankel Pfaffians via classical skew orthogonal polynomials
aa r X i v : . [ m a t h . C A ] J a n EVALUATIONS OF CERTAIN CATALAN-HANKEL PFAFFIANS VIACLASSICAL SKEW ORTHOGONAL POLYNOMIALS
BO-JIAN SHEN, SHI-HAO LI, AND GUO-FU YU
Abstract.
This paper is to evaluate certain Catalan-Hankel Pfaffians by the theory of skeworthogonal polynomials. Due to different kinds of hypergeometric orthogonal polynomials un-derlying the Askey scheme, we explicitly construct the classical skew orthogonal polynomialsand then give different examples of Catalan-Hankel Pfaffians with continuous and q -momentsequences. Introduction
Hankel determinants as a specific determinant has attracted much attention from researchers inmany different subjects (see, for example [5, 12, 29] and references therein). It is well known thatif { µ m } m ≥ is the moment sequence taking the form µ n = Z R x n ω ( x ) dx, then the Hankel determinant det( µ i + j ) n − i,j =0 has a nice integral formula due to the Andreiéf formula[14] det( µ i + j ) n − i,j =0 = 1 n ! Z R n Y ≤ i Key words and phrases. Catalan-Hankel Pfaffian; Classical orthogonal polynomials; Skew orthogonalpolynomials. where { µ n } n ≥ is a moment sequence related to discrete q -measure. The specific little q -Jacobicase were considered in [19] and some recent developments like the Al-Salam & Carlitz I case wasgiven in [20].Regarding evaluations of ( q -)Catalan-Hankel Pfaffian, one application is to give formula tothe weighted enumeration of some specific plane partitions. For example, in [19], the authorsconsidered the little q -Jacobi case and enumerated a special family of shifted reverse plane partitionswith weights that resemble the weight in the inner product of little q -Jacobi polynomials. Theimportance of these evaluations lies in the random matrix theory as well. In an earlier work ofMehta and Wang [25], they gave an evaluation of the Hankel PfaffianPf [( j − i )Γ( i + j + b )] N − i,j =0 , (1.3)which was related to the skew orthogonal Laguerre polynomials given in [1]. Moreover, the shiftedCatalan-Hankel PfaffianPf [( j − i ) µ i + j + r ] N − i,j =0 or Pf [([ j ] q − [ i ] q ) µ i + j + r ] N − i,j =0 is closely related to the adjacent families of skew-orthogonal polynomials considered in [23]. Theseevaluations may give more hints in exploring novel examples in integrable systems and randommatrices.As mentioned, one way to evaluate these Pfaffians is based on the de Bruijn formula and Sel-berg integral [19, 20]. In this paper, we will investigate another way, namely by using the the-ory of classical ( q -)skew orthogonal polynomials, to make these evaluations and demonstrate theeffectiveness—by simply considering different classsical weights underlying Askey scheme, we canget different examples of ( q -)Catalan-Hankel Pfaffians. Since the Hankel determinant is closelyrelated to the theory of classical orthogonal polynomials, then if we know normalisation constantsof these classical orthogonal polynomials, the evaluations of Hankel determinant can be made. Itis natural to ask whether we can apply the theory of classical ( q -)skew orthogonal polynomials intoevaluations of the certain ( q -)Catalan-Hankel Pfaffians. The answer is affirmative. If we know thenormalisation factors of skew orthogonal polynomials, then these Pfaffians can be explicitly givenby those normalisation factors. Due to different expressions in discrete and continuous cases (c.f.(1.1) and (1.2)), we discuss them separately. Continuous cases including Hermite-type, Laguerre-type and Jacobi-type skew orthogonal polynomials were firstly constructed by Adler et al [1] andwe give a brief review in section 2. Some evaluations of Catalan-Hankel Pfaffians related to theseweights including formula (1.3) will be demonstrated. Moreover, we consider a Catalan-HankelPfaffian with Cauchy weight in that part. Evaluations of q -Catalan-Hankel Pfaffians will be basedon the theory of classical q -skew orthogonal polynomials. With the help of discrete Pearson rela-tion, we construct different examples of classical ( q -)skew orthogonal polynomials, thus obtainingdifferent kinds of skew orthogonal polynomials and normalisation factors with respect to differentclassical q -weights, which enlarge the results given by [16, 20].2. Continuous measure Let’s consider the skew inner product related to (1.1) h φ ( x ) , ψ ( x ) i ,ω = 12 Z R [ φ ( x ) ψ ′ ( x ) − φ ′ ( x ) ψ ( x )] ω ( x ) dx, (2.1) VALUATION OF CERTAIN PFAFFIANS 3 then the skew moments are given by m i,j := h x i , x j i ,ω = 12 ( j − i ) Z R x i + j − ω ( x ) dx. The skew LU decomposition (or so-called skew Borel decomposition [2]) of the moment matrix ( m i,j ) i,j ∈ N could give rise to the monic skew orthogonal polynomials { Q j ( x ) } j ∈ N satisfying theskew orthogonal relation h Q n ( x ) , Q m +1 ( x ) i ,ω = u n δ n,m , h Q n ( x ) , Q m ( x ) i ,ω = h Q n +1 ( x ) , Q m +1 ( x ) i ,ω = 0 (2.2)for certain u n > . Moreover, these skew orthogonal polynomials have the following Pfaffianexpressions [8] Q n ( x ) = 1 τ n Pf (0 , · · · , n, x ) , Q n +1 ( x ) = 1 τ n Pf (0 , · · · , n − , n + 1 , x ) with τ n = Pf (0 , · · · , n − and the Pfaffian elements are given by Pf ( i, j ) = m i,j and Pf ( i, x ) = x i .By putting m i,j into the expression of τ n , we have τ n = 12 n Pf [( j − i ) µ i + j − ] n − i,j =0 , µ n = Z R x n ω ( x ) dx. (2.3)Interestingly, u n in the skew orthogonal condition (2.2) is the ratio τ n +2 /τ n , and therefore, if wecan explicitly compute { u n } n ∈ N in the skew orthogonal condition, then we directly have τ N = Q N − i =0 u i . In fact, normalisation factors { u n } n ∈ N can be explicitly computed when { Q n ( x ) } n ∈ N are classical skew orthogonal polynomials. For details, please refer to [1] and we give a brief reviewhere.Starting from the inner product h φ ( x ) , ψ ( x ) i ,ρ = Z R φ ( x ) ψ ( x ) ρ ( x ) dx, one can construct a family of monic orthogonal polynomials { p j ( x ) } j ∈ N satisfying the orthogonalrelation h p j ( x ) , p k ( x ) i ,ρ = h j δ j,k . We call these orthogonal polynomials classical if the weight function ρ ( x ) satisfies the followingPearson equation ρ ′ ( x ) ρ ( x ) = − g ( x ) f ( x ) with deg f ( x ) ≤ and deg g ( x ) ≤ . From the relation, one can construct an operator A = f ∂ x + f ′ − g , (2.4)such that h φ ( x ) , A ψ ( x ) i ,ρ = h φ ( x ) , ψ ( x ) i ,ω , ω ( x ) = ρ ( x ) f ( x ) , where the skew inner product h· , ·i ,ω was given in (2.1). This is the key formula to establishthe relation between classical orthogonal polynomials and classical skew orthogonal polynomials.Moreover, the normalisation factor u n could be computed via A p k ( x ) = − c k h k +1 p k +1 ( x ) + c k − h k − p k − ( x ) , u n = c n , (2.5) BO-JIAN SHEN, SHI-HAO LI, AND GUO-FU YU with h k the normalisation factor of orthogonal polynomials. Therefore, to obtain normalisationfactors of skew orthogonal polynomials, our attention should be paid to the computation of c k inthe above equation. The fastest method to compute c k is to compare the coefficients of x k +1 onboth sides and the following are the examples of the continuous classical weights including Hermite,Laguerre, Jacobi and Cauchy weights.From weight functions ρ ( x ) = e − x , Hermite ,x a e − x , Laguerre ,x a (1 − x ) b , Jacobi , (1 + x ) − a , Cauchy , one can obtain the Pearson pair ( f, g ) = (1 , x ) , Hermite , ( x, x − a ) , Laguerre , ( x (1 − x ) , ( a + b ) x − a ) , Jacobi , (1 + x , ax ) , Cauchy . This table was given by [13, eq. (5.58)] and it should be remarked that we use the weight functionof Jacobi as x a (1 − x ) b supported in [0 , to make the moments easily written down. Thereare two quantities obtained from this table. One is the Hankel sequences given in (2.3). From ω ( x ) = f ( x ) ρ ( x ) , one knows that the weights of classical skew orthogonal polynomials are ω ( H ) ( x ) = e − x , ω ( L ) ( x ) = x a +1 e − x , ω ( J ) ( x ) = x a +1 (1 − x ) b +1 , ω ( C ) ( x ) = (1 + x ) − a +1 . Therefore, we have the following moments form µ ( H ) n = 1 + ( − n (cid:18) n + 12 (cid:19) , µ ( L ) n = Γ( n + a + 2) ,µ ( J ) n = Γ( n + a + 2)Γ( b + 2)Γ( n + a + b + 4) , µ ( C ) n = 1 + ( − n n + 1) / a − − ( n + 1) / a − . On the other hand, we can obtain c k = h ( H ) k +1 , Hermite ,h ( L ) k +1 / , Laguerre , ( k + 1 + ( a + b ) / h ( J ) k +1 , Jacobi , ( a − − k ) h ( C ) k +1 , Cauchywhere { h k } k ∈ N are normalisation factors with respect to different weights [13, Chap. 5] h ( H ) k = π / − k k ! , h ( L ) k = Γ( k + 1)Γ( a + k + 1) ,h ( J ) k = 2 a + b +1+2 k Γ( k + 1)Γ( a + b + 1 + k )Γ( a + 1 + k )Γ( b + 1 + k )Γ( a + b + 2 k + 1)Γ( a + b + 2 k + 2) ,h ( C ) k = π k +2 a +2 Γ( k + 1)Γ( − k − a )Γ( − a − k − − a − k )(Γ( − a − k )) . By using above results, we could state the following proposition. VALUATION OF CERTAIN PFAFFIANS 5 Proposition 2.1. We have the following evaluations of certain Catalan-Hankel PfaffiansPf h ( j − i ) µ ( H ) i + j − i N − i,j =0 = 2 − N ( N − ( √ π ) N N − Y i =0 Γ(2 i + 2) , Pf h ( j − i ) µ ( L ) i + j − i N − i,j =0 = N − Y i =0 Γ(2 i + 2)Γ(2 i + a + 2) , Pf h ( j − i ) µ ( J ) i + j − i N − i,j =0 = 4 N Γ( N + ( a + b + 2) / a + b + 2) / N − Y i =0 Γ( a + 2 i + 2)Γ( b + 2 i + 2)Γ(2 i + 3)Γ( a + b + 4 i + 3) , Pf h ( j − i ) µ ( C ) i + j − i N − i,j =0 = ( − N N − N ( a − Γ( N + (1 − a ) / − a ) / N − Y i =0 Γ(2 a − i − i + 3)(Γ( a − i − . Remark 2.2. The second formula is exactly equation (1.3) with the shift a → a − . Discrete measure: q -case This part is devoted to the evaluations of q -Catalan Hankel Pfaffians given by formula (1.2).The q -case corresponds to a special discrete measure distributed on exponential lattices x ( i ) = q i with < q < and i ∈ Z . By using the definition of Jackson’s q -integral Z ∞ f ( x ) d q x = (1 − q ) ∞ X s = −∞ f ( q s ) q s , one can define the following inner product h φ ( x ) , ψ ( x ) i ,ρ := Z ∞ φ ( x ) ψ ( x ) ρ ( x ) d q x = (1 − q ) X s ∈ Z φ ( q s ) ψ ( q s ) ρ ( q s ) q s . (3.1)With this inner product, q -orthogonal polynomials { p n ( x ; q ) } n ∈ N are defined by the orthogonalrelation h p i ( x ; q ) , p j ( x ; q ) i ,ρ = h n ( q ) δ n,m (3.2)with respect to the weight ρ ( x ; q ) . Moreover, if ρ ( x ; q ) is a classical weight, we call the correspondingorthogonal polynomials the classical q -orthogonal polynomials. Classical q -orthogonal polynomialsinclude many interesting examples underlying the q -Askey scheme. For details, please refer to[3, 21, 22]. One important property of classical q -orthogonal polynomials is that the weight functionsatisfies an analogy of Pearson relation given by Nikiforov and Suslov [26] ρ ( qx ) ρ ( x ) = f ( x ) − q − (1 − q ) xg ( x ) f ( qx ) with deg f ( x ) ≤ and deg g ( x ) ≤ . (3.3)In the following, we will demonstrate how to connect q -inner product with q -skew inner product.Let’s define a q -analogy of the skew inner product (2.1) h φ ( x ) , ψ ( x ) i ,ω = Z ∞ [ φ ( x ) D q ψ ( x ) − D q φ ( x ) ψ ( x )] ω ( x ) d q x (3.4)with q -difference operator D q f ( x ) = f ( x ) − f ( qx )(1 − q ) x . The definition of q -integral on the interval [0 , ∞ ) is different from the one defined on [0 , , see [21]. However,these two cases can be treated similarly and we just consider the former one here. BO-JIAN SHEN, SHI-HAO LI, AND GUO-FU YU Then by defining an operator A q [16] A q = q − g ( x ) T q + q − f ( x ) D q − + f ( x ) D q , T q f ( x ) = f ( qx ) , (3.5)one can find such a connection formula h φ ( x ; q ) , A q ψ ( x ; q ) i ,ρ = h φ ( x ; q ) , ψ ( x ; q ) i ,ω , ω ( x ) = ρ ( qx ) f ( qx ) . (3.6)On the other hand, from the skew inner product (3.1), one has the following ( q -)skew moments m i,j := ([ j ] q − [ i ] q ) Z ∞ x i + j − ω ( x ) d q x, [ j ] q = 1 − q j − q . (3.7)Similarly, it is known that a family of monic skew orthogonal polynomials { Q i ( x ; q ) } i ∈ N could beconstructed from those moments if even-order moment matrices Pf ( m i,j ) n − i,j =0 = 0 for all n ∈ N + [16]. Furthermore, polynomials { Q i ( x ; q ) } i ∈ N admit the following Pfaffian expressions Q n ( x ; q ) = 1 τ n ( q ) Pf (0 , · · · , n, x ) , Q n +1 ( x ; q ) = 1 τ n ( q ) Pf (0 , · · · , n − , n + 1 , x ) τ n ( q ) = Pf (0 , · · · , n − , Pf ( i, j ) = m i,j , Pf ( i, x ) = x i , and they satisfy the following skew orthogonal relations h Q n ( x ; q ) , Q m +1 ( x ; q ) i ,ω = τ n +2 ( q ) τ n ( q ) := u n ( q ) δ n,m , h Q n ( x ; q ) , Q m ( x ; q ) i ,ω = h Q n +1 ( x ; q ) , Q m +1 ( x ; q ) i ,ω = 0 . Therefore, one knows thatPf (cid:20) ([ j ] q − [ i ] q ) Z ∞ x i + j − ω ( x ) d q x (cid:21) N − i,j =0 = Pf (0 , . . . , N − 1) = N − Y i =0 u i ( q ) . Interestingly, there is a method to evaluate the value of u n ( q ) quickly by taking advantage of q -Pearson relation.As an analogy of equation (2.5), there holds the formula A q p k ( x ; q ) = − c k ( q ) h k +1 ( q ) p k +1 ( x ; q ) + c k − ( q ) h k − ( q ) p k − ( x ; q ) (3.8)in the q -case due to the property of A q [16, eq. (4.28)]. Moreover, the quantity h k ( q ) is thenormalisation factor of the orthogonal relation (3.2) and c k ( q ) is closely related to u k ( q ) via therelation u k ( q ) = c k ( q ) . Therefore, it is the point to compute c k ( q ) from above equation and thenthe exact value of q -Catalan Hankel Pfaffian could be obtained. In the following part, we discussseveral different cases including the examples in [19, 20].3.1. Al-Salam & Carlitz I case. The first example considered here is the Al-Salam & Carlitz Icase with the weight function ρ ( x ; q ) = ( qx, a − qx ; q ) ∞ , a < . It is well known that the Al-Salam & Carlitz polynomials { U ( a ) n ( x ; q ) } n ≥ have the orthogonality Z a U ( a ) m ( x ; q ) U ( a ) n ( x ; q ) ρ ( x ; q ) d q x = ( − a ) n (1 − q )( q ; q ) n ( q, a, a − q ; q ) ∞ q ( n ) δ n,m := h n δ n,m , VALUATION OF CERTAIN PFAFFIANS 7 and canonical moments were given by [27] µ n = Z a x n ρ ( x ; q ) d q x = (1 − q ) (cid:0) q, a, a − q ; q (cid:1) ∞ n X i =0 " ni q a i Specifically, Al-Salam & Carlitz I polynomials have q -hypergeometric function expressions U ( a ) n ( x ; q ) = ( − a ) n q ( n ) φ q − n , x − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ; qxa ! . As was shown in [16, Sec. 4.4.1(1)], the Pearson pair in this case is ( f, g ) = (cid:18) x − (1 + a ) x + a, q / − q ( x − (1 + a )) (cid:19) , and the coefficient c n in (3.8) can be explicitly computed as c n = − q − n h n +1 / (1 − q ) . Thus themoment Pfaffian can be written asPf ( m i,j ) N − i,j =0 = N − Y k =0 a k +1 ( q ; q ) k +1 q ( k ) (cid:0) q, a, a − q ; q (cid:1) ∞ = a N q N ( N − N − (cid:0) q, a, a − q ; q (cid:1) N ∞ N − Y k =0 ( q ; q ) k +1 , where m i,j = ([ j ] q − [ i ] q ) R a x i + j − ω ( x ; q ) d q x . Since ω ( x ; q ) = f ( qx ; q ) ρ ( qx ; q ) = aρ ( x ; q ) , we have m i,j = a ([ j ] q − [ i ] q ) µ i + j − = ( q i − q j ) (cid:0) q, a, a − q ; q (cid:1) ∞ i + j − X k =0 " i + j − k q a k +1 . Dividing both sides by (cid:2) a ( q, a, a − q ; q ) ∞ (cid:3) N leads toPf ( q i − q j ) i + j − X k =0 " i + j − k q a k N − i,j =0 = a N ( N − q N ( N − N − N − Y k =0 ( q ; q ) k +1 . Remark 3.1. With a = − , Al-Salam & Carlitz I polynomials reduce to the q -Hermite I poly-nomials [22, Sec. 14.28], therefore the above mentioned method can be applied to the q -Hermite Icase as well. The moments of q-Hermite I polynomials take the form [27] µ n = (1 − q )( q, − , − q ; q ) ∞ − n (cid:0) q ; q (cid:1) n/ . Thus we have the following evaluation of q -Catalan-Hankel PfaffianPf (cid:18) ( q i − q j ) 1 + ( − i + j − q ; q ) i + j − (cid:19) N − i,j =0 = ( − N q N ( N − N − N − Y k =0 ( q ; q ) k +1 . Stieltjes-Wigert case. The Stieltjes-Wigert polynomials are well studied in random matrixtheory of the so-called Stieltjes-Wigert ensemble, which firstly appeared in the study of non-intersecting Brownian walkers, and subsequently in quantum many body systems etc.. For adetailed review, please refer to [15] and references therein. The weight of Stieltjes-Wigert polyno-mials was first given by Stieltjes as an example of indeterminate moment problems [28] and further BO-JIAN SHEN, SHI-HAO LI, AND GUO-FU YU studied by Wigert [30]. Usually there are several different expressions for the Stieltjes-Wigert’sweight function [11]. The original one corresponding to a continuous measure is w ( x ) = 1 √ π kx − k log x , x > with the moment µ n = R ∞ x n w ( x ) dx = e ( n +1) / k . If we set q = e − / k , then µ n can bewritten as q − ( n +1) / . With this measure, Wigert found the following expression for Stieltjes-Wigert polynomials [30] P n ( x ) = ( − n q n/ / p ( q ; q ) n n X k =0 " nk q ( − k q k + k/ x k , with orthogonality Z ∞ P n ( x ) P m ( x ) w ( x ) dx = δ mn . In [9], Chihara proposed a discrete weight on the exponential lattices admitting the form ξ ( x ) = ( √ qM q n + n / , x = q n , x = q n , M = ( − q √ q, − q − / , q ; q ) ∞ , n ∈ Z . Then the corresponding inner product becomes h p n ( x ) , p m ( x ) i ,ξ = 1 √ qM ∞ X k = −∞ p n ( q k ) p m ( q k ) q k + k / We can prove that this discrete weight is equivalent to the continuous one by showing that theyhave same moments. For completeness, we give a short proof to this fact.One can easily check that ξ ( qx ) = q xξ ( x ) , x ∈ R and therefore obtain a recurrence relation for the moments { µ n } n ∈ N µ n := h x n , i ,ξ = ∞ X k = −∞ q nk ξ ( q k ) = ∞ X k = −∞ q n ( k +1) ξ ( q k +1 )= q n + ∞ X k = −∞ q ( n +1) k ξ ( q k ) = q n + µ n +1 . As a result, we have µ n = q − n / − n µ . By making use of the Jacobi triple product identity [18,Thm 352] ∞ X n = −∞ ( − n q ( n ) x n = ( x, q/x, q ; q ) ∞ ,µ can be directly computed as µ = 1 √ qM ∞ X k = −∞ q k + k / = 1 √ q . Thus µ n = q − ( n +1)22 , which is the same as that of the continuous measure. VALUATION OF CERTAIN PFAFFIANS 9 For the present need, we would consider the discrete measure. Denote ρ ( x ; q ) = 1 √ qM x ln x q to be the weight function and define p n ( x ) = p ( q ; q ) n q n + n + P n ( x ) to be the monic Stieltjes-Wigert polynomials. We can rewrite the orthogonality in terms of ρ ( x ) and { p n ( x ) } n ≥ as Z ∞ p n ( x ) p m ( x ) ρ ( x ) d q x = (1 − q ) ( q ; q ) n q n +2 n + δ mn := h n δ n,m . By assuming f ( x ) = x and solving the Pearson equation (3.3), we get g ( x ) = ( q x − q / ) / ( q − which leads to c n ( q ) = q n + − q h n +1 . As a consequence, we have the following evaluation of the moment PfaffianPf ( m i,j ) N − i,j =0 = q − N ( N +1)(8 N − N − Y k =0 ( q ; q ) k +1 . On the other hand, we can compute the skew moments (3.7) as a scalar product of canonicalmoments µ n m i,j =([ j ] q − [ i ] q ) Z ∞ x i + j − ω ( x ; q ) d q x = ([ j ] q − [ i ] q ) Z ∞ x i + j +1 q ρ ( x ; q ) d q x =([ j ] q − [ i ] q ) q / µ i + j +1 = ([ j ] q − [ i ] q ) q − [( i + j +2) − / . By combining above results, we have the evaluationPf (([ j ] q − [ i ] q ) q − ( i + j +2) / ) N − i,j =0 = q − N (2 N +1)(8 N − N − Y k =0 ( q ; q ) k +1 . Little q -Jacobi case. Little q -Jacobi polynomials are important in many mathematical fieldssuch as polynomials theory [22] and quantum group [24]. These polynomials have the followingseries form (c.f. [24, eq. (2.21)]) p ( α,β ) n ( z ; q ) = X r ≥ ( q − n ; q ) r ( q α + β + n +1 ; q ) r ( q ; q ) r ( q α +1 ; q ) r ( qz ) r and they obey the following orthogonal relation [24, Prop. 3.9] Z p ( α,β ) m ( z ; q ) p ( α,β ) n ( z ; q ) z α ( qz ; q ) β d q z = δ n,m q ( α +1) n (1 − q )( q ; q ) α ( q ; q ) β + n ( q ; q ) n (1 − q α + β +2 n +1 )( q ; q ) α + n ( q ; q ) α + β + n . In this case, the canonical moments are given by [27] µ ( α,β ) n = Z z α + n ( qz ; q ) β d q z = (1 − q )( q α + β +1 ; q ) ∞ ( q ; q ) ∞ ( q α +1 ; q ) n (1 − q α + β +1 )( q α +1 ; q ) ∞ ( q β +1 ; q ) ∞ ( q α + β +2 ; q ) n . If we define monic little q -Jacobi polynomials { ˜ p ( α,β ) n ( z ; q ) } n ≥ ˜ p ( α,β ) n ( z ; q ) = ( − n q n ( n − ( q α +1 ; q ) n ( q α + β + n +1 ; q ) n p ( α,β ) n ( z ; q ) , then they satisfy the orthogonal relation Z ˜ p ( α,β ) n ( z ; q )˜ p ( α,β ) m ( z ; q ) z α ( qz ; q ) β d q z = q n ( n + α ) [ α + β + 2 n + 1] − q ( q ; q ) n ( q ; q ) n + α ( q ; q ) n + β ( q ; q ) n + α + β ( q ; q ) n + α + β δ n,m := h n δ n,m . (3.9)Regarding the weight of little q -Jacobi polynomials ρ ( z ; q ) = z α ( qz ; q ) β , by solving (3.3), it admits the Pearson pair ( f, g ) = (cid:16) − x + x, − q ([ α + β + 2] q x − [ α + 1] q ) (cid:17) . Therefore, the coefficient c n has the expression c n = q − n [2 n + 2 + α + β ] q h n +1 , where h n is the nomalization constant in the orthogonal relation of monic polynomials (3.9). Thenwe have the following expression for the moment PfaffianPf ( m ( α,β ) ij ) N − i,j =0 = q N (4 N − N +2)+ αN × N − Y k =0 (1 − q α + β +4 k +2 )( q, q α +1 , q β +1 , q α + β +1 ; q ) k +1 ( q ; q ) ∞ ( q α + β +1 ; q ) ∞ ( q α +1 , q β +1 ; q ) ∞ ( q α + β +1 ; q ) k +2 ( q α + β +1 ; q ) k +3 . Since the weight of the corresponding skew orthogonal little q -Jacobi polynomials is ω ( x ; q ) = f ( qx ; q ) ρ ( α,β ) ( qx ; q ) = q α +1 ρ ( α +1 ,β +1) ( x ; q ) = q α +1 x α +1 ( qx ; q ) β +1 , the skew moments { m ( α,β ) i,j } i,j ∈ N are related to the canonical moments by m ( α,β ) i,j =([ j ] q − [ i ] q ) Z x i + j − ω ( x ; q ) d q x = ([ j ] q − [ i ] q ) q α +1 µ ( α +1 ,β +1) i + j − =([ j ] q − [ i ] q ) q α +1 (1 − q )( q α + β +1 ; q ) ∞ ( q ; q ) ∞ ( q α +1 ; q ) n (1 − q α + β +1 )( q α +1 ; q ) ∞ ( q β +1 ; q ) ∞ ( q α + β +2 ; q ) n . After eliminating the constant (cid:2) q α +1 ( q α + β +1 ; q ) ∞ ( q ; q ) ∞ / (cid:0) (1 − q α + β +1 )( q α +1 , q β +1 ; q ) ∞ (cid:1)(cid:3) N , wehavePf (cid:18) ([ j ] q − [ i ] q ) (1 − q )( q α +2 ; q ) i + j − ( q α + β +4 ; q ) i + j − (cid:19) N − i,j =0 (3.10) = q N ( N − N +1)+ αN ( N − N − Y k =0 ( q ; q ) k +1 ( q α +2 , q β +2 ; q ) k ( q α + β +4 ; q ) k − ( q α + β +2 k +2 ; q ) k ( q α + β +2 k +2 ; q ) k +2 , which coincides with the result in [20, Thm. 5.1]. VALUATION OF CERTAIN PFAFFIANS 11 Big q -Jacobi case. Big q -Jacobi polynomials were introduced by Andrews and Askey asan infinite-dimentional version of the q -Hahn polynomials [4]. In addition, big q -Jacobi polyno-mials were also contained in the Bannai-Ito scheme of dual systems of orthogonal polynomialsas an infinite dimension analogue of the q-Racah polynomials [6]. These polynomials take thehypergeometric function form P n ( x ; a, b, c ; q ) = φ q − n , abq n +1 , xaq, cq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ; q ! , orthogonal with respect to the weight ρ ( a,b,c ) ( x ; q ) = ( a − x, c − x ; q ) ∞ ( x, bc − x ; q ) ∞ and have the orthogonality Z aqcq P m ( x ; a, b, c ; q ) P n ( x ; a, b, c ; q ) ρ ( a,b,c ) ( x ; q ) d q x = aq (1 − q ) (cid:0) q, a − c, ac − q, abq ; q (cid:1) ∞ ( aq, bq, cq, abc − q ; q ) ∞ (1 − abq )(1 − abq n +1 ) (cid:0) q, bq, abc − q ; q (cid:1) n ( abq, aq, cq ; q ) n (cid:0) − acq (cid:1) n q ( n ) δ mn . One can easily check that the normalisation constant for the monic polynomials is h n = aq (1 − q )( − acq ) n q ( n )( q, a − c, ac − q, abq ; q ) ∞ ( q, aq, bq, cq, abc − q ; q ) n (1 − abq n +1 )( aq, bq, cq, abc − q ; q ) ∞ ( abq ; q ) n ( abq n +1 ; q ) n and the canonical moments are given by [27] µ ( a,b,c ) n = Z aqcq x n ρ ( a,b,c ) ( x ; q ) d q x = aq (cid:0) abq , a − c, ac − q ; q (cid:1) ∞ ( aq, bq, cq, abc − q ; q ) ∞ n X m =0 ( − m " nm q q − nm + ( m +12 ) ( aq, cq ; q ) m ( abq ; q ) m . Solving equation (3.3) gives the Pearson pair ( f, g ) = (1 − xaq )(1 − xcq ) , q − q (cid:18) ( 1 acq − bc ) x + bc + 1 − aq − cq (cid:19)! . Then, by comparing the leading coefficients on both sides of (3.8) we have c n = abq n +2 − ac (1 − q ) q n +2 h n +1 , which leads to the following evaluation of the moment PfaffianPf ( m ( a,b,c ) ij ) N − i,j =0 =( − N c N ( N − a N q N ( N +1)(4 N − × N − Y k =0 ( q, a − c, ac − q, abq ; q ) ∞ ( q, aq, bq, cq, abc − q ; q ) k +1 ( aq, bq, cq, abc − q ; q ) ∞ ( abq ; q ) k + N )+1 . where m ( a,b,c ) i,j is given by m ( a,b,c ) i,j = ([ j ] q − [ i ] q ) Z aqcq x i + j − ω ( x ; q ) ( a,b,c ) d q x, and ω ( x ; q ) is expressed by ω ( x ; q ) ( a,b,c ) = f ( qx ; q ) ρ ( a,b,c ) ( qx ; q ) = (1 − x )(1 − bc − x ) ρ ( x ; q ) ( a,b,c ) = ρ ( qa,qb,qc ) ( qx ; q ) . Note that µ ( a,b,c ) n = Z aqcq x n ρ ( a,b,c ) ( x ; q ) d q x = q n +1 Z ac x n ρ ( a,b,c ) ( qx ; q ) d q x, we have the expression m ( a,b,c ) i,j =([ j ] q − [ i ] q ) q − ( i + j ) µ ( qa,qb,qc ) i + j − =([ j ] q − [ i ] q ) aq − ( i + j − (cid:0) abq , a − c, ac − q ; q (cid:1) ∞ ( aq, bq, cq, abc − q ; q ) ∞ n X m =0 ( − m " nm q q − nm + ( m +12 ) ( aq, cq ; q ) m ( abq ; q ) m . Dividing both sides by (cid:2) aq (cid:0) abq , a − c, ac − q ; q (cid:1) ∞ / (cid:0) aq, bq, cq, abc − q ; q (cid:1) ∞ (cid:3) N and changing variables ( a, b, c ) → ( q − a, q − b, q − c ) , we getPf ([ j ] q − [ i ] q ) i + j − X m =0 ( − m " i + j − m q q − ( i + j − m +1)+ ( m +12 ) ( aq, cq ; q ) m ( abq ; q ) m N − i,j =0 = ( − N ( ac ) N ( N − q N (4 N − N +11) N − Y k =0 ( q ; q ) ∞ ( q ; q ) k +1 ( a, b, c, abc − q ; q ) k ( abq ; q ) k + N ) − . Remark 3.2. With a = b = 1 , big q-Jacobi polynomials reduce to the big q-Legendre polynomials[23]. In particular, we have the following evaluation of q-Catalan-Hankel PfaffianPf ([ j ] q − [ i ] q ) i + j − X m =0 ( − m " i + j − m q q − ( i + j − m +1)+ ( m +12 ) ( q, cq ; q ) m ( q ; q ) m N − i,j =0 = ( − N ( c ) N ( N − q N (4 N − N +11) N − Y k =0 ( q ; q ) ∞ ( q ; q ) k +1 ( q, q, cq, c − q ; q ) k ( q ; q ) k + N ) − . Further remarks In this paper, we have developed a method based on the relation between classical ( q -)orthogonaland ( q -)skew orthogonal polynomials to evaluate certain q -Catalan-Hankel Pfaffians whose entriesare composed of the moments of classical orthogonal polynomials. Some examples are given toillustrate the approach including the continuous ones (e.g. Hermite, Laguerre, Jacobi and Cauchy)and discrete q -cases (e.g. Al-Salam & Carlitz I, little q -Jacobi, Stieltjes-Wigert and big q -Jacobipolynomials). Amoung those examples, the Al-Salam & Carlitz I and Little q -Jacobi case arecompared with the results obtained by M. Ishikawa and J. Zeng in [20] and we present alternativeproofs of [20, eq. (6.7) & Thm 5.2]. Besides, the examples in [20, Conjecture 7.1] seem to berelated to some discrete measure on the linear lattice. However, as mentioned in [16], the skewmoments defined in linear lattices are of the form µ i,j = ( j − i ) µ i + j − + ( (cid:0) j (cid:1) − (cid:0) i (cid:1) ) µ i + j − + · · · ,it is unclear to us whether it could be written in the Catalan-Hankel Pfaffian form. VALUATION OF CERTAIN PFAFFIANS 13 As mentioned before, q -analogues of Selberg integral can be used to evaluate q -Catalan-HankelPfaffians. For example, the formula (3.10) can be evaluated by the following Askey-Habsieger-Kadell formula [20] Z [0 , n Y i 2) Γ q (2 n + 1)Γ q (2 n − i + 5 / q (2 n − i + 2)Γ q (2 n − i/ . Moreover, combinatoric explanations of q -Catalan-Hankel Pfaffians are left to be further investi-gated. References [1] M. Adler, P. Forrester, T. Nagao and P. van Moerbeke. 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