Macdonald Polynomials of Type $C_n$ with One-Column Diagrams and Deformed Catalan Numbers

We present an explicit formula for the transition matrix $\mathcal{C}$ from the type $C_n$ degeneration of the Koornwinder polynomials $P_{(1^r)}(x\,|\,a,-a,c,-c\,|\,q,t)$ with one column diagrams, to the type $C_n$ monomial symmetric polynomials $m_{(1^{r})}(x)$. The entries of the matrix $\mathcal{C}$ enjoy a set of three term recursion relations, which can be regarded as a $(a,c,t)$-deformation of the one for the Catalan triangle or ballot numbers. Some transition matrices are studied associated with the type $(C_n,C_n)$ Macdonald polynomials $P^{(C_n,C_n)}_{(1^r)}(x\,|\,b;q,t)= P_{(1^r)}\big(x\,|\,b^{1/2},-b^{1/2},q^{1/2}b^{1/2},-q^{1/2}b^{1/2}\,|\,q,t\big)$. It is also shown that the $q$-ballot numbers appear as the Kostka polynomials, namely in the transition matrix from the Schur polynomials $P^{(C_n,C_n)}_{(1^r)}(x\,|\,q;q,q)$ to the Hall-Littlewood polynomials $P^{(C_n,C_n)}_{(1^r)}(x\,|\,t;0,t)$.


Introduction
The aim of this article is to investigate the transition matrix C, which describes the expansion of the type C n Macdonald polynomials [M2] P (Cn) (1 r ) = P (Cn) (1 r ) (x|b; q, t) with one column diagrams, in terms of the type C n monomial symmetric polynomials m (1 r ) (x). As for our convention of notation, see §3. On this course, we found that certain deformations appear, associated with the Catalan triangle or ballot numbers, and binomial coefficients. We refer the readers to [S] concerning the Catalan triangle numbers, and [FH] and [A] for the q-Catalan and q-ballot numbers.
(1.10) 1 A proof of this is presnted in § §2.4. The solution to the three term recursion relations (1.6), (1.7), (1.8) and (1.9) for C i,j given in terms of the function f [s] is presented in Proposition 7.3.
Corollary 1.2. When b = q and t = q, the Macdonald polynomials become the Schur polynomials s λ (x) = s (Cn) λ (x) of type C n . In this case we have f [t i+1 ] = 1 for i ≥ 0, indicating that the recursion relations (1.6)-(1.9) reduces to the ones for the ordinary Catalan triangle (or ballot) numbers. Therefore it holds that s (Cn) (1 r ) (x) = P (Cn) (1 r ) (x|q; q, q) = ⌊ r 2 ⌋ k=0 n − r + 1 n − r + k + 1 where m j = m(m − 1) · · · (m − j + 1) j! denote the ordinary binomial coefficient. Remark 1.4. To be precise, when ℓ(λ) = n, the polynomial P (Cn) λ (x|1; q, t) (or m λ ) has to be further decomposed in terms of the type D n Macdonald (or monomial) polynomials, since the Weyl group is smaller than the one for C n . Such a decomposition is easy but takes some space for a separate treatment. Therefore throughout in this paper, we do not go in detail in that direction, leaving the extra detail for the readers.
The first few terms of (1.11) and ( (1.14) As an application of our results obtained in this paper, we calculate the transition matrix from the Schur polynomials to the Hall-Littlewood polynomials, namely the Kostka polynomials, associated with one column diagrams.
Remark 1.7. Note that the K (Cn) (t)'s are essentially give by the t 2 -deformed ballot numbers [A] (the case n = r corresponds to the t 2 -deformation of the Catalan numbers [FH]), and the K (Cn) (t)'s by a version of t-deformed binomial numbers.

First few entries of
t 2 t 4 + t 8 t 6 + t 10 + t 12 + t 14 + t 18 1 t 2 + t 4 t 4 + t 6 + t 8 +t 10 + t 12 · · · 1 t 2 + t 4 + t 6 t 4 + t 6 + 2t 8 + t 10 +2t 12 + t 14 + t 16 1 t 2 + t 4 + t 6 + t 8 · · · . . . . . . (1.20) (1.21) The present article is organized as follows. In §2, several transition formulas obtained in this paper are summarized for the convenience of reading. Then we present a proof of our main result Theorem 1.1. In §3 and §4, we use Mimachi's kernel function identity to have a description of the BC n Koornwinder polynomials and type C n Macdonald polynomials with one column diagrams. The §5 and §6 are the core of the technical part of this article. In §5, Bressoud's matrix inversion is applied to invert the formula for the type C n Macdonald polynomials with one column diagrams. In §6, the four term relations for B [s, j] and B[s, j] are derived. In §7 is given the basic properties for the transition matrix C. In §8, we studied some degenerate cases, including the calculation of the Kostka polynomials. Some conjectures are presented in §9, concerning the asymptotically free type eigenfunctions for the type C n when b = t.
Throughout the paper, we use the standard notation (see [GR]) (1.24) 2. Collection of Transition Formulas and Proof of Theorem 1.1 In this section, we collect several transformation formulas which we need to establish Theorem 1.1, giving brief explanations about our ideas and methods for their derivations.
2.1. Koornwinder polynomials P (1 r ) (x|a, b, c, d|q, t) with one column diagrams. In [FHNSS], we studied some explicit formulas for the Koornwinder polynomials [K] with one-row diagrams. The results were interpreted as certain summation over the sets of tableaux of types C n and D n . While using the same technique as in [FHNSS], but replacing the Cauchy type kernel function by the dual-Cauchy type namely Mimachi's one (as to the kernel functions, see [Mi] and [KNS]), one can study an explicit formula for the Koornwinder polynomials with one column diagrams. Mimachi's kernel function [Mi] intertwines the action of the Koornwinder operator of type BC n to the one for BC 1 (namely for the Askey-Wilson operator) which in turn act on the Askey-Wilson eigenfunction. To perform the explicit calculations based on this idea, as was in the one-row diagram case, we need the fourfold summation formula for the Askey-Wilson eigenfunction [HNS]. The detail will be given in §3 and §4.
Specializing the parameters of the Koornwinder polynomials, we obtain the Macdonald polynomials of types C n and D n with one column diagram. In these particular limits, the fourfold summation (for the Askey-Wilson eigenfunction) reduces to a twofold one. In this way, we have explicit expressions for the Macdonald polynomials of types C n and D n with one column diagrams.
Let n ∈ Z >0 and x = (x 1 , . . . , x n ) be a set of variables. Let P (1 r ) (x|a, b, c, d|q, t) be the Koornwinder polynomial with one column diagram (1 r ) (r ∈ Z ≥0 ). (See §3, as to our notation.) Definition 2.1. Define the symmetric Laurent polynomial E r (x)'s by expanding the generating function E(x|y) as (2.1) Note that we have E 2n−r (x) = E r (x) for 0 ≤ r ≤ n and E r (x) = 0 for r > 2n.

Coefficients B[s, j] and
(2.8) Theorem 2.6. The formulas (2.5) and (2.6) can be recast as (see Theorem 5.7) (1 r−2j ) (x|b; q, t). (2.10) Definition 2.7. Let f [s] be the function defined in (1.10). Set for simplicity of display that where m j denotes the ordinary binomial coefficient. In view of this, we are naturally led to the following definition.
Definition 2.9. Let s ∈ C, and write s = t m+1 for simplicity. Let C[s, j] be the function in s defined by Theorem 2.10. The type C n Macdonald polynomial P (1 r ) (x|b; q, t) with one column diagram is expanded in terms of the monomial symmetric polynomials as Proof. It follows from (2.9) and (2.16).
2.4. Proof of Main Theorem. Now we are ready to present a proof of our main theorem. Proof of Theorem 1.1. The transition matrix C is even and upper triangular. In vew of Theorem 2.10 and C[t n−r+1 , j] = C n−r,n−r+2j , we have for any n > 0 and 0 ≤ r ≤ n P (Cn) indicating the stabilized transition formula (1.5). The three term recursion relation (1.6), (1.7), (1.8) and (1.9) are shown in Theorem 2.11.

Koornwinder's q-Difference Operator, Koornwinder Polynomials and Mimachi Kernel Function
We briefly recall some basic properties concerning the Koornwinder polynomials [K] and the Mimachi Kernel function identity [Mi].
3.1. Koornwinder's operator and Mimachi's Kernel function. Let (a, b, c, d; q, t) be a set of complex parameters. We assume that |q| < 1. Set α = (abcd/q) 1/2 for simplicity. Let Wn be the ring of W n -invariant Laurent polynomials in x. For a partition λ = (λ 1 , λ 2 , · · · , λ n ) of length n, i.e. λ i ∈ Z ≥0 and λ 1 ≥ · · · ≥ λ n , we denote by m λ = m λ (x) the monomial symmetric polynomial being defined as the orbit sums of monomials where we have used the notation The eigenvalue d λ is explicitly written as where we used the notations x = x 1/2 − x −1/2 and x; y = xy x/y = x + x −1 − y − y −1 for simplicity of display.
Definition 3.1. Define the involution * of the parameters by Theorem 3.2 ( [Mi] Lemma 3.2). Let n and m be positive integers, and let x = (x 1 , · · · , x n ), y = (y 1 , · · · , y m ) be two sets of independent indeterminates. Mimachi's kernel function enjoys the kernel function identity When we apply Mimachi's kernel function, the following Lemmas will be used. Recall that the generating function E(x|y) is introduced in Definition 2.1.
where m j denotes the ordinary binomial coefficient.
Proof. For an integer s satisfying 0 ≤ s ≤ n, we can find that the coefficient of the monomial Lemma 3.4. Let λ = (λ 1 , . . . , λ m ) be a partition satisfying the condition λ 1 ≤ n. We have Proof. Note that for any partitions λ and µ, we have m λ m µ = m λ+µ + lower terms. By using Lemma 3.3, we have E r (x) = m (1 r ) + lower terms. Hence we have (3.12).

3.2.
Asymptotically free eigenfunction f (x; s) for D x and reproduction formula. Let s = (s 1 , . . . , s n ) ∈ C n be a set of complex parameters. It is convenient to parametrize s by using another set of parameters λ = (λ 1 , . . . , λ n ) ∈ C n as s i = t −n+i q −λ i (i = 1, . . . , n). We use the shorthand notation where Q + denotes the positive octant of the root lattice of type BC n . To be more explicit, corresponding to the simple roots Assuming the genericity of the eigenvalue, one can show that the f (x; s) is determined uniquely.
Definition 3.7. Define the involution * of the parameters by Write for simplicity the composition of the involutions as * = * , namely we have Theorem 3.9. Let n ≥ m be positive integers, and x = (x 1 , . . . , x n ), y = (y 1 , . . . , y m ) be sets of independent indeterminates. Let λ = (λ 1 , . . . , λ m ) be a partition satisfying ℓ(λ) ≤ m and λ 1 ≤ n. Set (3.20) Let f (y; s) be the formal series in y uniquely characterized by c 0 (s) = 1 and Then we have where the notation [· · · ] 1,y denotes the constant term in y, and λ ′ is the conjugate diagram of λ.
Proof. Firstly, we show that the product Ψ(x; y)V (y) f(y; s) has a non vanishing constant tern in y. Write for short. Noting that we have ℓ(λ ′ ) ≤ n from the assumption λ 1 ≤ n, we have In the last step, we have used the Lemma 3.4. Next, we can show that the constant term satisfies the eigenvalue equation as Here we have used Theorem 3.2, Proposition 3.8, and the property x; y + y; z = x; z .
To check that the eigenvalue is the desired one, we prepare some lemmas.

Koornwinder polynomial with one column diagram
When we apply Theorem 3.9 for the simplest case m = 1, namely when we plug the BC 1 asymptotically free eigenfunction f (y; s) into the formula (3.23), we have the Koornwinder polynomials P (1 r ) (x) with one column diagrams. To execute the explicit calculation based on this, we need to recall the fourfold series expansion of the Askey-Wilson polynomials [HNS].
(4.9) 4.1. Koornwinder polynomial with one column diagram P (1 r ) (x|a, b, c, d|q, t). We move on to the proof of Theorem 2.2. Recall that n is a positive integer, x = (x 1 , . . . , x n ) is a set of variables, and P (1 r ) (x|a, b, c, d|q, t) denotes the Koornwinder polynomial with one column diagram (1 r ).

Bressoud's Matrix
Inversion. An infinite-dimensional matrix (f ij ) i,j∈Z ≥0 is said to be lower-triangular if f ij = 0 unless i ≥ j. Two infinite-dimensional lower-triangular matrices (f ij ) i,j∈Z ≥0 and (g ij ) i,j∈Z ≥0 are said to be mutually inverse if i≥j≥k f ij g jk = δ i,k . A matrix (f ij ) i,j∈Z ≥0 is said to be even if the following parity condition holds: i+j is odd implies f ij = 0.
Proposition 5.1 ( [B], p.1, Theorem, [L], p.5, Corollary). Let M[u, v; x, y; q] be the infinite even lower-triangle matrix with nonzero entries given by If two infinite matrices (f ij ) and (g ij ) are mutually inverse, then the conjugated ones (f ij d i /d j ) and (g ij d i /d j ) are also mutually inverse for any sequence (d r ) with nonzero entries.

(5.3)
Let M[u, v; x, y; t] denotes the conjugation of the matrix M[u, v; x, y; t 2 ] by the (d r ) with entries Proof. It follows from Bressoud's matrix inversion (5.2).
Now we turn to (6.3). Set for simplicity. Then we can write We have LHS of (6.16) On the other hand, we have RHS of (6.16) ; t 2 , t 2 = RHS of (6.17). Proposition 6.2. The four terms relations in Theorem 6.1 imply that We prove this by induction. The case i = 0 is clearly correct. Suppose that it is valid for i − 1. Then we have (6.23) Lemma 6.3. We have Proof. The (6.24) follows from the definition of B[s, i]. By noting where we used the notation (6.11) and (6.12), we have (6.25) from the identity Proposition 7.2. We have Hence the three term relation (7.3) for s = 1 reads Proof. We have C[1, 0] = 1. From Lemma 6.3, we have for j > 0 7.3. Solution to the Deformed Catalan triangle recursion relations.
Proposition 7.3. We have C[t r+1 , 0] = 1 for r ∈ Z ≥0 , and for i ∈ Z >0 , r ∈ Z ≥0 we have where P[r, i] denotes the finite set defined by We prepare some lemmas.
Lemma 7.4. For r ∈ Z ≥0 , we have (7.9) The case r = 0 holds since C[t, i + 1] = F [1, −1]C[t 2 , i]. Then we can show the induction step as Lemma 7.5. We have Proof of Proposition 7.3. We prove (7.8) by induction on i. It holds for i = 0, since we have C[t r+1 , 0] = 1 (r ∈ Z ≥0 ). The induction step is shown as follows. Lemmas 7.4 and 7.5 and the induction hypothesis give us 8. Some Degenerations of Macdonald Polynomials of Types C n and D n with One Column Diagrams and bv Polynomials This section is devoted to the study of several degenerations of our formulas for the Macdonald polynomial P (1 r ) (x|b; q, t). 8.1.1. C n case.
Theorem 8.9. We have (1 r−2j ) (x|t; 0, t), (8.27) t j 1 + t n−r 1 + t n−r+2j n − r + 2j j t 2 P (1 r−2j ) (x|0, t). (8.29) Then, applying the the formulas for the Schur polynomials in Corollary 8.8, we can calculate the Kostka polynomials (i.e. the transition coefficients from the Schur polynomials to the Hall-Littlewood polynomials) of types C n and D n associated with one column diagrams as follows.
Hence we have Theorem 2.4.
Remark 8.11. The expansion coefficient of (8.30) (times t −2j ) is identified with the q-ballot (when m = 0, q-Catalan) number [FH] [A] (8.32) by the replacement m → n − r, q → t 2 . The case m = 0 gives us the q-Catalan number. It is known that the q-Catalan or q-ballot number is a polynomial in q with positive integral coefficients (see [A] and [FH]). The expansion coefficient of (8.31) is identified with the following version of the q-binomial number q j 1 + q m−2j 1 + q m m j q 2 = q m−j m − 1 j − 1 q 2 + q j m − 1 j q 2 , (8.33) by the replacement m → n−r + 2j, q → t. Note that this is also a polynomial in q with positive integral coefficients.
9. Some Conjectures about Macdonald Polynomials of Type C n 9.1. Asymptotically free eigenfunctions for the Macdonald operator of type A n−1 . First we recall some facts about the asymptotically free eigenfunctions for the case A n−1 . Let n ∈ Z >0 , and q, t ∈ C be generic parameters. Let x = (x 1 , . . . , n n ) be a set of independent indeterminates. Macdonald's difference operator of type A n−1 is defined by For a partition λ with ℓ(λ) ≤ n, the Macdonald symmetric polynomial P λ (x; q, t) ∈ C[x 1 , . . . , x n ] Sn exists uniquely characterized by the conditions: 2) D (A n−1 ) P λ = n i=1 q λ i t n−i · P λ . (9.3) Let s 1 , s 2 , · · · , s n ∈ C be a set of complex variables. Let M (n) be the set of strict upper triangular matrices with entries in Z ≥0 , namely for θ (n) = (θ (n) ij ) i,j∈Z ≥0 ∈ M (n) i ≥ j implies θ (n) ij = 0.