CLASSICAL SKEW ORTHOGONAL POLYNOMIALS in a TWO-COMPONENT LOG-GAS 3 Polynomials [29, §5]

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Starting with (1.1) in the cases β = 1, 2 and 4, a different generalisation to regarding β as continuous is to consider a two-component interpolation between β =1 and 4. This is motivated by the fact that the logarithmic potential is the solution of the Poisson equation in two-dimensions, allowing the particles in the classical statistical mechanical interpretation of (1.1) to be thought of as charges. With the inverse temperature fixed to unity, the particles in (1.1) for β = 1 have charge +1, while those in (1.1) for β =4 have charge +2 — the pair potential between particles of charges q and q confined to the unit circle embedded in two-dimensions is q q log eiθ1 eiθ2 . 1 2 − 1 2 | − | A natural two-component generalisation is thus the PDF proportional to [15]

N1 N2 eiθk eiθj eiφα eiφβ 4 eiθj eiφα 2, (1.3) | − | | − | | − | 1 2 j=1 α=1 1≤j

N1 particles of charge +1 and N2 particles of charge +2. Since the charges are of the same sign, this two-component log-gas may also be referred to as a two-component plasma. It is immediate that when N2 =0 (resp. N1 =0), this model corresponds to (1.1) with β =1 (resp. β =4). It is furthermore the case that (1.3) corresponds to the ground state wave function of a two-component generalisation of the quantum many body Hamiltonian (1.2) [16]. Explicit evaluations of the partition function and the two-point charge–charge correlation functions corresponding to (1.3) were given in [15]. These explicit formulas allowed the free energy in the thermodynamic limit to be computed, as well as the evaluation of the bulk scaling limit of the two-point charge–charge correlation functions; see also [17]. Subsequently, it was found that the general (k1, k2)-point correlation function could be expressed as a quaternion determinant (or equivalently a Pfaﬃan) [18, Equation (6.168)]. Well established theory relating to the correlation functions for (1.1) with β = 1 and 4 [18, §6] (see also Section 1.2 below) implies that there is a family of skew orthogonal polynomials underpinning this result. Appreciating this point, the authors of [33] considered a variant of (1.3) on the real line speciﬁed by the Boltzmann factor

N1 N2 α α β β 4 α β 2e−V (αj )−2V (βk) (1.4) | k − j | | k − j | | j − k| 1 2 j=1 1≤jHermite polynomials with coeﬃcients being particular Laguerre polynomials. Asymptotic results relating to the partition function and one-point correlation functions were given; see too the recent work [4]. This two-component log- gas plays a key role in the recent study [14] relating to small gaps for the Gaussian orthogonal ensemble. Furthermore, excluding the weight factors, it is noted in [9] that (1.4) results as the eigenvalue Jacobian for real symmetric matrices with a prescribed number of double degeneracies. In the present work, it is the skew orthogonal polynomials which are our focus.

1.2. Skew orthogonal polynomials arising from log-gases. Since this paper is about the skew orthogonal polynomials arising from two-component log-gases (1.4), it is appropriate to ﬁrst revise how skew orthogonal polynomials appear in the corresponding one-component cases N1 =0 and N2 = 0. In these cases, (1.4) is proportional to the eigenvalue PDF for invariant ensembles of real symmetric matrices (N2 = 0), and self dual Hermitian matrices (N1 = 0). With this interpretation, the skew orthogonal polynomials most simply occur as the averaged characteristic CLASSICAL SKEW ORTHOGONAL POLYNOMIALS IN A TWO-COMPONENT LOG-GAS 3 polynomials [29, §5]. For β =1, the monic skew orthogonal polynomials p(1)(x) ∞ are given by { j }j=0 (up to a shift with arbitrary constant cn)

p(1)(x)= det(x M) (1), p(1) (x)= (x + TrM + c ) det(x M) (1), (1.5) 2n h − i2n 2n+1 h n − i2n while for β =4, the monic skew orthogonal polynomials p(4)(x) ∞ are given by { j }j=0 p(4)(x)= det(x M) (4), p(4) (x)= (x + TrM + c ) det(x M) (4). (1.6) 2n h − in 2n+1 h n − in The label in these latter averages relates to the size of M regarded as having quaternion entries. In practice each quaternion is represented as a particular 2 2 complex matrix (see e.g. [18, §1.3.2]), × so the sizes are consistent with the β = 1 case. The quantities in these averages depend only on the eigenvalues of M. Consequently, they can each be written in terms of integrals over the corresponding eigenvalue PDFs. Thus,

2n (1) 1 −V (xl) p2n (x)= xk xj (x xl)e dxl, CN,1 In | − | − Z 1≤j

1.3. Main results. The family of skew orthogonal polynomials corresponding to the Boltzmann factor (1.4) is skew orthogonal with respect to a particular skew symmetric inner product , 1 2 , h· ·iX,V ,V which is a linear combination of the , 1 and , 2 . Since in [33] it has been shown that h· ·i1,V h· ·i4,V for the Gaussian weight, the skew orthogonal polynomials with , 1 2 can be written as a h· ·iX,V ,V linear combination of Hermite polynomials with coefficients expressed by Laguerre polynomials (the latter have argument in terms of the fugacity—below a grand canonical formalism will be introduced), we pose the question as to whether all of the classical skew orthogonal polynomials under , 1 2 can be related to classical orthogonal polynomials. An affirmative answer is h· ·iX,V ,V given in Proposition 2.6, showing that classical skew orthogonal polynomials under , 1 2 can h· ·iX,V ,V be constructed from the classical orthogonal polynomials due to the Pearson equation. There are four cases to consider: the Gaussian, Laguerre, Jacobi and (generalised) Cauchy weight. We show that the hypergeometric orthogonal polynomials in the Askey-scheme appear when expressing the coefficients of the linear combination, extending the result of the Gaussian case for which the coefficients are given by Laguerre polynomials (i.e. 1F1 hypergeometric function). In particular, special attention is paid to the so-called generalised Cauchy weight in the form (1 ix)c(1 + ix)c¯ − with x R and c C. The model with the generalised Cauchy weight on the real line can be ∈ ∈ transformed into a generalised two-component log-gases with Jacobi weight on the unit circle under a stereographic projection, and therefore relates to the original work [15].

2. Skew orthogonal polynomials arising from a solvable mixed charge model

In this section, we present the general formalism showing how skew orthogonal polynomials enter into the computation of the partition function for the mixed charge model (1.4). As has been pointed out above, the Gaussian weight case has previously been considered in [33]; our formalism works for general classical weights.

2.1. A solvable mixed charge model. Let us consider a statistical mechanical system of particles corresponding to two-dimensional charges (logarithmic potential) confined to a line. Specif- ically, let there be L particles of charge +1 and M particles of charge +2, so that the total charge is L +2M = K. For convenience, we will assume that the total charge K = 2N is even, which implies L is even too. The configuration space ξ of this 1-dimensional system is a subspace of RL RM , defined as ξ = (ξ ; ξ ) = (α , , α ; β , ,β ), ξ RL, ξ RM , where α L × { 1 2 1 ··· L 1 ··· M 1 ∈ 2 ∈ } { i}i=1 and β M represent the locations of the charge +1 and +2 particles respectively. The particles { j}j=1 are each to subject to an external confining potential. The total energy of this model is given by

E(ξ ; ξ )= log α α +4 log β β 1 2 | k − j | | k − j | 1≤j

The partition function ZL,M of a ﬁxed pair of species numbers (L,M) is speciﬁed by

1 −E(ξ1;ξ2) L M ZL,M = e dµ (α)dµ (β) L!M! RL RM Z × CLASSICAL SKEW ORTHOGONAL POLYNOMIALS IN A TWO-COMPONENT LOG-GAS 5 with Boltzmann factor L M e−E(ξ1;ξ2) = α α β β 4 α β 2e−V1(αj )−2V2(βk) | k − j | | k − j | | j − k| 1≤j

The second line in (2.1) follows from a confluent form of the Vandermonde determinant identity; see [15]. To most clearly exhibit solvable structures, we do not consider the partition function directly, but rather first form a particular grand canonical ensemble, obtained by simultaneously considering all systems with species numbers L and M such that L +2M = 2N. To distinguish the species number making up the grand canonical ensemble, we introduce a fugacity (generating function parameter) X in the specification of the corresponding grand canonical partition function,

L L X −E(ξ1;ξ2) L M ZN (X)= X ZL,M = e dµ (α)dµ (β), L!M! RL×RM XhAi XhAi Z where here = (L,M) L +2M =2N,L,M Z . hAi { | ∈ ≥0} First we will demonstrate that the partition function ZN (X) has a Pfaﬃan representation.

Proposition 2.1. The partition function ZN (X) can be written as a Pfaﬃan

N 2N−1 ZN (X)=2 Pf(mi,j (X))i,j=0 , (2.2) where the moments mi,j (X) are deﬁned as 2 X i j −V1(y)−V1(z) j i i+j−1 −2V2(y) mi,j (X)= y z sgn(z y)e dµ(y)dµ(z)+ − y e dµ(y). 2 R R − 2 R Z × Z (2.3)

Proof. We begin by performing a Laplace expansion (see e.g. [38, §3.3]) of the conﬂuent Vander- monde determinant in (2.1) to express the determinant as sums in terms of determinants of species α and β respectively,

−E(ξ1;ξ2) σ(η) −V1(αj ) e = sgnσ det αj e · j, η = 1, ··· , L. σ X ′ ′ ′ ′ σ (η ) −V2(β ′ ) ′ ′ σ (η )−1 −V2(β ′ ) ′ j ′ j ′ det βj e , σ (η )βj e j = 1, ··· ,M, , × ′ ··· η = 1, , 2M where σ is injective map from 1, ,L 1, , 2N satisfying σ(1) < < σ(L) and the { ··· } 7→ { ··· } ··· image of σ′ is Im(σ′) = 1, , 2N Im(σ) satisfying σ′(1) < < σ′(2M). Moreover, the sum { ··· }\ ··· L σ represents the sum of all the possibilities of L integer partitions, contributing a factor C2N . Since the determinant has been expanded into terms involving species α and β independently, P we can similarly factor the integrations L L X −E(ξ1;ξ2) L M X σ(j) −V1(αj ) L e dµ (α)dµ (β)= sgnσ det αj e dµ (α) L!M! RL RM · L! RL × σ Z X Z 1 ′ ′ ′ ′ σ (j ) −V2(β j′ ) ′ ′ σ (j )−1 − V2(βj′ ) M det βj′ e , σ (j )βj′ e dµ (β). × M! RM Z

6 PETER J. FORRESTER AND SHI-HAO LI

By making use of integration formulas due to de Bruijn [7] for both factors, the right hand side of the above equation is equal to

2 L 2M sgnσ Pf(X m˜ ) Pf(ˆm ′ ′ ) , · σ(i),σ(j) i,j=1 σ (i),σ (j) i,j=1 σ X where

i−1 j−1 −V1(y)−V1(z) m˜ i,j = y z sgn(z y)e dµ(y)dµ(z), R R − Z × i+j−3 −2V2(y) mˆ i,j = (j i) y e dµ(y). − R Z We can now make use of [36, Lemma 4.2] to sum the Pfaﬃan according to

2 L 2M 2 2N sgnσ Pf(X m˜ ) Pf(ˆm ′ ′ ) = Pf(X m˜ +m ˆ ) . · σ(i),σ(j) i,j=1 σ (i),σ (j) i,j=1 i,j i,j i,j=1 σ XhAi X 2 By denoting 2mi+1,j+1(X)= X m˜ i,j +m ˆ i,j , we complete the proof.

2.2. Skew orthogonal polynomials arising from the solvable mixed charge model. In keeping with Proposition 2.1, we introduce the skew symmetric inner product

, 1 2 : C[z] C[z] C h· ·iX,V ,V × → i j 2 i j i j (z , z ) m (X)= X z ,z 1 + z ,z 2 . 7→ i,j h i1,V h i4,V Here, with weight function V , V speciﬁed in (2.3), the skew inner products , with β =1, 2, 4 1 2 h· ·iβ are deﬁned by

1 −V (y)−V (z) φ(z), ψ(z) 1, V = φ(y)ψ(z)sgn(z y)e dµ(y)dµ(z), h i 2 R R − Z × −2V (z) φ(z), ψ(z) 2, V = φ(z)ψ(z)e dµ(z), h i R Z 1 −2V (z) φ(z), ψ(z) 4, V = (φ(z)∂zψ(z) ∂zφ(z)ψ(z))e dµ(z). h i 2 R − Z Following the procedure of skew symmetric tri-diagonalisation [2, 20], we know that the skew 2N−1 −1 −⊤ symmetric moment matrix M := (mi,j (X))i,j=0 can be decomposed into M = S JS , where S is a lower triangular matrix with diagonals 1 and J is a block matrix of the form

0 u0 0 uN−1 τ2n+2 2n−1 J = diag , , , un = , τ2n = Pf(mi,j )i,j=0 (2.4) u 0 ··· u 0 τ2n ( − 0 ! − N−1 !) with τ = 1. Due to the decomposition, we can introduce a family of polynomials Q (z) 2N−1 0 { n }n=0 such that Q (z) = (Sχ(z)) , χ(z) = (z0,z1, ,z2N−1)⊤, n n+1 ··· where (Sχ(z))n+1 denotes the (n + 1)-th component of vector (Sχ(z)). These polynomials skew diagonalise , 1 2 . h· ·iX,V ,V Proposition 2.2. The family of polynomials Q (z) 2N−1 are skew orthogonal with respect to the { n }n=0 inner product , 1 2 . In other words, we have h· ·iX,V ,V Q2n(z),Q2m+1(z) X,V1,V2 = unδn,m, h i (2.5) Q (z),Q (z) 1 2 = Q (z),Q (z) 1 2 =0 h 2m 2n iX,V ,V h 2m+1 2n+1 iX,V ,V CLASSICAL SKEW ORTHOGONAL POLYNOMIALS IN A TWO-COMPONENT LOG-GAS 7 for n, m =0, 1, ,N 1. ··· − Proof. By denoting Q(z) = (Q (z), ,Q (z))⊤ = Sχ(z), one can easily ﬁnd 0 ··· 2N−1 ⊤ ⊤ ⊤ ⊤ Q(z),Q (z) 1 2 = S χ(z),χ (z) 1 2 S = SMS = J. h iX,V ,V h iX,V ,V

Remark 2.3. Since the skew orthogonal relation (2.5) is equivalent to a linear system, one can obtain a Pfaﬃan form of the skew orthogonal polynomials following the procedure in [8, Section 2]. Since we mainly consider the relation between the skew orthogonal polynomials and orthogonal polynomials in this paper, we omit the Pfaﬃan forms here.

The skew symmetric inner product deﬁned in (2.3) is a linear combination of skew symmetric inner products induced by random matrices with orthogonal and symplectic invariance (i.e. β =1, 4 respectively). As is known from [1] , the skew orthogonal polynomials with β =1, 4 can be expressed as a linear combination of orthogonal polynomials in the classical cases (a precise meaning of ‘classical’ is given in the paragraph above Proposition 2.6). Therefore, we seek a rela- tionship between the skew orthogonal polynomials arising from (2.3) and orthogonal polynomials. Important for this purpose is the following proposition.

Proposition 2.4. For specific coefficients ζ ∞ , if p (z) ∞ are monic polynomials satisfying { j }j=0 { j }j=0 (1) p (z),p (z) ζ p (z) 1 = hn δ , h 2m 2n+1 − n 2n−1 i1,V n,m  p2m(z),p2n(z) 1,V1 = p2n+1(z) ζnp2n−1(z),p2m+1(z) ζmp2m−1(z) 1,V1 =0, h i h (4) − (4) − i (2.6)  p2m(z),p2n+1(z) 4,V2 = h δn,m h δm−1,n,  h i 2n − 2m−1 p (z),p (z) 2 = p (z),p (z) 2 =0, h 2m 2n i4,V h 2m+1 2n+1 i4,V  then there exist a family of monic skew orthogonal polynomials Q (z) ∞ { j }j=0 m m 1 1 Q2m(z)= αkp2k(z),Q2m+1(z)= ξk(p2k+1(z) ζkp2k−1(z)), (2.7) αm ξm − Xk=0 Xk=0 2 skew orthogonal under the skew symmetric inner product , 1 2 = X , 1 + , 2 . h· ·iX,V ,V h· ·i1,V h· ·i4,V Moreover, the coefficients α ∞ and ξ ∞ satisfy the recurrence relations { j}j=0 { j }j=0 h(4) α X2h(1) + h(4) + ζ h(4) α + ζ h(4) α =0 (2.8a) 2j+1 j+1 − j 2j j 2j−1 j j 2j−2 j−1 ζ h(4)ξ X2h(1) + h(4) + ζ h(4) ξ + h(4) ξ =0 (2.8b) j+1 2j j+1 − j 2j j 2j−1 j 2j−1 j−1 with initial values α−1 = ξ−1 =0 and α0 = ξ0 =1.

Proof. Since p (z), p (z) ζ p (z) ∞ form a basis of L2(dµ), we can write the monic { 2j 2j+1 − j 2j−1 }j=0 polynomials Q (z) ∞ as linear combinations of this basis { j }j=0 m m−1 Q (z)= α(m)p (z)+ β(m)(p (z) ζ p (z)), α(m) =1, (2.9a) 2m k 2k k 2k+1 − k 2k−1 m k=0 k=0 Xm X m Q (z)= ξ(m)(p (z) ζ p (z)) + η(m)p (z), ξ(m) =1. (2.9b) 2m+1 k 2k+1 − k 2k−1 k 2k m Xk=0 Xk=0 8 PETER J. FORRESTER AND SHI-HAO LI

The skew orthogonal relations (2.5) can be written as 1

Q (z),p (z) ζ p (z) 1 2 =0, for j

Q (z),p (z) 1 2 =0, for j m; (2.10b) h 2m 2j iX,V ,V ≤

Q (z),p (z) ζ p (z) 1 2 =0, for j

Q (z),p (z) 1 2 =0, for j < m. (2.10d) h 2m+1 2j iX,V ,V From (2.9a) and (2.9b), the equation (2.10a) is equivalent to m (m) α p (z),p (z) ζ p (z) 1 2 =0, for jrecurrence relation (2.8a) with initial data

α−1 =0, α0 =1 (2.12) and m m−1 1 (m) Q2m(z)= αkp2k(z)+ βk (p2k+1(z) ζkp2k−1(z)). αm − Xk=0 Xk=0 Now substituting the above form of Q2m(z) into (2.10b), one can see m−1 β(m) p (z) ζ p (z),p (z) =0, k h 2k+1 − k 2k−1 2j iX,V Xk=0 which leads us to m−1 m−1 X2 β(m)h(1)δ β(m)(h(4)δ h(4) δ ) − k k k,j − k 2k k,j − 2j−1 j−1,k Xk=0 Xk=0 m−1 + β(m)ζ (h(4) δ h(4) δ )=0, for j m. k k 2k−2 k−1,j − 2j−1 k,j ≤ Xk=0 Taking j from m to 0 iteratively we ﬁnd β(m) = = β(m) = 0. Therefore, from (2.10a) and m−1 ··· 0 (2.10b), we obtain m 1 Q2m(z)= αkp2k(z), αm kX=0 where α satisfy (2.8a) with initial values α =0 and α =1. { k} −1 0 1 For the third condition, the skew orthogonal relation only hold for j < m since we haven’t assume Q2m+1(z) is independent of Q2m(z) yet. CLASSICAL SKEW ORTHOGONAL POLYNOMIALS IN A TWO-COMPONENT LOG-GAS 9

Now we turn to equations (2.10c) and (2.10d). First consider (2.10c). From (2.9a) and (2.9b), this is equivalent to m (m) η p (z),p (z) ζ p (z) 1 2 =0, for j < m. k h 2k 2j+1 − j 2j−1 iX,V ,V Xk=0 Again, by the use of conditions (2.6), one can show that this implies

h(4) η(m) X2h(1) + h(4) + ζ h(4) η(m) + ζ h(4) η(m) =0, for j < m, 2j+1 j+1 − j 2j j 2j−1 j j 2j−2 j−1 demonstrating that η(m) admit the same recurrence relation with α(m) and therefore { j } { j } m Q (z)= ξ(m)(p (z) ζ p (z)) + c Q (z) 2m+1 k 2k+1 − k 2k−1 m 2m kX=0 with a constant cm. A basic property of skew orthogonal polynomials, is that the transformation Q (z) Q (z)+ c Q (z) does not change the skew orthogonality (this explains the 2m+1 7→ 2m+1 m 2m arbitrary constant cn in (1.5) and (1.6)), and therefore we are free to set cm =0, giving m Q (z)= ξ(m)(p (z) ζ p (z)). 2m+1 k 2k+1 − k 2k−1 kX=0 We now make use of (2.10d) by substituting this form to obtain m m m X2 ξ(m)h(1)δ ξ(m)(h(4)δ h(4) δ )+ ξ(m)ζ (h(4) δ h(4) δ )=0 − k k k,j − k 2k k,j − 2j−1 k,j−1 k k 2k−2 k−1,j − 2j−1 k,j Xk=0 Xk=0 Xk=0 (m) for j

Corollary 2.5. We have j (4) −1 h2k−2 ξj = cj αj , where cj = ζk . (2.13) h(4) kY=1 2k−1 −1 ˆ Proof. Note that if we make the transformation of variables ξj = cj ξj , the recurrence relation (2.8b) reads (4) cj ˆ 2 (1) (4) (4) ˆ (4) cj ˆ ζj+1h2j ξj+1 X hj + h2j + ζj h2j−1 ξj + h2j−1 ξj−1 =0. cj+1 − cj−1 To be consistent with (2.8a), it is further required (4) cj (4) (4) cj (4) ζj+1h2j = h2j+1, h2j−1 = ζj h2j−2. cj+1 cj−1 Hence j (4) h2k−2 −1 cj = C ζk , and ξj = cj αj , h(4) kY=1 2k−1 for some constant C. By noting α0 = ξ0 =1, it follows that C =1, thus completing the proof. 10 PETER J. FORRESTER AND SHI-HAO LI

The condition of proposition (2.4) is very general. Thus, starting with two arbitrary skew symmetric inner products , s1 and , s2 , if a family of (not necessarily orthogonal) polynomials ∞ h· ·i h· ·i pj(z) j=0 satisfy the condition (2.6), then we can define a new skew symmetric inner product { } 2 , = X , 1 + , 2 with corresponding skew orthogonal polynomials specified by (2.7), h· ·inew h· ·is h· ·is (2.8a) and (2.8b). The task then is to specify families of skew symmetric inner products and polynomials p (z) ∞ { j }j=0 satisfying (2.6). In fact we find that all of the classical monic orthogonal polynomials p (x) ∞ { j }j=0 with weight e−2V , characterised by the Pearson equation 2V ′(z)= g(z)/f(z) with deg g 1 and ≤ deg f 2, satisfy the condition (2.6) with ≤ 1 1 V = V + log f, V = V log f. (2.14) 1 2 2 − 2 It is remarkable that for the classical weights, the procedure to find the relations between β =1, 4 and β =2 is to construct a linear operator [1] f ′ g = f∂ + − A z 2 −1 such that φ, ψ 2,V = φ, ψ 1,V1 and φ, ψ 2,V = φ, ψ 4,V2 . In this case, the inner product h A i −h i h A i 2 −h 1 i , 1 2 can be constructed from the operator X + , but it needs more computations h· ·iX,V ,V A A which are not considered in this paper.

Proposition 2.6. All classical orthogonal polynomials, in the sense of the requirement on the Pearson equation as speciﬁed above, satisfy (2.6), where the relations (2.14) are assumed.

Proof. The proof of this claim is based on the Pearson equation. The ﬁrst two relations have been demonstrated in [1]—the key idea is to use the Pearson pair (f,g) to ﬁnd the skew orthogonal polynomials with β = 1; a detailed procedure can be found in the Appendix B. Moreover, the quantities ζ ∞ and h(1) ∞ are computed from the procedure. The last two relations can also { j }j=0 { j }j=0 be obtained from the Pearson pair. In [3], it has been demonstrated that the classical orthogonal polynomials p (z) ∞ which are characterised by the Pearson pair (f,g) must satisfy { n }n=0 ′ f(z)pn(z)= anpn+1(z)+ bnpn(z)+ cnpn−1(z), where a ∞ , b ∞ , c ∞ are three sequences independent of z. Therefore, one can obtain { n}n=0 { n}n=0 { n}n=0

p (z),p (z) 2 = p (z),p (z) 2 =0, h 2m 2n i4,V h 2m+1 2n+1 i4,V 1 ′ 1 ′ p (z),p (z) 2 = p (z),f(z)p (z) f(z)p (z),p (z) h 2m 2n+1 i4,V 2h 2m 2n+1 i2,V − 2h 2m 2n+1 i2,V 1 1 = (c h a h )δ (c h a h )δ , 2 2n+1 2n − 2n 2n+1 n,m − 2 2m 2m−1 − 2m−1 2m m,n+1 where h = p (z),p (z) . We complete the proof by deﬁning h(4) = 1 (c h a h ). n h n n i2,V j 2 j+1 j − j j+1

In conclusion to this section, we demonstrate that the partition function ZN (X) in (2.2) can be calculated by use of the skew orthogonal polynomials constructed in Proposition 2.4, thus exhibiting the solvability of the model. To begin, note that

N−1 N (2.2) N (2.4) N (2.5) N Z (X) = 2 τ = 2 u = 2 Q ,Q 1 2 . N 2N j h 2j 2j+1iX,V ,V j=0 j=1 Y Y CLASSICAL SKEW ORTHOGONAL POLYNOMIALS IN A TWO-COMPONENT LOG-GAS 11

Substituting the skew orthogonal polynomials Q (z) ∞ in terms of orthogonal polynomials via { j }j=0 (2.7) shows

N−1 N−1 j j N N 1 2 Q2j ,Q2j+1 X,V1,V2 =2 αk1 ξk2 p2k1 ,p2k2+1 ζk2 p2k2−1 X,V1,V2 . h i αj ξj h − i j=0 j=0 1 2 Y Y kX=0 kX=0 Making the use of (2.6), we see this is equal to

j j j N−1 1 2N α ξ (X2h(1) + h(4) + ζ h(4) ) α ξ ζ h(4) α ξ h(4) . α ξ k k k 2k k 2k−1 − k−1 k k 2k−2 − k k−1 2k−1 j=0 j j ! Y Xk=0 Xk=1 Xk=1 By taking into account the recurrence relation (2.8a), we conclude

N−1 N (4) ZN (X)=2 αN h2j+1. j=0 Y In the classical cases, h(4) ∞ are known normalisations, expressible in terms of quantities { j }j=0 relating to the corresponding orthogonal polynomials [1]. We will show below that in these cases

αN , and thus the partition function, can be written in terms of hypergeometric orthogonal polynomials. Interestingly, this class of polynomials has appeared in a recent work in the moments of the spectral density for the classical weights with β =2 [10].

3. The classical cases

We know from Proposition 2.6 that the classical weights allow for a structured form of the skew orthogonal polynomials generated in terms of the classical orthogonal polynomials. Here we will show that the coeﬃcients α ∞ and ξ ∞ can be expressed in terms of hypergeometric or- { j }j=0 { j }j=0 thogonal polynomials from the Askey-Wilson table. There are four classical weights: the Gaussian, Laguerre, Jacobi and generalised Cauchy. These will be considered in turn.

3.1. Gaussian case. The two-component plasma for mixed charges with Gaussian weight has already been considered in [33]. We restate the main results for the completeness of our work. It is a standard result that the monic Hermite polynomials p (z) ∞ satisfy the orthogonal relation { k }k=0 p (z),p (z) =2−k√πΓ(k + 1)δ , h k j i2,V k,j 2 where the weight function is given by e−2V (z) = e−z . From the weight function, we can compute the Pearson pair as (f,g)=(1, 2z), and therefore e−V1(z) = e−V2(z) = z2/2, meaning that the external potentials of these two diﬀerent particles are exactly the same. Moreover, from [1, Equation 3.2], we have

−2n p ,p np 1 =2 √πΓ(2n + 1)δ , h 2m 2n+1 − 2n−1i1,V n,m and from the derivative formula ∂zpk(z)= kpk−1(z), one knows 1 1 p ,p 2 = (2n + 1) p ,p 2 (2m) p ,p 2 h 2m 2n+1i4,V 2 h 2m 2ni2,V − 2 h 2m−1 2n+1i2,V =2−2n−1√πΓ(2n + 2)δ 2−2m√πΓ(2m + 1)δ . n,m − m−1,n Therefore, the coeﬃcients of the Hermite polynomials with regard to the relations (2.6) are

(1) −2j (4) −j−1 ζj = j, hj =2 √πΓ(2j + 1), hj =2 √πΓ(j + 2). (3.1) 12 PETER J. FORRESTER AND SHI-HAO LI

Substituting them into the recurrence relation (2.8a), one finds that the coefficients α satisfy { j } (2j + 2)(2j + 1)α 2(2X2 +4j + 1)α +4α =0, j+1 − j j−1 2j j! which, upon scaling αj =2 (2j)! αˆj , reduces to 1 1 (j +1)ˆα (X2 +2j + )ˆα + (j )ˆα =0. j+1 − 2 j − 2 j−1 This equation, in combination with the initial conditions (2.12), can be recognised as the three term recurrence satisfied by the Laguerre polynomials with parameter 1/2, and so αˆ = L−1/2( X2), − j j − where La (z) ∞ denotes the standard (not monic!) Laguerre polynomials with parameter a { n }n=0 through the orthogonal relation ∞ Γ(a + n + 1) La (z)La (z)zae−zdx = δ . n m Γ(n + 1) n,m Z0 Therefore, we can conclude that

j! − 1 α =22j L 2 ( X2). j (2j)! j − With regard of the coeﬃcients ξ ∞ , since { j }j=0 (4) −2j−1 −2j−2 (4) ζj+1h2j = (j + 1)2 √πΓ(2j +2)=2 √πΓ(2j +3)= h2j+1, one sees that α ∞ and ξ ∞ satisfy the same recurrence relation with same initial values. { j}j=0 { j}j=0 Therefore,

j! − 1 α = ξ =22j L 2 ( X2), j j (2j!) j − as demonstrated in [33, Section 4].

3.2. Laguerre case. The Laguerre case is to consider the weight function e−2V (z) = zae−z, supported on [0, ). The Pearson pair of the Laguerre weight is (f,g) = (z,z a), and therefore, ∞ − according to (2.14) the external potentials are e−V1(z) = z(a−1)/2e−z/2 and e−V2(z) = z(a+1)/2e−z/2. For the monic Laguerre polynomials 2 p(a)(z) ∞ , we have { k }k=0 p(a),p(a) = Γ(n + 1)Γ(n + a + 1)δ , h m n i2,V n,m d (3.2) p(a−1)(z)= p(a)(z)+ np(α) (z), p(a)(z)= np(a+1)(z). n n n−1 dx n n−1 From [1, Equation 3.6] or [30, Equation 4.31], one knows

(a) (a) (a) p ,p 2n(2n + a)p 1 = 2Γ(2n + 1)Γ(2n + a + 1)δ . h 2m 2n+1 − 2n−1i1,V n,m Properties of the Laguerre polynomials (3.2) tell us

(a) (a) 1 (a) (a+1) 1 (a+1) (a) p ,p 2 = (2n + 1) p ,p 2 (2m) p ,p 2 h 2m 2n+1i4,V 2 h 2m 2n+1 i2,V − 2 h 2m−1 2n i2,V 1 (a+1) (a+1) (a+1) 1 (a+1) (a+1) (a+1) = (2n + 1) p +2mp ,p 2 (2m) p ,p + (2n + 1)p 2 2 h 2m 2m−1 2n i2,V − 2 h 2m−1 2n+1 2n i2,V 1 1 = Γ(2n + 2)Γ(2n + a + 2)δ Γ(2m + 1)Γ(2m + a + 1)δ . 2 n,m − 2 m−1,n

2 (a) − k (a) The monic Laguerre polynomials are given in terms of the Laguerre polynomials by pk (z) = ( 1) k!Lk (z) CLASSICAL SKEW ORTHOGONAL POLYNOMIALS IN A TWO-COMPONENT LOG-GAS 13

Therefore, the coeﬃcients in (2.6) for the Laguerre case are 1 ζ =2j(2j + a), h(1) = 2Γ(2j + 1)Γ(2j + a + 1), h(4) = Γ(j + 2)Γ(j + a + 2). (3.3) j j j 2 Substituting in the recurrence relation (2.8a) shows

(2j + 1)(2j + 2)(2j + a + 1)(2j + a + 2)αj+1 (4X2 + (2j + 1)(2j + a +1)+2j(2j + a))α + α =0. − j j−1 Scaling α = 1 αˆ gives that αˆ ∞ satisfy j (2j)! j { j }j=0 a a 1 1 a 1 a 1 (j + + 1)(j + + )ˆα X2 + (j + )(j + + )+ j(j + ) αˆ + j(j )ˆα =0. 2 2 2 j+1 − 2 2 2 2 j − 2 j−1 Together with the initial condition (2.12), we recognise this three term recurrence as that satisﬁed by a particular continuous dual Hahn polynomial [25, Section 1.3] a +1 1 αˆ = S˜ X2; , , 0 j j − 2 2 or hypergeometric function

a+1 a+1 j, 2 X, 2 + X αˆj = 3F2 − a+2− a+1 1 . , ! 2 2

Hence, for the coeﬃcients α ∞ we have { j }j=0 1 a +1 1 α = S˜ X2; , , 0 . j (2j)! j − 2 2 (4) (4) As for the Laguerre case we can check ζj+1h2j = h2j+1, and thus the recurrence relations of ξ ∞ are the same with α ∞ . Therefore, { j }j=0 { j}j=0 1 a +1 1 ξ = S˜ X2; , , 0 . j (2j)! j − 2 2 3.3. Jacobi case. The weight function in the Jacobi case is e−2V (z) = (1 z)a(1 + z)b supported − on the interval ( 1, 1). This is more general than the Gaussian and Laguerre weights, in the − sense that the latter can be obtained as limiting cases. For the Pearson pair, we have (f,g) = 1 z2, (a + b)z + (a b) , and therefore the external potentials are e−V1(z) = (1 z)(a−1)/2(1 + − − − z)(b−1)/2 and e−V2(z) = (1 z)(a+1)/2(1 + z)(b+1)/2. − The monic Jacobi polynomials p(a,b)(z) ∞ have the well known properties { k }k=0 (a,b) (a,b) (a,b) d (a,b) (a+1,b+1) pm ,pn 2,V = hn δn,m, pn (z)= npn−1 (z), h i dz (3.4) (a,b) (a+1,b+1) (a,b) (a+1,b+1) (a,b) (a+1,b+1) pn (z)= pn (z)+ µn pn−1 (z)+ ǫn pn−2 (z), where the coeﬃcients are Γ(n + 1)Γ(n + a + 1)Γ(n + b + 1)Γ(n + a + b + 1) h(a,b) =2a+b+2n+1 , n Γ(2n + a + b + 1)Γ(2n + a + b + 2) 2n n + b +1 n + a µ(a,b) = + , (3.5) n (2n + a + b + 1) −2n + a + b +2 2n + a + b 4n(n 1)(n + a)(n + b) ǫ(a,b) = − . n −(2n + a + b + 1)(2n + a + b)2(2n + a + b 1) − 14 PETER J. FORRESTER AND SHI-HAO LI

From [1, Equation 3.14], one knows

(a,b) (a,b) (a,b) (a,b) (1) p ,p ζ p 1 = h δ h 2m 2n+1 − n 2n−1i1,V n n,m with 8n(2n + a)(2n + b)(2n + a + b) ζ(a,b) = , n (4n + a + b 1)(4n + a + b)(4n + a + b + 1)(4n + a + b + 2) − Γ(2n + 1)Γ(2n + a + 1)Γ(2n + b + 1)Γ(2n + a + b + 1) h(1) =2a+b+4n+2 . n Γ(4n + a + b + 1)Γ(4n + a + b + 3) Furthermore, from the properties (3.4), one can compute

(a,b) (a,b) 1 (a,b) (a+1,b+1) 1 (a+1,b+1) (a,b) p ,p 2 = (2n + 1) p ,p 2 (2m) p ,p 2 h 2m 2n+1i4,V 2 h 2m 2n i2,V − 2 h 2m−1 2n+1i2,V 1 (a+1,b+1) (a,b) (a+1,b+1) (a,b) (a+1,b+1) (a+1,b+1) = (2n + 1) p + µ p + ǫ p ,p 2 2 h 2m 2m 2m−1 2m 2m−2 2n i2,V 1 (a+1,b+1) (a+1,b+1) (a,b) (a+1,b+1) (a,b) (a+1,b+1) (2m) p ,p + µ p + ǫ p 2 (3.6) − 2 h 2m−1 2n+1 2n+1 2n 2n+1 2n−1 i2,V 1 = (2n + 1)h(a+1,b+1) (2n)ǫ(a,b) h(a+1,b+1) δ 2 2n − 2n+1 2n−1 n,m 1 (2m)h(a+1,b+1) (2m 1)ǫ(a,b)h(a+1,b+1) δ , − 2 2m−1 − − 2m 2m−2 m−1,n (a,b) (a,b) where hn and ǫn are defined in (3.5). As a consequence, 8j(2j + a)(2j + b)(2j + a + b) ζ = , j (4j + a + b 1)(4j + a + b)(4j + a + b + 1)(4j + a + b + 2) − Γ(2j + 1)Γ(2j + a + 1)Γ(2j + b + 1)Γ(2j + a + b + 1) h(1) =2a+b+4j+2 , j Γ(4j + a + b + 1)Γ(4j + a + b + 3) Γ(j + 2)Γ(j + a + 2)Γ(j + b + 2)Γ(j + a + b + 2) h(4) =2a+b+2j+2 . j Γ(2j + a + b + 2)Γ(2j + a + b + 4) Substituting into the recurrence relation (2.8a), one finds that the coefficients α ∞ satisfy the { j }j=0 recurrence (2j + 1)(2j + a + 1)(2j + b + 1)(2j + a + b + 1) 2j(2j + a)(2j + b)(2j + a + b) X2 + + α (4j + a + b + 1)(4j + a + b + 3) (4j + a + b 1)(4j + a + b + 1) j − (2j + 1)2(2j + a + 1)2(2j + b + 1)2(2j + a + b + 1)2 1 =4 αj+1 + (4j + a + b 2)(4j + a + b)αj−1, (4j + a + b + 1)3(4j + a + b + 3)3 4 − where the Pochhammer symbol (a)k is defined by

(a) =1, and (a) = a(a + 1) (a + k 1), k =1, 2, 3, . 0 k ··· − ··· Making the substitution (4j + a + b + 1)! α =2−2j αˆ j (2j)!(2j + b)! j gives the modiﬁed recurrence

2 1 a+1 b+1 a+b+1 a b a+b X (j + 2 )(j + 2 )(j + 2 )(j + 2 ) j(j + 2 )(j + 2 )(j + 2 ) + a+b+3 a+b+1 + a+b+1 a+b−1 αˆj 4 (2j + 2 )(2j + 2 ) (2j + 2 )(2j + 2 ) ! a+2 a+1 a+b+2 a+b+1 1 b b−1 (j + 2 )(j + 2 )(j + 2 )(j + 2 ) j(j 2 )(j + 2 )(j + 2 ) = a+b+3 a+b+1 αˆj+1 + − a+b+1 a+b−1 αˆj−1. (2j + 2 )(2j + 2 ) (2j + 2 )(2j + 2 ) CLASSICAL SKEW ORTHOGONAL POLYNOMIALS IN A TWO-COMPONENT LOG-GAS 15

This modiﬁed recurrence, together with the initial conditions (2.12), is recognised as being satisﬁed by a special case of the Wilson polynomials [25, Section 1.1], telling us that X2 a +1 b +1 1 αˆ = W˜ ; , , , 0 j j − 4 2 2 2 or in terms of a hypergeometric function

a+b+1 a+1−X a+1+X j, j + 2 , 2 , 2 αˆj = 4F3 − a+1 a+2 a+b+2 1 . , , ! 2 2 2

Hence

(4j + a + b + 1)! X2 a +1 b +1 1 α =2−2j W˜ ; , , , 0 . j (2j)!(2j + b)! j − 4 2 2 2 In contrast to the Gaussian and Laguerre cases, in the Jacobi case ζ h(4) = h(4) . This means j 2j 6 2j+1 the recurrence relation for ξ ∞ is different to that for α ∞ . However, one can make use of { j }j=0 { j }j=0 Corollary 2.5 to conclude that if we change ξˆ = c ξ , then ξˆ ∞ admits the same recurrence as j j j { j}j=0 α ∞ . In the Jacobi case, c can be computed via (2.13) and we find { j }j=0 j j (4) j h2k−2 4k + a + b 2 2+ a + b cj = ζk = − = . h(4) 4k + a + b +2 4j + a + b +2 kY=1 2k−1 kY=1 −1 Therefore, ξj = cj αj and (4j + a + b + 2)! X2 a +1 b +1 1 ξ =2−2j W˜ ; , , , 0 . j (2j)!(2j + b)!(2 + a + b) j − 4 2 2 2 3.4. Generalised Cauchy case. In this subsection, we consider the remaining of the classical weights on the line, namely the generalised Cauchy weight e−2V (z) = (1 iz)c(1 + iz)c¯ with z R, − ∈ c C. In the one-component case, this weight is in fact related to the circular Jacobi ensemble ∈ [21, 26], whose probability density function (PDF) is defined on [0, 2π)N . Thus if we consider the stereographic projection 1 ix eiθj = − j , 1+ ixj then we can transform the eigenvalue PDF on the unit circle

N (1 eiθj )βa(1 e−iθj )βa¯ eiθk eiθj βdθ dθ − − | − | 1 ··· N j=1 Y 1≤j

N (1 ix )c(1 + ix )c¯ x x βdx dx . − j j | k − j | 1 ··· N j=1 Y 1≤j

˜(c,c¯) ˜(c+1,c¯+1) c ˜(c+1,c¯+1) c ˜(c+1,c¯+1) In (z)= In (z)+ µnIn−1 (z)+ ǫnIn−2 (z)

c (a,b) c 2 (a,b) (a,b) (a,b) with coeﬃcients µn = iµn a→c, b→c¯ and ǫn = i ǫn a→c, b→c¯, where µn and ǫn are given | | (4) in (3.5). Therefore, in analogy with the computation in (3.6), h2j in the Routh-Romanovski case can be computed as

1 h(4) = (2j + 1)h(c−1,c¯−1) 2nǫc h(c−1,c¯−1) 2j 2 2j − 2j+1 2j−1 Γ(2j + 2)Γ(2p 4j 1)Γ(2 p 4j 3) =24j−2p+3π − − − − . Γ(2p 2j 1)Γ(p 1 iq 2j)Γ(p 1+ iq 2j) − − − − − − − (1) (4) (4) Taking ζj , hj and hj into the equation (2.8a) and dividing hj on both sides, one can obtain

(2j + 1)(2p 2j 1)( c 2j 1)( c¯ 2j 1) 2j(2p 2j)( c 2j)( c¯ 2j) X2 + − − − − − − − − + − − − − − α (2p 4j 1)(2p 4j 3) (2p 4j 1)(2p 4j + 1) j − − − − − − − (2j + 1) (2p 2j 2) ( c 2j 2) ( c¯ 2j 2) =22 2 − − 2 − − − 2 − − − 2 α +2−2(2p 4j)(2p 4j + 2)α . (2p 4j 3) (2p 4j 5) j+1 − − j−1 − − 3 − − 3 If we make the change of variable

( c 2j 1)! α = ( 1)j 2−2j − − − αˆ , j − (2j)!(2p 4j 2)! j − − the above equation is transformed into

2 1 1 c+1 c¯+1 c c¯ X (j + 2 )(j p + 2 )(j + 2 )(j + 2 ) j(j p)(j + 2 )(j + 2 ) + − + − αˆj − 4 (2j p + 3 )(2j p + 1 ) (2j p + 1 )(2j p 1 ) − 2 − 2 − 2 − − 2 1 c¯+1 c¯+2 1 c c−1 (j p + 1)(j p + 2 )(j + 2 )(j + 2 ) j(j 2 )(j + 2 )(j + 2 ) = − − αˆj+1 + − αˆj−1. (2j p + 3 )(2j p + 1 ) (2j p + 1 )(2j p 1 ) − 2 − 2 − 2 − − 2 The solution of this recurrence relation can also be expressed as a Wilson polynomial

X2 c¯+1 c +1 1 αˆ = W˜ ; , , , 0 j j 4 2 2 2 in analogy with the Jacobi case, and hence

Γ( c 2j) X2 c¯+1 c +1 1 α = ( 1)j2−2j − − W˜ ; , , , 0 . j − Γ(2j + 1)Γ(2p 4j 1) j 4 2 2 2 − − The computation of ξ is based on the Corollary 2.5 and equation (2.13). One can show { j } 2p 4j 2 ( 1)j2−2j Γ( c 2j) X2 c¯+1 c +1 1 ξ = − − α = − − − W˜ ; , , , 0 . j 2p 2 j (2p 2)Γ(2j + 1)Γ(2p 4j 2) j 4 2 2 2 − − − − CLASSICAL SKEW ORTHOGONAL POLYNOMIALS IN A TWO-COMPONENT LOG-GAS 17

4. Concluding remarks

The focus of the paper has been on the skew orthogonal polynomial theory arising from the two-component plasma for the mixed charge on the line in the case of classical weight functions, and so on giving a uniﬁed framework including the Gaussian case considered in [33]. It has been shown that for all of the classical cases the skew orthogonal polynomials can be written as a linear combination of the classical orthogonal polynomials, with coeﬃcients and partition functions being expressed in terms of Askey-scheme hypergeometric orthogonal polynomials, which we summarise in Table 1.

Table 1. Coeﬃcients with regard to classical weights

Classical weights Coeﬃcients/Partition functions

Gaussian Laguerre polynomials (1F1)

Laguerre continuous Hahn polynomials (3F2)

Jacobi Wilson polynomials (4F3)

Generalised Cauchy Wilson polynomials (4F3)

Still, a number of questions remain for further investigation:

(1) In random matrix theory, one motivation for knowledge of the expansion of the classical skew orthogonal polynomials in terms of classical orthogonal polynomials is that the Christoffel-Darboux kernel for the skew orthogonal polynomials has a simpler form in terms of orthogonal polynomials, as shown in [1, 39]; see also [22] in the setting of a combinatorial model. However, an analogous evaluation of the kernel for the two-component plasma on a line is not known; (2) In the present paper, we only consider the case that the measure set is continuously supported in R. One would like to develop a theory of the two-component plasma on the discrete (or q-) lattice, to include (perhaps) a broader class of the Askey-Wilson scheme. The difficulty here is to find the discrete (or q-) skew orthogonal polynomials with β =1, and thus to obtain the first two equations in relation (2.6) in a discrete setting (Note: the second two equations are naturally satisfied due to the discrete Pearson equation). Although a discrete orthogonal ensemble is given by [5], or in terms of discrete Selberg integral by [6], the skew orthogonal polynomials with β =1 are still unknown to us; (3) The model we consider in this paper can be viewed as a mixture of β = 1 and β = 4 ensembles in random matrix theory. Another known mixture model, defined on the circle and now interpolating between β =2 and β =4, is given by the Boltzmann factor

N1 N2 eiθk eiθj 2 eiφα eiφβ 4 eiθj eiφα 2. | − | | − | | − | 1 2 j=1 α=1 1≤j

Acknowledgements

This work is part of a research program supported by the Australian Research Council (ARC) through the ARC Centre of Excellence for Mathematical and Statistical frontiers (ACEMS). PJF also acknowledges partial support from ARC grant DP170102028. S.-H. Li would like to thank ZiF, University of Bielefeld for their generous support when he attended the summer school ‘Ran- domness in Physics and Mathematics: From Stochastic Processes to Networks’, where this work was begun.

Appendix A. An introduction to the Routh-Romanovski polynomials

The Routh-Romanovski polynomials were introduced by Routh [34] and Romanovski [35] and have subsequently shown to be of relevance to several studies in physics [32]. This family of polynomials are orthogonal with respect to the weight function

−p az + b W (p,q)(z; a,b,c,d)= (az + b)2 + (cz + d)2 exp 2q arctan . cz + d In this paper, we restrict ourselves to the case e−2V (z) := (1 iz)c(1 + iz)c¯ = (1+ z2)ℜ(c) exp(2 (c) arctan(z)) = W (ℜ(c),ℑ(c))(z;1, 0, 0, 1), − ℑ where (c) and (c) denote the real and imaginary part of c. The weight function is then simply ℜ ℑ related to the Jacobi weight and one can show that correspondingly the monic Routh-Romanovski (c,c¯) ∞ polynomials I˜n (z) can be connected with the monic Jacobi polynomials (see Section 3.4) { }n=0 via the relation ˜(c,c¯) −n (c,c¯) In (z)= i pn (iz). (A.1) If we denote c = p + iq, then we have the orthogonal relation − Γ(n + 1)Γ(2p 2n)Γ(2p 2n 1) I˜(c,c¯), I˜(c,c¯) =22n−2p+2π − − − δ := h(c,c¯)δ . h n m i2,V Γ(2p n)Γ(p iq n)Γ(p + iq n) n,m n n,m − − − − It should be noted that this family of orthogonal polynomials are valid for n N and n

Appendix B. Construction of skew orthogonal polynomials with β =1 relating to the Routh-Romanovski polynomials

A procedure to construct the classical skew orthogonal polynomials for the Hermite, Laguerre and Jacobi weights with β = 1, 4 has been given in [1]; a comprehensive review can be found in [18, §6]. Later, the remaining classical weight function — the generalised Cauchy weight — was considered in [21] during studies of the circular ensemble. In this appendix, we will give a brief review of the construction of the skew orthogonal polynomials with β =1 in the classical cases and apply it to the generalised Cauchy weight, to obtain the relations (2.6) in the Routh-Romanovski case. Throughout the equations (2.14) will be assumed. For the classical weights, we can start with the Pearson pair (f,g) and construct an operator := f∂ + (f ′ g)/2 of order at most 1, such that A z − c c p (z)= j p (z)+ j−1 p (z). A j − p ,p j+1 p ,p j−1 h j+1 j+1i2,V h j−1 j−1i2,V CLASSICAL SKEW ORTHOGONAL POLYNOMIALS IN A TWO-COMPONENT LOG-GAS 19

This operator has the signiﬁcant property of inter-relating the β =2 inner product with the β =1 and β =4 skew inner products,

−1 φ, ψ = φ, ψ 2 , φ, ψ = φ, ψ 1 . h A i2,V h i4,V h A i2,V −h i1,V Specific attention is paid to the case β = 1. Denoting γ = c /( p ,p p ,p ), one j j h j+1 j+1i2,V h j ji2,V finds the relation between the monic orthogonal polynomials p (z) ∞ and the skew orthogonal { j }j=0 polynomials q (z) ∞ with β =1 as { j }j=0 γ2j−1 q2j (z)= p2j (z), q2j+1(z)= p2j+1(z) p2j−1(z), − γ2j and for the normalisation q (z), q (z) 1 =1/γ . h 2j 2j+1 i1,V 2j Therefore, from the Pearson pair (A.2) of Routh-Romanovski polynomials, we can construct = (1+ z2)∂ + (1 p)z iq. Since we consider the monic Routh-Romanovski polynomials A z − − throughout the paper, it is easy to compute the coefficients c = (p 1 j) I˜(c,c¯), I˜(c,c¯) . j − − h j+1 j+1 i2,V Therefore, one can obtain

(c,c¯) (c,c¯) (c,c¯) (1) I˜ , I˜ ζ I˜ 1 = h δ h 2m 2n+1 − n 2n−1i1,V n n,m where 16n(p n)(p iq 2n)(p + iq 2n) ζ = − − − − , n (2p 4n 2)(2p 4n 1)(2p 4n)(2p 4n + 1) − − − − − − (B.1) Γ(2n + 1)Γ(2p 4n)Γ(2p 4n 2) h(1) =24n−2p+3π − − − . n Γ(2p 2n)Γ(p + iq 2n)Γ(p iq 2n) − − − − References

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School of Mathematical and Statistics, ARC Centre of Excellence for Mathematical and Sta- tistical Frontiers, The University of Melbourne, Victoria 3010, Australia E-mail address: [email protected]