EXPANDING THE FOURIER TRANSFORM OF THE SCALED CIRCULAR JACOBI β ENSEMBLE DENSITY

A bstract . The family of circular Jacobi β ensembles has a singularity of a type associated with Fisher and Hartwig in the theory of Toeplitz determinants. Our interest is in the Fourier transform of the corresponding bulk scaled spectral density about this singularity, expanded as a series in the Fourier variable. Various integrability aspects of the circular Jacobi β ensemble are used for this purpose. These include linear differential equations satisfied by the scaled spectral density for β = 2 and β = 4, and the loop equation hierarchy. The polynomials in the variable u = 2/ β which occur in the expansion coefficents are found to have special properties analogous to those known for the structure function of the circular β ensemble, specifically in relation to the zeros lying on the unit circle | u | = 1 and interlacing. Comparison is also made with known results for the expanded Fourier transform of the density about a guest charge in the two-dimensional one-component plasma.


Introduction
In random matrix theory, the joint eigenvalue probability density function (PDF) proportional to (1.1) is referred to as the generalised circular Jacobi β ensemble.This reduces to the PDF specifying the circular β ensemble (see [24, §2.8]) upon setting p = q = 0.With q = 0 but p ̸ = 0, an interpretation of the factor |1 + e iθ | βp is as being due to a spectrum singularity of degeneracy p at θ = ±π, and the resulting PDF has been termed the circular Jacobi ensemble; see [24, §3.9].With q ̸ = 0 also, the factor e qθ is discontinuous comparing values at θ = ±π, extending the spectrum singularity to the class introduced by Fisher and Hartwig in the theory of asymptotics of Toeplitz determinants [23].We remark that generally the study of Fisher-Hartwig singularities in random matrix theory and its applications is active to this present day, with recent references including [20,38,4,63,66,12,19,7,14,8,1].The naming "circular Jacobi" comes from the fact that the orthogonal polynomials in z = e iθ with respect to the inner product ( f , g) = π −π f (θ)g(θ)w (cJ) (θ)| β=2 dθ are given by the hypergeometric polynomials { 2 F 1 (−n, b, b + b; 1 − z)}, where b = p − iq, which in term can be written in terms of Jacobi polynomials [59].
The PDF (1.1) is an example of a β ensemble with the eigenvalues supported on the unit circle in the complex plane.In contrast, a β ensemble on the real line is specified by a joint eigenvalue PDF of the form Generally an Hermitian matrix H can be constructed from a unitary matrix U by way of the Cayley transform This was introduced into random matrix theory by Hua [42].The corresponding mapping of the eigenvalues is 1 + e iθ = tan(θ/2), which is recognised as specifying a stereographic projection between the unit circle and the real line.Applying the inverse of this mapping, up to proportionality the PDF (1.1) transforms to the functional form (1.2) with (1.4) w which is then referred to as the generalised Cauchy β ensemble; see [24,Eq. (3.124)].
It has recently been pointed out that there is also a relationship between the generalised Cauchy β ensemble and the Jacobi β ensemble, defined by the PDF (1.2) with weight (1.5) w(x) = x λ 1 (1 − x) λ 2 χ 0<x<1 , where χ A = 1 for A true, χ A = 0 otherwise.Thus for f a multivariable polynomial, one has that [35] (1.6) ⟨ f (1 − ix 1 , . . ., 1 − ix N )⟩ (Cy) = ⟨ f (2x 1 , . . ., 2x N )⟩ (J) where the average on the RHS is to be understood in the sense of analytic continuation.The identity (1.6) was used in [35] to obtain linear differential equations satisfied by the density in the Cauchy ensemble with β = 1, 2 and 4, and to study the corresponding moments.This line of study was initiated in [2].Another application of (1.6) was given in [27], where it was used to study the distribution of the trace in the Cauchy β ensemble; see also [3].
Applying the mapping (1.3) and recalling the theory of the previous paragraph gives the relationship between averages in the circular Jacobi and Jacobi ensembles (1.7) f 1 1 + e iθ 1 , . . ., 1 1 + e iθ N (cJ) = ⟨ f (x 1 , . . ., x N )⟩ (J)   λ 1 =−β(N+p−1)/2−1+iq λ 2 =−β(N+p−1)/2−1−iq .Note that unlike (1.6), (1.7) no longer inter-relates averages of polynomials.On the other hand, there is a further inter-relation which does have this property, namely [24, corollary of Prop.3.9.1] ⟨ f (−e iθ 1 , . . ., −e iθ N )⟩ (cJ) = ⟨ f (x 1 , . . ., x N )⟩ (J) , again valid in the sense of analytic continuation.Note that shifting the integration variables θ j → θ j + π on the LHS of (1.8) removes the minus signs in the arguments of f to give (1.9) ⟨ f (e iθ 1 , . . ., e iθ N )⟩ ( cJ) = ⟨ f (x 1 , . . ., x N )⟩ (J) , where cJ refers to the circular Jacobi β ensemble now defined on 0 ≤ θ j < 2π with weight (1.10) It is clear from the above discussion that known properties of the Cauchy and Jacobi β ensembles have application in the study of the generalised circular Jacobi β ensemble.For example, some time ago [34] the full set of k-point correlation functions for (1.1) in the case q = 0 and β = 1, 2, 4, and their scaling limit about the spectrum singularity, were deduced from knowledge of the corresponding k-point correlation functions in the case of the Cauchy ensemble.More recently [33] this analysis was repeated with the restriction to q = 0 removed, and moreover the leading finite size corrections to the scaling limit were determined.
In the present paper we focus our attention on integrable structures in relation to the density for the generalised circular Jacobi β ensemble, ρ (1),N (θ; β, p, q) say, and the analogue of the corresponding moments.The latter are the coefficients {c k (N, β, p, q)} of the Fourier series for the density, (1.11) ρ Of particular interest is the scaled limit which is expected to be well defined for general real τ.The integrable structures come from various sources, including the interplay with the Cauchy and Jacobi β ensembles.
To see why this is of interest, we begin by noting that the density in the neighbourhood of the spectrum singularity admits a well defined large N limit, first studied for q = 0 and β = 1, 2 and 4 in [34], and extended to all even β and general q in [49,33].For general β > 0 this can also be regarded as a bulk scaling limit, with origin at the spectrum singularity, specified by taking N → ∞ and the angular coordinates θ j scaled θ j → 2πx j /L so that the mean spacing between eigenvalues L/N is a constant, which we take to be unity by choosing L = N.The Fourier series for finite N, upon subtracting the constant c then tends to a well defined Fourier transform relating to c ( cJ) ∞ .Thus the scaling limit of the density in the neighbourhood of the spectrum singularity is specified by (1.13) ρ or in the setting of the weight (1.10) ,N (2πχ x<0 + 2πx/N; β, p, q).
Applying this limit to the Fourier series (1.11) we obtain the Fourier transform expression where c ∞ is given by (1.12), and thus by Fourier inversion (1),∞ (x; β, p, q) − 1 e iτx dx.
In the case q = 0, p = 1, (1.16) can be identified with the bulk structure function for the limiting statistical state defined by (1.1) with p = q = 0 [29,25,13].Thus for finite N the structure function S N (k; β) can be defined by the covariance of the pair of linear statistics ∑ N j=1 e ikλ j , ∑ N j=1 e −ikλ j , (1.17) where the averages are with respect to (1.1) in the case p = q = 0.In the bulk scaling limit, this can be written in terms of the truncated (connected) two-particle correlation function But by the definition of the truncated two-particle correlation function in the case p = q = 0 we have (1),∞ (x; β, 1, 0) − 1, where the RHS comes about since one of the fixed eigenvalues involved in the definition of the two-particle correlation induces a spectrum singularity of degeneracy 1. Comparing (1.18) with the substitution (1.19) to (1.16) then tells us that In relation to the finite N structure function for general random matrix ensembles, one remarks that the two-point quantity ⟨| ∑ N j=1 e ikλ j | 2 ⟩ on the RHS of (1.17) has been the subject of some recent attention in literature applying random matrix theory to the study of many body quantum chaos [16,60] and the scrambling of information in black holes [15,10]; many more references could be given.Of interest is its graphical shape as a function of k, which assuming a non-constant spectral density exhibits dip-ramp-plateau features as k varies as a function of N, with corresponding dynamical significance.Following on from the applications, several theoretical works have quantified dip-ramp-plateau for various model random matrix ensembles.With such concerns being in common with present work when viewed broadly, we give a comprehensive list of references: [55,25,26,51,13,61,62,30] Returning now to (1.18), we know from [29] that the quantity extends to an analytic function of τ for |τ| < min (2π, πβ) and furthermore satisfies the functional equation The power series expansion of f (τ; β) in (1.22) is known to have the form [29,65,25] (1.23) Thus the coefficient of τ j is a palindromic or anti-palindromic polynomial in u = 2/β.Note that this structure is consistent with (1.22).The explicit form of p j (u) has been determined up to and including j = 9 in [29,65], while the explicit form of p 10 (u) was calculated recently in [25].
It follows from these equations together that h j alternates pure imaginary then real, while hj alternates real and then pure imaginary.Further motivation for our interest in (1.11) is that the case q = 0 has previously been the subject of some literature for the corresponding two-dimensional generalisation of (1.1), which is up to proportionality specified by the PDF (1.28) The PDF (1.28) has the interpretation as the Boltzmann factor for the two-dimensional one-component plasma, consisting of N mobile two-dimensional unit charges, repelling via the pair potential − log |z − z ′ |, with a smeared out uniform neutralising background of charge density −1 in the region |z| < √ N/π; see [9, §4] for a recent review.At the origin is a so-called host charge (or impurity) of strength Q.
Let ρ OCP (1),∞ (r; Q) denote the density at point r in the limit N → ∞ of (1.28).Charge neutrality of the screening cloud implies Two distinct derivations have been given for a sum rule specifying the second moment of the screening cloud [57,43] (1.30) Most recently, under the assumption of analyticity in Q, the fourth moment of the screening cloud has been shown to obey [58] (1.31) Earlier [44,11] the polynomial b 0 (β)/β 2 appeared as the coefficient of k| 6 in the expansion of the structure function S OCP ∞ (k; β).From the spherical symmetry of ρ OCP (1) (r; Q) − 1 we have for its Fourier transform At order |k| 6 there is evidence that the polynomial structure in 4/β breaks down, at least for the coefficient of Q as the latter is equivalent to the coefficient of |k| 8 in the small |k| expansion of the structure function S OCP ∞ (k; β) ( [58]; see also Section 5 below) which is expected to consist of an infinite series in powers of 4/β [44].In contrast, in [58] it is argued that the coefficient of |k| 2j is always a polynomial in Q of degree j + 1.
Our study of the Fourier transform (1.15) and its small τ expansion (1.25) begins in Section 2 where we specialise to β = 2.By doing this use can be made of a third order linear differential equation which then characterises ρ (cJ) (1),∞ .Our main finding is a second order difference equation for the odd indexed expansion coefficients of the non-analytic terms in (1.25) -see Proposition 2.12.For p a positive integer the implied sequence terminates for indices great than 2p − 1.The even indexed expansion coefficients of the non-analytic terms all vanish as does the coefficients of the analytic terms vanish except for h0 .The considerations of Section 2 are repeated in Section 3, but now for β = 4.This is possible due to the derivation of an explicit fifth order linear differential equation characterising ρ (1),∞ .Proposition 3.2 then gives a fourth order difference equation satisfied by the expansion coefficients in (1.25), which in general are all non-zero.A different approach to studying the expansion coefficients of the Fourier transform is introduced in Section 4. This is to derive and apply a hierarchical set of equations referred to as loop equations for the generating function of the Fourier coefficients.In this method β > 0 is arbitrary.However the computational complexity increases with the order of the expansion coefficient, which restricts the number of terms which can be computed.Five orders are specified in Proposition 4.2.A discussion of the explicit functional forms obtained by the loop equation analysis, in the context of known properties of the expansion coefficients of the structure function S ∞ (τ; β), is given in Section 5.
As relevant for the bulk scaling defined in the paragraph including (1.13), in this we replace {θ j } in favour of {x j } and further define The case β = 2 of (2.1) and (2.2) is special.The k-point correlation function then has a determinantal structure, with the elements of the determinant moreover independent of k.Specifically, in relation to (2.2) we have , where [5, after the change of variables 1/x l → πx l ] (see also the introduction of [21] for further references, and the recent work [33] for an independent derivation) Here 1 F 1 (a; c; z) denotes the confluent hypergeometric function in standard notation.In particular, taking the limit y → x using L'Hôpital's rule gives for the bulk scaled density shows that (2.5) is unchanged by the mapping x → −x, q → −q as noted in the discussion above (1.26).

2.2.
The case q = 0 -Bessel function asymptotics.Use of the expression for a particular 1 F 1 in terms of a Bessel function, shows [52] (2.7) Taking the limit y → x and making use of Bessel function identities then gives [24, Eq. (7.49) As a check, it follows from (2.8) and trigonometric formulas for the Bessel functions at half integer order that (2.9) We see that this is as required by the rotation invariance of the PDF (1.1) for p = q = 0 and the normalisation implied by bulk scaling.Setting now p = 1, shows This is in keeping with (1.19) upon recalling the standard fact that in relation to (1.1) with p = q = 0, ρ (2),∞ (x, 0; β, p, q)| β=2 is given by the RHS of (2.10); see e.g.[24, Eq. ( 7.2) with ρ = 1].Generally, substituting any positive integer for p in (2.8) reduces the Bessel function to a trigonometric form involving sin 2πx, cos 2πx.This is further illustrated by the next simplest example after (2.10), which is to set p = 2 with the result In the notation of (1.16) and (1.25) Starting with (2.5), we don't know how to give an explicit special function evaluation of the integral in (2.12).Nonetheless, we can use (2.5), expanded for large x, to deduce the h j (β, p, q) for j odd in the corresponding small τ expansion.Our ability to do this comes about from Fourier transform theory [48].Thus if (2.13) ρ where only non-oscillatory terms are recorded on the RHS, then Generally this approach can access the terms in (2.12) singular in τ only.The analytic terms are not accessible via this method.According to (2.5), the expansion (2.13) as applies to the case q = 0 can be deduced from knowledge of the large x asymptotic expansion of the Bessel function.The latter is given by [53, Eq. (10.17 (2.15) where with (u) Proposition 2.2.Consider the large x asymptotic expansion of ρ (cJ) (1),∞ (x; β, p, q)| β=2,q=0 restricted to non-oscillatory terms.This is given by (2.13) with c 2n = c 2n (p), where (2.17) with {a k (ν)} as in (2.16).
Proof.We substitute (2.15) in (2.8).Applying simple trigonometric identities allows the non-oscillatory terms to be separated from the oscillatory terms.Finally, the coefficient of 1/x 2n is extracted using the formula for multiplication of power series.□ Remark 2.3.The use of computer algebra to compute (2.17) for small values of n and general p suggests the simplified form where α n is a rational number.Regarding the latter, the computer algebra computation gives We note in particular that (2.18) implies c 2n (p) = 0 for n > p, and that c 2n (p) is a polynomial in p 2 of degree n − 1.In fact this simplified form, together with the explicit value of α n , can be obtained by taking a different approach to the large x asymptotic expansion of ρ (cJ) (1),∞ (x; β, p, q)| q=0,β=2 , namely via a differential equation, which is to be done next.

2.3.
The case q = 0 -a third order linear differential equation.In the work [35, §3.1.2]the scaled density (1/π)ρ cJ (1),∞ (x/π; β, p, q)| β=2,q=0 was shown to satisfy the third order linear differential equation (2.20) This can be used to provide an alternative approach to the computation of the coefficients in the expansion (2.13).
With r(x) := R(x) − 1/π = O(x −2 ) as implied by (2.22), there is a well defined meaning to taking the Fourier transform of (2.20).Upon integration by parts and an additional differentiation with respect to τ this reads (2.24) Proposition 2.6.Let and specify {c 2n } by Proposition 2.4.The small τ expansion substituted in (2.24) has the unique solution Proof.Substituting (2.26) in (2.24) gives The value of b 0 is not restricted by the differential equation, nor is the value of b 1 , while for n even and positive the recurrence gives b n = 0.For odd subscripts, replacing n by 2n − 1 allows the recurrence to be related to that in (2.23), which implies (2.27).□ Remark 2.7.
1.The implied coefficient of the third derivative in (2.24) has a zero at τ = 0 and τ = ±2, telling us that the radius of convergence of (2.26) is 2.
2. Since r(x) as relates to (2.24) is an even function of x, r(τ) must be an even function of τ.Hence for τ < 0 (2.26) is now a power series in |τ|, as is consistent with (2.12).In particular the coefficient of τ must then relate to the coefficient of 1/x 2 in the large x expansion of r(x) as implied by the relation between (2.13) and (2.14).This justifies the choice of b 1 in (2.25) which as already mentioned otherwise is not determined by (2.24).

3.
The log-gas interpretation of the factor |1 + e iθ | βp in the circular Jacobi weight w (cJ )(θ) of (1.1), otherwise interpreted as a spectrum singularity of degeneracy p at θ = π, is as a fixed charge of strength p. From this log-gas picture, perfect screening of the fixed charge (see e.g.[24, §14.1]) implies and thus from (2.25) that b 0 = −p.
2.4.Third order linear differential equation for the case q ̸ = 0.A third order linear differential equation satisfied by the scaled density (1/π)ρ cJ (1),∞ (x/π; β, p, q)| β=2 for general p, q can be obtained by following the same procedure as used in [35] to deduce (2.20), suitably generalised to involve the extra parameter q.Proposition 2.9.The scaled density (1/π)ρ cJ (1),∞ (x/π; β, p, q)| β=2 obeys the third order linear differential equation Proof.For the Cauchy ensemble with weight (1.4) and β = 2 let ρ (Cy) (1),N (t) denote the corresponding density.With g(t) = (1 + t 2 )ρ (Cy) (1),N (t) we know from [35, Eq. (3.33) with r(t) = g(t), From the theory of the second paragraph of the Introduction it follows 2g(t)| t=tan(θ/2) = ρ (cJ) (1),N (θ).We want to scale θ about the spectrum singularity at θ = π, and as a normalisation we require that at large distances in the scaled variable the density is 1/π.These requirements are met by choosing θ = π + 2X/N.With t = tan(θ/2) and N large this implies t ∼ −N/X.We make this substitution in (2.31).Equating terms at leading order in N in the resulting equation gives (2.30).□ For q ̸ = 0, it follows from the functional form (1.1) that ρ (cJ) (1),∞ (x; β, p, q) is not an even function of x, in distinction to the case q = 0.In place of (2.13) we now expect (2.32) ρ for the asymptotic form of the non-oscillatory terms.The analogue of (2.14) for the nonanalytic terms in the small τ form of the Fourier transform is then [48] (2.33) As for the result of Proposition 2.4, the differential equation (2.30) can be used to determine the coefficients d n in (2.32).
Proposition 2.10.The expansion (2.32) holds with (2.34) Proof.As with (2.22), to relate the expansion (2.32) to an expansion of R(x) in (2.30) requires a scaling x → x/π to give The result now follows by substituting (2.35) in (2.30) and equating like (negative) powers of x. □ We note that setting q = 0 in (2.34) reclaims the result of (2.21).However unlike the situation with q = 0 the recurrence in (2.34) does not admit a simple functional form for its solution.Structural points of interest are that d 2n is equal to (p 2 + q 2 ) times a polynomial in p 2 and q 2 of degree n − 1, while for n ≥ 2, d 2n−1 is equal to q(p 2 + q 2 ) times a polynomial in p 2 and q 2 of degree n − 2. Specifically, as some low order examples (2.36) And as we know from (2.33), knowledge of {d n } gives the explicit form of the coefficients {h n (β, p, q)| β=2 } in the functional form (1.25) according to (2.37) Remark 2.11.
1. Starting from (2.5), and using the appropriate generalisation of (2.15) as can be found in [53], we have independently verified that with respect to non-oscillatory terms as is consistent with Proposition 2.10.
3. Difference equations satisfied by moments of the spectral density for classical ensembles is a rich theme in random matrix theory; references include [41,46,47,50,64,18,17,2,39,31,32].A feature of these settings is a global scaling for which the spectral density has compact support.However in the present setting of the non-compactly supported bulk scaled spectral density about a spectrum singularity, there is no literal meaning of the moments due to the slow decay at infinity.Hence the non-analytic terms in the expansion (1.25) of the Fourier series.Note that this is in contrast to the two-dimensional case as exhibited in (1.33), even though the support there is also non-compactly supported.
It remains to investigate the coefficients { hj (β, p, q)| β=2 } in the expansion (1.25) for j ≥ 1.For this purpose we introduce r(x) = R(x) − 1/π and proceed to take the Fourier transform of (2.30) to deduce as the q ̸ = 0 generalisation of (2.24) The following result now follows by direct substitution.
Proposition 2.12.The small τ expansion substituted in (2.40) implies the recurrence for {e n } (2.42) The values of e 2 , e 3 , . . .implied by (2.42) are independent of e 0 but depend on e 1 .However in keeping with the circumstances of Proposition 2.6 the value of e 1 is not determined by the differential equation (2.40).We note that writing e n = dn+1 i n+1 /n! in (2.42) shows { dn } satisfies the same recurrence relation as {d n } in Proposition 2.10.In the context of the expansion (1.25) this implies that for β = 2 the sequences {h j (β, p, q)} and { hj (β, p, q)} satisfy the same recurrence.While all terms in the sequence {h j (β, p, q)| β=2 } are known according to Proposition 2.10 and (2.37), the sequence members hj (β, p, q)| β=2 for q ̸ = 0 and j ≥ 2 depend on the value of h1 (β, p, q)| β=2 for q ̸ = 0, which is not known from the above considerations.However results from Section 4 -specifically by taking the imaginary part of α 1 | β=2 in (4.17) -allow us to deduce that we have h1 (β, p, q)| β=2 = 0 for general q and hence hj (β, p, q)| β=2 = 0 for j ≥ 1.

The case β = 4
The scaled statistical state in the neighbourhood of the spectrum singularity for β = 4 is a Pfaffian point process, rather than the simpler determinantal point process for β = 2 [34,33].Perhaps surprisingly then, the functional form of the of the β = 4 density is functionally related to the β = 2 density.This is simplest to state with q = 0, for which from [34,Eq. (3.34) with a = p and X = Y] reads (3.1) where for convenience it is assumed x > 0 (as previously remarked, for q = 0 the density is an even function).An analogous result for q ̸ = 0 is given in [33].However for our interest in the expansion (1.25) we will not make use of such an explicit expression.Rather the key feature of the β = 4 density for our purposes is that it, like the β = 2 density, satisfies a linear differential equation albeit now of degree 5.
Proposition 3.1.Let p = p − 2p 2 .The scaled density (1/π)ρ cJ (1),∞ (x/π; β, p, q)| β=4 satisfies the differential equation Proof.A fifth order linear differential equation for the β = 4 Jacobi ensemble density has been given in [56,Th. 2].Making use of the relation between the Jacobi and Cauchy averages (1.6) we can deduce from this a fifth order differential equation for the quantity (1 + t 2 )ρ (Cy) (1),N (t), where ρ (Cy) (1),N (t) denotes the β = 4 Cauchy ensemble density corresponding to the weight (1.4).As in the proof of Proposition 2.9, the computation is concluded by substituting t = −N/X as corresponds to a hard edge scaling and equating to leading order in N. Due to the large number of terms involved, the required steps were all carried out using computer algebra.□ As for the differential equation (2.30), the differential equation of Proposition 3.1 has a unique solution of the form (2.35).Proposition 3.2.Substituting the expansion (2.35) in the differential equation of Proposition 3.1, but with the coefficients renamed from {d n } to {g n } for distinction, one obtains that with the higher order coefficients g 5 , g 6 , . . .then determined by the fourth order recurrence relation As with going from (2.32) to (2.33), then to (2.37), knowledge of {g n } tells us that Remark 3.3.

1.
A fifth order linear differential equation is also known for the density of the Jacobi ensemble in the case β = 1 [56].However this is not independent of the corresponding differential equation in the case β = 4. Writing g n = g n (β, p, q), where {g n } is used in the same sense as Proposition 3.2, the dependence implies This functional equation is generalised in (4.29) below.

2.
The fifth order differential equation of Proposition 3.1 can be transformed into a fifth order differential equation for the Fourier transform of (R(x) − 1/π).Now substituting the small τ expansion (2.41) gives a fifth order recurrence for the coefficients {e n }.Upon the substitution e n = gn+1 i n+1 /n!, this recurrence becomes that of Proposition 3.2 with {g n } replaced by { gn }.As in the discussion below Proposition 2.12 in the case β = 2, this implies that for β = 4 the sequences {h j (β, p, q)} and { hj (β, p, q)} of the expansion (1.25) satisfy the same recurrence.According to (3.2) and Proposition 3.2 the first of these is fully determined.However the latter requires specification of g2 , g3 , g4 as initial conditions, which are not determined by the differential equation.However, we will see that these initial conditions can be accessed using result deduced from results of the next section (specifically (4.27) and (4.28)), which imply where C 1 denotes the unit circle in the complex plane.Expanding for large x shows that this quantity relates to the Fourier components of the density as specified in (1.11), (4.2) A primary aim of this paper is to obtain the small τ expansion of the Fourier transform c (cJ) ∞ (τ; β, p, q) as specified in (1.16).To see how W 1 (x; N, β, p, q) relates to this aim, suppose that this quantity admits a 1/N expansion Suppose furthermore that each W l 1 (x; β, p, q) when expanded for large x analogous to (4.2) has coefficients which are polynomials in k of degree l − 1. Denote the leading coefficient in the polynomial by α l−1 (β, p, q).It then follows from the definition (1.12) that ( In fact this general approach has been used to determine {α l (β, p, q)| p=1,q=0 } up to and including l = 10 in [65], thereby reclaiming the explicit form of the polynomials {p j (y)} in (1.23) for j = 1, . . ., 9, first obtained in [29].
In the loop equation formalism, the computation of the coefficients in the series of (4.3) up to the j-th requires knowledge of the large N expansion of particular truncations of the multipoint correlators (4.5) W n (x 1 , . . ., These truncations, denoted W n , have for n = 2 the simple covariance form and similarly for n = 3.For general n ≥ 2 the truncations, referred to as connected correlators, are symmetric linear combinations of {W m (x j 1 , . . ., x j m ; N, β, p, q)} m=1,...,n with 1 ≤ j 1 < • • • < j m ≤ n defined so that they are generating functions for the mixed cumulants; see e.g.[28, Eq. (1.17)].A key feature of the connected correlators is that their order of decay in a 1/N expansion increases as n increases, ( W n (x 1 , . . ., see [6,65].To compute W l 1 (x; β, p, q) in (4.4) up to order l = j, we will require knowledge of the expansion (4.7) of W n for n = 2, . . ., j + 1 up to order l = n − j − 1.

4.2.
The loop equation hierarchy.The connected correlators have previously been analysed in the case of the Jacobi β ensemble based on (1.2) with the weight (1.5).They satisfy a hierarchy of equations -referred to as loop equations -specified by [36,Eq. (4.6)] Here the subscripts J on the W J m indicate the connected correlations with respect to the Jacobi β ensemble, κ := β/2 and J n := {x 2 , . . ., x n } with J 1 = ∅.Also, the tuple (x 1 , . . ., xk , . . ., x n ) consists of x 1 , . . ., x n in order with x k excluded.
The mapping (1.9) allows the loop equations (4.8) for the Jacobi β ensemble to be transformed to the loop equations for the circular Jacobi β ensemble with weight (1.10) by a simple substitution of λ 1 , λ 2 .With this done, the large N expansion (4.7) can be substituted and like powers of N equated to obtain a triangular system of equations for {W In addition, an equation analogous to (4.13) holds which expresses W l+1 n for n ≥ 3 in terms of lower order terms in the triangular system.
A hand calculation readily suffices to solve the above triangular system up to and including W 2 1 (x).Thus we find (4.14) with the intermediate results The evaluations of W 0 1 (x), W As commented in the sentence above (1.13),this last expression, with the restriction to k = 0 removed, is to be subtracted from the Fourier coefficient 2π 0 ρ (1),N (θ; β, p, q)e ikθ dθ to arrive at (1.16).With this done, according to the definition of {α l } given above (4.4),we read off from the second and third equations in (4.14) that (4.17) where κ := β/2.The use of computer algebra allows the evaluations (4.15) to be extended, and thus similarly for (4.17).Specifically, in relation to the latter, this shows where α := κ p + iq.We are now in a position to specify the coefficients in (1.25) up to and including index label j = 5.
Explicity, for indices up to and including j = 3, Now setting n = 1, expanding in 1/N as in (4.3), then proceeding from this to (4.4) gives the functional equation or equivalently (4.29) The functional equations (4.29) are readily exhibited on the explicit functional forms (4.24)-(4.28).

Discussion
Setting q = 0 we read off from (1.16), (1.25) and the results of Proposition 4.2 that It is of interest to compare this expansion to the analogue for the two-dimensional onecomponent plasma (1.33).With Q identified as p, we see that the first term on the RHS, which corresponds to the perfect screening of the external charge, is the same for both.In two-dimensions the generalised Fourier transform of the logarithmic potential V OCP (r) = − log |r| is VOCP (k) = 2π/|k| 2 and the dimensionless thermodynamic pressure is βP OCP = 1 − β/4.Hence the second term in the expansion (1.33) can be written In relation to the circular β ensemble (CβE), the generalised Fourier transform of the logarithmic potential V CβE (r) = − log |r| is VCβE (τ) = π/|τ| and the dimensionless thermodynamic pressure is βP CβE = 1 − β/2.Hence the second term in the expansion (5.1) can be written The structural similarity between (5.2) and (5.3) is evident.However when it comes to comparing the third terms on the RHSs of (5.1) and (1.33) structural differences show.Specifically the former is Q times a quadratic polynomial in Q, while the latter is p times a linear polynomial in p.This is perhaps not unsurprising as the argument used in [58] to predict the Q 2 term relies in part on the screening cloud having a fast decay with well defined moments, which is not true in one-dimension.The dependence on β is quadratic in 1/β in both cases, although in (5.1) there is a stand alone factor of (1/κ − 1) with no analogue in (1.33).
A linear response argument is used in [58] to deduce that (5.4) lim where the second equality follows from the two-dimensional analogue of (1.20).As noted in [58], this sum rule is readily checked to be a feature of the expansion (1.34).The same linear response argument can be applied to the one-dimensional log-gas to obtain Using the results of Proposition 4.2 we can check the validity of (5.6) in a small τ expansion up to and including order τ 4 .Linear response also applies in the limit q → 0. To facilitate this, define V(x) = πsgn(x).Noting as an improper integral that allows us to take the inverse Fourier transform to conclude V(τ) = 2πi/τ.With this knowledge we can deduce the analogues of (5.5) and (5.6).Specifically, in relation to the latter we obtain (5.7) lim ∞ (τ; β, 1, 0) .
As with (5.6), using the results of Proposition 4.2 we can check the validity of (5.7) in a small τ expansion up to and including order τ 4 .In Proposition 4.2 let us replace q by κq.Then h j and hj are each polynomials of degree j in 1/κ.As remarked in the Introduction below (1.24), in the case p = 1, q = 0 these polynomials have the property of being palindromic or anti-palindromic in u := 1/κ.Moreover, it has been observed from the explicit form of these polynomials that all the zeros lie on the unit circle |u| = 1 in the complex u-plane and moreover exhibit an interlacing property [29,25], which is a feature too of the polynomials appearing in the 1/N expansion of the moments of the spectral density for the Gaussian β ensemble [64].(More on this has been communicated to the senior author by Michael A. La Croix, who among other things highlights the references [22,40,45]; one should add too the recent work [54], although any sought of explanation is still lacking.)Our results of Proposition 4.2 reveal these same properties for the polynomials in u := 1/κ which result as coefficients by expanding h j or hj in a power series in p for q = 0, or in q for p = 0.For example, from (4.21) and (4.21) we compute that in relation to h 5 (β, p, q)| q=0 the coefficients of p, . . ., p 5 in order are, up to proportionality, the (anti-)palindromic polynomials (u − 1)(30 − 91u + 124u 2 − 91u 3 + 30u 4 ), 32 − 75u + 90u 2 − 75u 3 + 32u 4 , (u − 1)(11 − 20u + 11u 2 ), 5 − 9u + 5u 2 , u − 1, and which indeed can be checked to have successively interlacing zeros all on the unit circle in the complex u-plane.
The results of Proposition 4.2 take on a simpler form in the case q = 0 with β → ∞ (low temperature limit), (5.8)where p := p(p − 1).The factor of p in all terms but the first makes sense as the ground state is an equally spaced lattice for both p = 0 and p = 1, which is when p vanishes.Also, the leading term in p of each power of τ is suggestive of the summation in τ , which makes explicit the form of the singularity in τ as |τ| → (2π) − .In fact for p → 0 (and thus p → −p → 0) this can be anticipated from the first equality in (5.5).The required input is the large β form of S CβE ∞ (τ; β), which we know from [29, §5] to be given by the RHS of (5.9) with p replaced by 2/β.Substituting in the first equality of (5.5) gives a result equivalent to (5.9) for p → 0.
Remark 5.1.With q = 0 a result of [37] tells us that with bulk spectrum singularity scaling, in the limit β → ∞ the eigenvalues crystallise at the zeros of the Bessel function J p−1/2 (πx).
and so the results (2.21)-(1.31)give the small |k| expansion up and including order |k| 4 , and exhibit a polynomial structure in Q and 4/β,(1.34)