Christoffel transformations for (partial-)skew-orthogonal polynomials and applications
aa r X i v : . [ m a t h - ph ] A ug CHRISTOFFEL TRANSFORMATIONS FOR(PARTIAL-)SKEW-ORTHOGONAL POLYNOMIALS AND APPLICATIONS
SHI-HAO LI AND GUO-FU YU
In celebration of Professor Peter J Forrester’s th birthday Abstract.
In this article, we consider the Christoffel transformations for skew-orthogonal poly-nomials and partial-skew-orthogonal polynomials. We demonstrate that the Christoffel trans-formations can act as spectral problems for discrete integrable hierarchies, and therefore wederive certain integrable hierarchies from these transformations. Some reductional cases are alsoconsidered. Introduction
The theory of orthogonal polynomials is an important topic in modern analysis. In particular, ithas many different applications in contexts of mathematical physics such as random matrix theoryand integrable system, for example, see [11, 12, 14, 23, 31]. One of the key features of an orthogonalpolynomial sequence is the three term recurrence. This relation, also referred to as the spectralproblem, is connected with integrable systems when time evolutions are permitted. Apart from thecontinuous spectral problem, attention is also paid to the discrete spectral transformation, namelythe Christoffel transformation. The Christoffel transformation is given in terms of an adjacentfamily of orthogonal polynomials [27, 30] P ( m ) n ( x ) = 1 τ ( m ) n − det c m · · · c n + m ... ... c n + m − · · · c n + m − · · · x n , τ ( m ) n − = det( c m + i + j ) i,j =0 , ··· ,n − . By directly using determinant identities (see [27, Eqs. (1.1.22a-b)]), one can find the recurrence P ( m ) n ( x ) = xP ( m +1) n − ( x ) − τ ( m +1) n − τ ( m ) n − τ ( m +1) n − τ ( m ) n − P ( m ) n − ( x ) ,P ( m ) n ( x ) = xP ( m +2) n − ( x ) − τ ( m +1) n − τ ( m +1) n − τ ( m ) n − τ ( m +2) n − P ( m +1) n − ( x ) . This kind of spectral transformation is useful in finding a relationship with discrete integrablesystems since the discrete index m naturally appears in the adjacent orthogonal polynomials. Inthe literature, there are many applications of Christoffel transformations to classical integrablesystems. For example, in [8, 28], the Christoffel transformations were applied to finding discreteToda systems with higher analogues, which are related to the qd and qqd algorithms. In [4], Mathematics Subject Classification.
Key words and phrases.
Christoffel transformations; Skew-orthogonal polynomials; partial-skew-orthogonal poly-nomials; Pfaffian tau-functions.
Christoffel transformation for matrix orthogonal polynomials was considered, and its connectionwith non-abelian 2D Toda lattice hierarchy was found. Moreover, the Christoffel transformationfor multivariate orthogonal polynomials was considered in [6], and its connection with integrablesystem was considered in [5].In this work, we mainly consider the Christoffel transformations for skew orthogonal polynomi-als (SOPs) and partial-skew-orthogonal polynomials (PSOPs) with applications in the theory ofclassical integrable systems. SOPs are well known in the studies of random matrix theory, as theyare the characteristic polynomials of celebrated orthogonal and symplectic ensembles with theirspecified Pfaffian structures. Besides, these polynomials are also applicable to integrable system.A connection between SOPs and the so-called Pfaff lattice was firstly considered in [1, 3], and laterconsidered in the geometric setting [18, 19]. In [24], the discrete Pfaff lattice was considered byusing the discrete spectral transformation of SOPs with Pfaffian tau functions.We emphasise that Pfaffian tau functions are not only important in integrable systems [2]but play a significant role in quantum field theory such as 2D Ising model, dimer models and1D XY chain [7, 26]. Therefore, Pfaffian tau functions are very worthy of study. We remarkthat in addition to the above mentioned SOPs and even-order Pfaffian tau functions, one canobtain odd-order Pfaffian tau functions from a generalised Wick’s theorem. One can thereforenaturally ask about the odd-order Pfaffian tau functions and corresponding polynomials theory.In [9], the concept of PSOPs was proposed and the reason to call these polynomials PSOPs isthat these odd-order polynomials are not skew orthogonal with the even ones. Though not skew-orthogonal, by making use of these polynomials, many interesting integrable lattice were found withapplications in convergence acceleration algorithms, vector Padé approximation and condensationalgorithms for Pfaffians [21]. More importantly, one specified PSOPs is related to the Buresensemble with potential application in quantum information theory [13]. Therefore, the Christoffeltransformations for SOPs and PSOPs are not merely important in orthogonal polynomials theoryitself but with potential and established applications in many other subjects.In Section 2, we firstly give a brief review of SOPs and PSOPs. By employing Pfaffian identities,we give their Christoffel transformations. For SOPs { P ( m ) n ( z ) } n,m ∈ N , we have the transformations P ( m )2 n +1 ( z ) − A mn P ( m )2 n ( z ) = z (cid:16) P ( m +1)2 n ( z ) − B mn P ( m +1)2 n − ( z ) (cid:17) ,P ( m )2 n +2 ( z ) − C mn P ( m )2 n ( z ) = z (cid:16) P ( m +1)2 n +1 ( z ) − D mn P ( m +1)2 n ( z ) (cid:17) , with proper coefficients A mn , B mn , C mn and D mn . This Christoffel transformation is slightly differentfrom the one in [24, Thm. 3] since here the formula only involves two adjacent families of SOPs.Regarding PSOPs { Q ( m ) n ( z ) } n,m ∈ N , the Christoffel transformation could be identically written as Q ( m ) n +1 ( z ) + ξ mn Q ( m ) n ( z ) = z (cid:16) Q ( m +1) n ( z ) + η mn Q ( m +1) n − ( z ) (cid:17) , with coefficients ξ mn and η mn properly chosen. Moreover, we find a multi-component version ofodd-order PSOPs, and therefore give a multi-component Christoffel transformation as well.In Section 3, we manifest how to make use of the Christoffel transformation of SOPs. Byexpanding the SOPs in terms of monomial with coefficients expressed by Pfaffian tau functions, onecan easily get the DKP (or Pfaff-lattice) hierarchy from the Christoffel transformation. Moreover,we consider a reductional case—Laurent type SOPs [25], by which the Christoffel transformation is HRISTOFFEL TRANSFORMATIONS FOR SOPS AND PSOPS 3 reduced to a three term recurrence relation and the corresponding integrable hierarchy is reducedto the 1d-Toda hierarchy with wave function expressed as SOPs.In Section 4, some considerations are taken into the Christoffel transformation of PSOPs. Themost general case is firstly given and then some reductional cases are considered. We demonstratethree different examples to show how to make use of moment constraint approach to obtain lowerdimensional integrable lattices, generalising the moment constraint approach proposed in [20].Concluding remarks are given in Section 5.2.
Christoffel transformations of skew orthogonal polynomials and partial skeworthogonal polynomials
The main purpose of this part is to derive the Christoffel transformation for the skew-orthogonalpolynomials (SOPs) and partial-skew-orthogonal polynomials (PSOPs) as an analogy of the thatfor orthogonal polynomials. Such transformations can be regarded as the spectral transformations,thus being prepared for the later discussion about how to connect with integrable systems afterthe involvement of time. To this end, we firstly need to give introductions to SOPs and PSOPs.Following [9], we start with a skew symmetric inner product, and then give some brief derivationsabout SOPs and PSOPs from a unified framework.2.1.
Skew symmetric inner product, SOPs and PSOPs.
Let us consider a skew symmetricinner product h· , ·i from R [ z ] × R [ z ] → R satisfying the skew symmetry property h f ( z ) , g ( z ) i = −h g ( z ) , f ( z ) i , and define the skew symmetric bi-moments µ i,j = h z i , z j i = −h z j , z i i = − µ j,i . Then we investigate the (skew-)orthogonality under the skew symmetric inner product.For monic polynomials { P n ( z ) } n ∈ N , if we consider the orthogonal conditions h P n ( z ) , z i i = 0 , ≤ i ≤ n − , then as that discussed in [9, Sec. 2], only the even family of polynomials are well-defined andthe odd ones are not. Therefore, how to set up the inner product condition to make the odd-order polynomials well defined is a key point at this stage. One suggestive way is to consider theconditions h P n ( z ) , z i i = 0 , ≤ i ≤ n, h P n +1 ( z ) , z i i = α n +1 ,i , ≤ i ≤ n + 1 where { α n +1 ,i } n +1 i =0 are n + 2 parameters satisfying det µ , · · · µ n, µ n +1 , − α n +1 , µ , · · · µ n, µ n +1 , − α n +1 , ... ... ... µ , n +1 · · · µ n, n +1 µ n +1 , n +1 − α n +1 , n +1 = 0 . (2.1)By differently choosing { α n +1 ,i } n +1 i =0 , we get different families of odd-order polynomials. SHI-HAO LI AND GUO-FU YU
Skew-orthogonal polynomials { P n ( z ) } n ∈ N . The choices α n +1 ,i = − τ n +2 τ n δ i, n , ≤ i ≤ n + 1 give rise to the concept of SOPs, where τ n = Pf (0 , · · · , n − and Pf ( i, j ) = µ i,j . Therefore, onecan get SOPs { P n ( z ) } n ∈ N by requiring the skew orthogonal relations h P n ( z ) , P m ( z ) i = h P n +1 ( z ) , P m +1 ( z ) i = 0 , h P m ( z ) , P n +1 ( z ) i = τ n +2 τ n δ m,n . (2.2)The condition (2.2) is indeed a consistent linear system for the coefficients of polynomials. Bysolving it and applying a Jacobi identity, one can find Pfaffian expressions for SOPs [1, 9] P n ( z ) = 1 τ n Pf (0 , · · · , n, z ) , P n +1 ( z ) = 1 τ n Pf (0 , · · · , n − , n + 1 , z ) with Pf ( i, z ) = z i .2.1.2. Partial-skew-orthogonal polynomials { Q n ( z ) } n ∈ N . Except the choice demonstrated above,there is another choice to introduce n + 2 quantities { β i } n +1 i =0 such that α n +1 ,i = − β i τ n +2 τ n +1 , τ n +1 = Pf ( d, , · · · , n ) with Pf ( d, i ) = β i and Pf ( i, j ) = µ i,j . Here the quantities { β i } n +1 i =0 are chosen so that τ n +1 = 0 .Verifications of the condition (2.1) is based on a Jacobi identity (see [9] for more details). In thiscase, the skew orthogonal relation can be formulated as follows h Q n ( z ) , z i i = τ n +2 τ n δ n +1 ,i , h Q n +1 ( z ) .z i i = − β i τ n +2 τ n +1 , ≤ i ≤ n + 1 . Moreover, these relations admit the following Pfaffian expressions Q n ( z ) = 1 τ n Pf (0 , · · · , n, z ) , Q n +1 ( z ) = 1 τ n +1 Pf ( d, , · · · , n + 1 , z ) with Pf ( d, z ) = 0 . It is remarkable that both even- and odd-order PSOPs are uniquely determined,although the odd ones can be arbitrarily chosen due to the freedom of { β j } n +1 j =0 . Therefore,by assuming that there are ℓ different sets { β ( k ) j } n +1 j =0 for k = 1 , · · · , ℓ such that for each k , τ n +1 ,k = Pf ( d k , , · · · , n + 1) = 0 with Pf ( d k , i ) = β ( k ) i , we can define ℓ -component PSOPs of oddorder satisfying the relations Q n +1 ,k ( z ) = 1 τ n +1 ,k Pf ( d k , , · · · , n + 1 , z ) , h Q n +1 ,k ( z ) , z i i = − β ( k ) i τ n +2 τ n +1 ,k , k = 1 , · · · , ℓ. Christoffel transformations of SOPs.
It is important to develop the Christoffel trans-formations of orthogonal polynomials since such transformations can act as the spectral problemand characterise the property of polynomials. Some previous results about the discrete spectraltransformations of SOPs are based on the evolution of the functional [24], but we emphasise on theadjacent families of SOPs and consider corresponding Christoffel transformations. It is remark-able that the adjacent SOPs here are the special µ = 0 case in [24, Thm. 2], but the Christoffeltransformation is only between two different families of polynomials and different from the knownresults. HRISTOFFEL TRANSFORMATIONS FOR SOPS AND PSOPS 5
Definition 2.1.
For m ∈ N , the m -th adjacent family of SOPs is defined by P ( m )2 n = 1 z m τ ( m )2 n Pf ( m, · · · , m + 2 n, z ) , P ( m )2 n +1 = 1 z m τ ( m )2 n Pf ( m, · · · , m + 2 n − , m + 2 n + 1 , z ) , where τ ( m )2 n = Pf ( m, · · · , m + 2 n − . One of the most significant features of the adjacent family of polynomials is that they inheritthe skew-orthogonality under the modified inner product h z m · , z m ·i . To be precise, we have D z m P ( m )2 n ( z ) , z m P ( m )2 l ( z ) E = D z m P ( m )2 n +1 ( z ) , z m P ( m )2 l +1 ( z ) E = 0 , D z m P ( m )2 l ( z ) , z m P ( m )2 n +1 ( z ) E = τ ( m )2 n +2 τ ( m )2 n δ l,m . The existence and uniqueness of the adjacent family of SOPs are equivalent to the condition τ ( m )2 n = 0 , and there are many physically interesting examples such as partition functions of or-thogonal/symplectic ensembles satisfying such a condition. Proposition 2.2.
The Christoffel transforms for SOPs has the form P ( m )2 n +1 ( z ) − A mn P ( m )2 n ( z ) = z (cid:16) P ( m +1)2 n ( z ) − B mn P ( m +1)2 n − ( z ) (cid:17) , (2.3a) P ( m )2 n +2 ( z ) − C mn P ( m )2 n ( z ) = z (cid:16) P ( m +1)2 n +1 ( z ) − D mn P ( m +1)2 n ( z ) (cid:17) , (2.3b) with coefficients A mn = P ( m )2 n +1 (0) P ( m )2 n (0) , B mn = τ ( m )2 n +2 τ ( m +1)2 n − τ ( m )2 n τ ( m +1)2 n , C mn = τ ( m )2 n τ ( m +1)2 n +2 τ ( m )2 n +2 τ ( m +1)2 n , D mn = P ( m − n +3 (0) P ( m − n +2 (0) . Proof.
Starting from the Pfaffian identityPf ( m, ∗ ) Pf ( ∗ , n + m, n + m + 1 , z ) = Pf ( ∗ , n + m ) Pf ( m, ∗ , n + m + 1 , z ) − Pf ( ∗ , n + m + 1) Pf ( m, ∗ , n + m, z ) + Pf ( ∗ , z ) Pf ( m, ∗ , n + m, n + m + 1) with {∗} = { m + 1 , · · · , n + m − } and recognising the fact that P ( m )2 n (0) = τ ( m +1)2 n τ ( m )2 n , P ( m )2 n +1 (0) = 1 τ ( m )2 n Pf ( m + 1 , · · · , n + m − , n + m + 1) (2.4)we get the identity (2.3a).The identity (2.3b) can be obtained from the Pfaffian identityPf ( m, ∗ , n + m + 1 , n + m + 2 , z ) Pf ( ∗ ) = Pf ( m, ∗ , n + m + 1) Pf ( ∗ , n + m + 2 , z ) − Pf ( m, ∗ , n + m + 2) Pf ( ∗ , n + m + 1 , z ) + Pf ( m, ∗ , z ) Pf ( ∗ , n + m + 1 , n + m + 2) with {∗} = { m + 1 , · · · , n + m } , and the term Pf ( m, · · · , n + m, n + m + 2) can be written interms of SOPs with the help of (2.4). (cid:3) SHI-HAO LI AND GUO-FU YU
Christoffel transformations of PSOPs.
Similar to the adjacent family of SOPs, we nowconsider the adjacent family of PSOPs. Here we focus on the multi-component case for odd-orderpolynomials since the one-component m = 1 case was implicitly given in [9, Sec. 3.1]. Definition 2.3.
The m -th adjacent family of PSOPs are defined by Q ( m )2 n ( z ) = 1 z m τ ( m )2 n Pf ( m, · · · , m + 2 n, z ) , Q ( m )2 n +1 ,k ( z ) = 1 z m τ ( m )2 n +1 ,k Pf ( d k , m, · · · , m + 2 n + 1 , z ) , where τ ( m )2 n = Pf ( m, · · · , m + 2 n − and τ ( m )2 n +1 ,k = Pf ( d k , m, · · · , m + 2 n ) . The skew inner product properties of the adjacent families are easily obtained as follows h z m Q ( m )2 n ( z ) , z m + i i = τ ( m )2 n +2 τ ( m )2 n δ n,i − , h z m Q ( m )2 n +1 ,k ( z ) , z m + i i = − β ( k ) m + i τ ( m )2 n +2 τ ( m )2 n +1 ,k , and the existence and uniqueness of the adjacent PSOPs are equivalent to the facts that τ ( m )2 n = 0 and τ ( m )2 n +1 ,k = 0 . Theorem 2.4.
The Christoffel transformation for one-component PSOPs is given by Q ( m ) n +1 ( z ) + ξ mn Q ( m ) n ( z ) = z (cid:16) Q ( m +1) n ( z ) + η mn Q ( m +1) n − ( z ) (cid:17) (2.5) with coefficients ξ mn = τ ( m ) n τ ( m +1) n +1 τ ( m ) n +1 τ ( m +1) n , η mn = τ ( m ) n +2 τ ( m +1) n − τ ( m ) n +1 τ ( m +1) n . Proof.
In fact, identity (2.5) is composed of two different situations. When n is even, we take theindex set {∗} = { m + 1 , · · · , n + m } and make use of the Pfaffian identityPf ( d, m, ∗ , n + m + 1 , z ) Pf ( ∗ ) = Pf ( d, m, ∗ ) Pf ( ∗ , n + m + 1 , z ) − Pf ( d, ∗ , n + m + 1) Pf ( m, ∗ , z ) + Pf ( d, ∗ , z ) Pf ( m, ∗ , n + m + 1) , then we get Q ( m )2 n +1 ( z ) + τ ( m )2 n τ ( m +1)2 n +1 τ ( m )2 n +1 τ ( m +1)2 n Q ( m )2 n ( z ) = z Q ( m +1)2 n ( z ) + τ ( m )2 n +2 τ ( m +1)2 n − τ ( m )2 n +1 τ ( m +1)2 n Q ( m +1)2 n − ( z ) ! . If n is odd, we need to shift the index {∗} = { m + 1 , · · · , m + 2 n + 1 } , and use the Pfaffian identityPf ( d, m, ∗ , m + 2 n + 2) Pf ( ∗ , z ) = Pf ( d, ∗ ) Pf ( m, ∗ , m + 2 n + 2 , z ) − Pf ( m, ∗ ) Pf ( d, ∗ , m + 2 n + 2 , z ) + Pf ( ∗ , m + 2 n + 2) Pf ( d, m, ∗ , z ) to get Q ( m )2 n +2 ( z ) + τ ( m )2 n +1 τ ( m +1)2 n +2 τ ( m )2 n +2 τ ( m +1)2 n +1 Q ( m )2 n +1 ( z ) = z Q ( m +1)2 n +1 ( z ) + τ ( m )2 n +3 τ ( m +1)2 n τ ( m )2 n +2 τ ( m +1)2 n +1 Q ( m +1)2 n ( z ) ! . Combining these results we obtain (2.5). (cid:3)
HRISTOFFEL TRANSFORMATIONS FOR SOPS AND PSOPS 7
The procedure stated above implies that the Christoffel transformation for PSOPs can be ex-tended to the multi-component case, i.e Q ( m )2 n +1 ,k ( z ) + E mn,k Q ( m )2 n ( z ) = z (cid:16) Q ( m +1)2 n ( z ) + F mn,k Q ( m +1)2 n − ,k ( z ) (cid:17) , (2.6a) Q ( m )2 n +2 ( z ) + G mn,k Q ( m )2 n +1 ,k ( z ) = z (cid:16) Q ( m +1)2 n +1 ,k ( z ) + H mn,k Q ( m +1)2 n ( z ) (cid:17) , (2.6b)where the coefficients E mn,k , F mn,k , G mn,k , H mn,k are given by E mn,k = τ ( m )2 n τ ( m +1)2 n +1 ,k τ ( m )2 n +1 ,k τ ( m +1)2 n , F mn,k = τ ( m )2 n +2 τ ( m +1)2 n − ,k τ ( m )2 n +1 ,k τ ( m +1)2 n , G mn,k = τ ( m )2 n +1 ,k τ ( m +1)2 n +2 τ ( m )2 n +2 τ ( m +1)2 n +1 ,k , H mn,k = τ ( m )2 n +3 ,k τ ( m +1)2 n τ ( m )2 n +2 τ ( m +1)2 n +1 ,k . Applications of SOPs’ Christoffel transformation
In this part, we introduce the commuting time flows, and make use of the SOPs’ Christoffeltransformation to obtain integrable lattices. The concept of commuting flows were proposed in [1]by considering the evolutions of moment matrices U := ( µ i,j ) i,j ∈ N such that ∂ t n U = Λ n U + U Λ ⊤ n ,where Λ is the shift operator whose off-diagonals are and the others are . Such evolutions holdvalid for each bi-moment, so we have ∂ t n µ i,j = µ i + n,j + µ i,j + n . (3.1)One of the most important property under the commuting flow is to find explicitly the derivativerelationship between P ( m )2 n ( z ) and P ( m )2 n +1 ( z ) . Proposition 3.1.
With time evolution (3.1) , it holds that ( z + ∂ t )( τ ( m )2 n P ( m )2 n ) = τ ( m )2 n P ( m )2 n +1 ( z ) . (3.2) Proof.
Noting that ( z + ∂ t )( τ ( m )2 n P ( m )2 n ) = ( z + ∂ t ) z − m Pf ( m, m + 1 , · · · , m + 2 n, z ) , and expanding the right hand side in terms of z , one can find z n +1 Pf ( m, · · · , m + 2 n − − n − X k =0 ( − z ) k +1 Pf ( m, · · · , \ m + k, · · · , m + 2 n )+ n X k =0 ( − z ) k h Pf ( m, · · · , \ m + k − , · · · , m + 2 n ) + Pf ( m, · · · , \ m + k, · · · , \ m + 2 n, m + 2 n + 1) i . Eliminating the last term in the first line and the first term in the last implies (3.2). (cid:3)
The proof of the case m = 0 case was given in [1, Lemma 3.6]. However, Proposition 3.1 showsthat this property holds for all adjacent families of SOPs. Moreover, the coefficients of SOPs canbe written in terms of Schur polynomials acting on the normalisation factor (i.e. tau function) [3,Sec. 3] P ( m )2 n ( z ) = 1 τ ( m )2 n n X k =0 z n − k s k ( − ˜ ∂ t ) τ ( m )2 n , (3.3)where { s k ( t ) } k ∈ N are the Schur polynomials given by exp ∞ X ℓ =1 t ℓ z ℓ ! = ∞ X k =0 s k ( t ) z k , SHI-HAO LI AND GUO-FU YU and ˜ ∂ t = ( ∂ t , ∂ t / , ∂ t / , · · · ) . Substituting expressions (3.2) and (3.3) into (2.3a)-(2.3b) andrecognising A mn = ∂ t log τ ( m +1)2 n , B mn = τ ( m )2 n +2 τ ( m +1)2 n − τ ( m )2 n τ ( m +1)2 n , C mn = τ ( m )2 n τ ( m +1)2 n +2 τ ( m )2 n +2 τ ( m +1)2 n , D mn = ∂ t log τ ( m )2 n +2 , by comparing with the coefficients of monomials, one can immediately get the following bilinearidentities τ ( m +1)2 n s n +1 − ℓ ( − ˜ ∂ t ) τ ( m )2 n + τ ( m +1)2 n ∂ t s n − ℓ ( − ˜ ∂ t ) τ ( m )2 n − ∂ t τ ( m +1)2 n s n − ℓ ( − ˜ ∂ t ) τ ( m )2 n = τ ( m )2 n s n +1 − ℓ ( − ˜ ∂ t ) τ ( m +1)2 n − τ ( m )2 n +2 s n − − ℓ ( − ˜ ∂ t ) τ ( m +1)2 n − . (3.4)The first nontrivial case is the case of ℓ = 2 n − . In this case, (3.4) has the form ( D t + D t ) τ ( m +1)2 n · τ ( m )2 n = 2 τ ( m )2 n +2 τ ( m +1)2 n − . Regarding the classification results of Kyoto School, this is exactly the KP equation of D ∞ type[16], and later recognised as Pfaff lattice hierarchy [1, 2, 3]. Remark 3.2.
A natural integrable discretisation of the t -flow has been considered in [24] µ ℓ +1 i,j = µ ℓi +1 ,j +1 + λµ ℓi,j +1 + λµ ℓi +1 ,j + λ µ ℓi,j , where ℓ is a discrete index. Combining the Christoffel transformation (2.3a) - (2.3b) and the discreteevolution gives rise to the fully discrete DKP equation. Geronimus transformation, Laurent type SOPs, and Toda lattice.
Despite the abovediscussed Christoffel transformation, Geronimus transformation is another important discrete trans-formation in the orthogonal polynomials theory. The aim of the Geronimus transformation is toexpress the adjacent family of polynomials in terms of the original ones, namely P ( m ) n ( z ) = n X i =0 α ( m ) i P ( m +1) i ( z ) . (3.5)Since these polynomials are monic. we naturally have α ( m ) n = 1 . Usually, the essential idea to findthe Geronimus transformation is to consider the relation h zP ( m ) n ( z ) , z i i by utilising orthogonal-ity. However, we can not get enough information to express the Geronimus transformation whenthe inner product is skew symmetric only. A possible case in which we can find the Geronimustransformation is the Laurent type SOPs proposed recently in [25]. It requires the moments µ i,j = µ i − ,j − or U = Λ U Λ ⊤ . (3.6)This condition is equivalent to the identity h z i , z j i = h z i − , z j − i , and thus one can prove thefollowing proposition. Proposition 3.3.
Under the assumption (3.6) , the following holds P ( m ) n ( z ) = P ( m +1) n ( z ) . (3.7) The operator D t is usually called as the Hirota’s bilinear operator, defined by D t f · g = ∂∂s f ( t + s ) g ( t − s ) | s =0 . HRISTOFFEL TRANSFORMATIONS FOR SOPS AND PSOPS 9
Proof.
We prove the identity P ( m )2 n ( z ) = P ( m +1)2 n ( z ) , while the odd-order case can be establishedsimilarly. With use of the assumption (3.5), we can explicitly express zP ( m )2 n ( z ) = n − X i =0 α ( m ) i z m Pf ( m + 1 , · · · , m + 2 i + 1 , z ) + β ( m ) i z m Pf ( m + 1 , · · · , m + 2 i, m + 2 i + 2 , z ) ! + 1 z m Pf ( m + 1 , · · · , m + 2 n + 1 , z ) . Taking the skew symmetric inner product on both sides with z j and note that h zP ( m )2 n ( z ) , z j i = h P ( m )2 n ( z ) , z j − i = 0 , if j − , · · · , n, we conclude that α ( m ) i = β ( m ) i = 0 for i = 0 , · · · , n − . (cid:3) Remark 3.4.
This result can also be obtained directly from (3.3) by substituting τ ( m )2 n = τ ( m +1)2 n .The fact that τ ( m )2 n = τ ( m +1)2 n could be verified by expanding the Pfaffians with use of (3.6) . The substitution of (3.7) into Christoffel transformations (2.3a)-(2.3b) yields the following iden-tities (see [25, Prop. 2]) P n +1 ( z ) − A n P n ( z ) = z ( P n ( z ) − B n P n − ( z )) , (3.8a) P n +2 ( z ) − P n ( z ) = z ( P n +1 ( z ) − A n +1 P n ( z )) (3.8b)where A n = ∂ t log τ n , B n = τ n − τ n +2 τ n . (3.9)Moreover, comparing the coefficients of these polynomials, one can get the reduction of (3.4) D t τ n · s n − ℓ ( − ˜ ∂ t ) τ n = τ n +2 s n − − ℓ ( − ˜ ∂ t ) τ n − . (3.10)The first nontrivial example of (3.10) is the following one D t τ n · τ n = 2 τ n − τ n +2 , which is indeed a Toda lattice. It is not surprising that the Toda lattice has a Pfaffian tau functionsince there is a one-to-one correspondence between the Toeplitz-type Pfaffian and the Hankeldeterminant [29, Prop. 2.3]Pf ( µ j − i ) ni,j =1 = det ( x i,j ) ni,j =1 , x i,j = µ | i − j | +1 + · · · + µ i + j +1 , and thus the Hankel determinant solution of the Toda lattice has a Pfaffian version when theevolution is properly chosen. In recent paper [25], the author showed that how to write down aHankel determinant in terms of Toeplitz-type Pfaffian.Moreover, it is of interest to obtain the Lax pair of Toda lattice in terms of SOPs. For thispurpose, we study the evolution of the eigenvectors under the t -flow. Besides identity (3.2), onecan establish the following proposition. Proposition 3.5.
The following identity holds τ n ( z + ∂ t ) (cid:16) τ n P n +1 ( z ) (cid:17) = P n +2 ( z ) + ( A n + A n +1 ) P n +1 ( z ) − D n P n ( z ) + B n P n − ( z ) , (3.11) where A n and B n are given in (4.27) and D n = s ( − ˜ ∂ t ) τ n +2 τ n +2 + s ( ˜ ∂ t ) τ n τ n , s ( t ) = t + 12 t . Proof.
The proof are based on three steps. The first one is to show ( z + ∂ t )( τ n P n +1 ( z )) = Pf (0 , · · · , n − , n + 2 , z ) + Pf (0 , · · · , n − , n, n + 1 , z ) . This step is an analogue of Proposition 3.1, and we omit the details here. Then, by using thePfaffian identityPf ( ∗ , n, n + 1 , n + 2 , z ) Pf ( ∗ ) = Pf ( ∗ , n, n + 1) Pf ( ∗ , n + 2 , z ) − Pf ( ∗ , n, n + 2) Pf ( ∗ , n + 1 , z ) + Pf ( ∗ , n, z ) Pf ( ∗ , n + 1 , n + 2) , with {∗} = { , · · · , n − } , one can obtain thatPf (0 , · · · , n − , n + 2 , z ) = τ n P n +2 ( z ) + A n +1 P n +1 ( z ) − s ( − ˜ ∂ t ) τ n +2 τ n +2 P n ( z ) ! . Moreover, the Pfaffian identityPf ( ∗ , n − Pf ( ∗ , n, n + 1 , z ) = Pf ( ∗ , n ) Pf ( ∗ , n − , n + 1 , z ) − Pf ( ∗ , n + 1) Pf ( ∗ , n − , n, z ) + Pf ( ∗ , z ) Pf ( ∗ , n − , n, n + 1) with {∗} = { , · · · , n − } leads toPf (0 , · · · , n − , n, n + 1 , z ) = τ n A n P n ( z ) − s ( ˜ ∂ t ) τ n τ n P n ( z ) + B n P n − ( z ) ! . Combining these results gives (3.11). (cid:3)
Therefore, with the help of (3.8a)-(3.8b), one can get the time evolutions for the Laurent SOPs ∂ t P n ( z ) − B n ∂ t P n − ( z ) = −B n P n − ( z ) + A n − B n P n − ( z ) ,∂ t P n +1 ( z ) − A n +1 ∂ t P n ( z ) = ( A n A n +1 − D n + 1) P n ( z ) + B n P n − ( z ) . The compatibility condition of the spectral problem and time evolutions give rise to the Todalattice ∂ t B n = B n ( C n − C n − ) , ∂ t C n = B n +1 − B n , where C n = A n +1 − A n . Remark 3.6.
It is not surprising that Toda lattice has a Pfaffian tau function with wave vectorSOPs. In the earlier work of Kodama and Pierce [18] , the authors showed that after some certainmoment constraints, SOPs { P n ( z ) } n ∈ N are connected with standard OPs { p n ( z ) } n ∈ N such that P n ( z ) = p n ( z ) , and the Pfaff lattice becomes Toda lattice. Applications of PSOPs’ Christoffel transformation
In this part, we derive integrable hierarchies with regards to the adjacent family of PSOPs.
HRISTOFFEL TRANSFORMATIONS FOR SOPS AND PSOPS 11
General case.
In the most general case, we consider the commuting flows such that ∂ t n µ i,j = µ i + n,j + µ i,j + n , ∂ t n β ( k ) j = β ( k ) j + n . (4.1)By [20, Prop. 4.4], the coefficients of the multi-component PSOPs can be expressed in terms ofSchur polynomials acting on the normalisation factors. Proposition 4.1.
With time evolutions (4.1) , the multi-component PSOPs have the form Q ( m )2 n ( z ) = 1 τ ( m )2 n n X ℓ =0 z n − ℓ s ℓ ( − ˜ ∂ t ) τ ( m )2 n , Q ( m )2 n +1 ,k ( z ) = 1 τ ( m )2 n +1 ,k n +1 X ℓ =0 z n +1 − ℓ s ℓ ( − ˜ ∂ t ) τ ( m )2 n +1 ,k . (4.2)Substituting the series sum into the Christoffel transformations (2.6a)-(2.6b), one can obtainthe following bilinear integrable hierarchy τ ( m +1)2 n s n +1 − ℓ ( − ˜ ∂ t ) τ ( m )2 n +1 ,k + τ ( m +1)2 n +1 ,k s n − ℓ ( − ˜ ∂ t ) τ ( m )2 n = τ ( m )2 n +1 ,k s n +1 − ℓ ( − ˜ ∂ t ) τ ( m +1)2 n + τ ( m )2 n +2 s n − ℓ ( − ˜ ∂ t ) τ ( m +1)2 n − ,k ,τ ( m +1)2 n +1 ,k s n +2 − ℓ ( − ˜ ∂ t ) τ ( m )2 n +2 + τ ( m +1)2 n +2 s n +1 − ℓ ( − ˜ ∂ t ) τ ( m )2 n +1 ,k = τ ( m )2 n +2 s n +2 − ℓ ( − ˜ ∂ t ) τ ( m +1)2 n +1 ,k + τ ( m )2 n +3 ,k s n +1 − ℓ ( − ˜ ∂ t ) τ ( m +1)2 n , (4.3)where the first nontrivial example is the case of ℓ = 2 n , ℓ = 2 n + 1 τ ( m )2 n +2 τ ( m +1)2 n − ,k = D t τ ( m +1)2 n · τ ( m )2 n +1 ,k + τ ( m +1)2 n +1 ,k τ ( m )2 n , (4.4a) τ ( m )2 n +3 ,k τ ( m +1)2 n = D t τ ( m +1)2 n +1 ,k · τ ( m )2 n +2 + τ ( m +1)2 n +2 τ ( m )2 n +1 ,k . (4.4b)This integrable hierarchy involves two different families of Pfaffian tau functions, namely, the evenand odd ones, and it was called the large BKP hierarchy in [17, 32]. The one-component case ofthis hierarchy leads to the so-called semi-discrete generalised Lotka-Volterra lattice τ ( m ) n +2 τ ( m +1) n − = D t τ ( m +1) n · τ ( m ) n +1 + τ ( m ) n τ ( m +1) n +1 . (4.5)Note that the one-component PSOPs has been considered in [9] with t -flow involved. Moreover,with the help of adjacent family of PSOPs, one can give a Lax-type pair for the integrable lattice.For simplicity, we consider the one-component case. However, the result can be easily extended tothe multi-component case. Proposition 4.2.
The time-dependent PSOPs have the following properties ( z + ∂ t ) Q ( m ) n ( z ) = Q ( m ) n +1 ( z ) + K mn Q ( m ) n ( z ) − J mn Q ( m ) n − ( z ) , (4.6) where K mn = ∂ t log τ ( m ) n +1 τ ( m ) n , J mn = τ ( m ) n +2 τ ( m ) n − τ ( m ) n τ ( m ) n +1 . (4.7) Proof.
The odd and even cases are almost the same, so we consider the odd case. First, as in theproof of Prop. 3.1, we have τ ( m )2 n +1 ( z + ∂ t )( τ ( m )2 n +1 Q ( m )2 n +1 ( z )) = z − m Pf ( d, m, · · · , m + 2 n, m + 2 n + 2 , z ) . (4.8) Choosing {∗} = { m, · · · , m + 2 n } and using Pfaffian identityPf ( d, ∗ ) Pf ( ∗ , m + 2 n + 1 , m + 2 n + 2 , z ) = Pf ( ∗ , m + 2 n + 1) Pf ( d, ∗ , m + 2 n + 2 , z ) − Pf ( ∗ , m + 2 n + 2) Pf ( d, ∗ , m + 2 n + 1 , z ) + Pf ( ∗ , z ) Pf ( d, ∗ , m + 2 n + 1 , m + 2 n + 2) , one finds that the right-hand side of equation (4.8) is indeed Q ( m )2 n +2 ( z ) + ∂ t log τ ( m )2 n +2 Q ( m )2 n +1 ( z ) − J m n +1 Q ( m )2 n ( z ) . Substituting it into (4.8) provides (4.6)-(4.7) for odd n . (cid:3) Therefore, if we denote Φ ( m ) = (cid:16) Q ( m )0 ( z ) , Q ( m )1 ( z ) , · · · (cid:17) ⊤ , then the Christoffel transformationof PSOPs (2.5) can be rewrite as follows z Φ ( m +1) = L ( m ) Φ ( m ) , L ( m ) = ( I + Λ η ( m ) ) − (Λ ⊤ + ξ ( m ) I ) (4.9)where Λ is the shift operator, η ( m ) = diag ( η m , η m , · · · ) , ξ ( m ) = diag ( ξ m , ξ m , · · · ) . The equation(4.6) can now be rewritten as ( z + ∂ t )Φ ( m ) = M ( m ) Φ ( m ) , M ( m ) = Λ ⊤ + K ( m ) I − Λ J ( m ) (4.10)with K ( m ) = diag ( K m , K m , · · · ) and J ( m ) = diag ( J m , J m , · · · ) . The compatibility condition of(4.9) and (4.10) gives us ∂ t L ( m ) = M ( m +1) L ( m ) − L ( m ) M ( m ) . Moreover, for the discrete t -flow, µ t +1 i,j = µ ti +1 ,j +1 + λµ ti,j +1 + λµ ti +1 ,j + λ µ ti,j , β t +1 j = β tj +1 + λβ tj , the corresponding discrete PSOPs, adjacent family of PSOPs, and integrable lattice were given in[9] as well. In the following, some reductional cases are emphasised.Inspired by the fact that the odd-order tau functions are independent on the even ones, namely,for each k ∈ { , · · · , ℓ } , { τ n +1 ,k } in (4.4a)-(4.4b) solve (4.5). Moreover, since the single moments { β kj } j ∈ N are independent with bi-moments { µ i,j } i,j ∈ N , we can expect that there are some relationsbetween these moments. Such relations are called as moment constraints . Using these constraintswe can impose reductions on PSOPs and corresponding integrable lattices. We start with theone-component case which can be easily extended to the multi-component case.4.2. Moment constraint I.
Consider the Laurent type PSOPs satisfying the relations µ i,j = µ i − ,j − , β j = β j − . Similar to the Laurent SOPs case, one can check that τ ( m )2 n = τ ( m +1)2 n and τ ( m )2 n +1 = τ ( m +1)2 n +1 . Also no-tice that { β j } j ∈ N are the same functions dependent on t , which means the multi-component case nolonger exists. Therefore, from the expressions (4.2), one knows that the Christoffel transformation(2.5) is just a three term recurrence relation Q n +1 ( z ) + Q n ( z ) = z ( Q n ( z ) + η n Q n − ( z )) , η n = τ n − τ n +2 τ n τ n +1 . Moreover, if we assume the evolution ∂ t n µ i,j = µ i + n,j + µ i,j + n , ∂ t n β j = β j , HRISTOFFEL TRANSFORMATIONS FOR SOPS AND PSOPS 13 holds, then (4.6) is independent on the index m . So one has ( z + ∂ t ) Q n ( z ) = Q n +1 ( z ) + ξ n Q n ( z ) − η n Q n − ( z ) , ξ n = ∂ t log τ n +1 τ n . From the above identities, one can derive the time evolutions for the Laurent type PSOPs. Thuswe have ∂ t Q n + η n ∂ t Q n − = ( ξ n + η n − Q n + η n ( ξ n − − Q n − ( z ) − η n − η n Q n − ( z ) . Since { Q n ( z ) } n ∈ N are monic PSOPs, ξ n + η n − is equal to . This fact is due to the reduction ofthe integrable hierarchy (4.3). Under the reduction, one has τ n s n +1 − ℓ ( − ˜ ∂ t ) τ n +1 + τ n +1 s n − ℓ ( − ˜ ∂ t ) τ n = τ n +1 s n +1 − ℓ ( − ˜ ∂ t ) τ n + τ n +2 + s n − ℓ ( − ˜ ∂ t ) τ n − . (4.11)The famous Lotka-Volterra lattice τ n − τ n +2 = ( D t + 1) τ n · τ n +1 is the first example of (4.11). Therefore, a reduction of the Lax pair (4.9) and (4.10) can beobtained. That is, if we denote Φ = ( Q ( z ) , Q ( z ) , · · · ) ⊤ , then z Φ = ( I + Λ η ) − ( I + Λ ⊤ )Φ , ∂ t Φ = ( η + Λ ⊤ ) − ( a + b Λ)Φ . where η = diag ( η , η , · · · ) , a = diag ( η ( ξ − , η ( ξ − , · · · ) and b = diag ( η η , η η , · · · ) .Note that the two families of tau functions τ n = Pf (0 , , · · · , n − , τ n +1 = Pf ( d, , · · · , n ) with Pf ( i, j ) = µ i,j and Pf ( d, i ) = β are two different solutions of the 1d-Toda hierarchy, andthus the integrable hierarchy (4.11) acts as the Bäcklund transformation of the 1d-Toda hierarchy(3.10), which is a classical result in soliton theory.4.3. Moment constraint II.
The case we consider here is a rank two shift condition µ i,j +1 + µ i +1 ,j = β i +1 β j − β i β j +1 or U Λ + Λ ⊤ U = Λ ⊤ ββ ⊤ − ββ ⊤ Λ , (4.12)where U = ( u i,j ) i,j ∈ N , Λ is the shift operator, and β = ( β , β , · · · ) ⊤ . One can check that underthis assumption, the evolutions ∂ t n β i = β i + n and ∂ t n µ i,j = µ i + n,j + µ i,j + n are consistent. Ifwe consider only one family of PSOPs, namely, for fixed m we consider { Q ( m ) n ( z ) } n ∈ Z , then it isrelated to the so-called B-Toda lattice. The details of the corresponding integrable lattice andLax pair was discussed in [9], and the corresponding integrable hierarchy was given in [20]. Forcompleteness, we give a brief review here.From the moment constraint (4.12), one can find that the derivative of t -flow has two differentexpressions. One expression is the commuting flow, and the other one is in terms of single moments.If two new labels d and d are introduced such that ∂ t Pf ( i, j ) = Pf ( i + 1 , j ) + Pf ( i, j + 1) = Pf ( d , d , i, j ) , ∂ t Pf ( d , i ) = Pf ( d , i + 1) = Pf ( d , i ) , then one gets ∂ t ( τ ( m )2 n Q ( m )2 n ( z )) = z − m τ ( m )2 n Pf ( d , d , m, · · · , m + 2 n, z ) ,∂ t ( τ ( m )2 n +1 Q ( m )2 n +1 ( z )) = z − m τ ( m )2 n +1 Pf ( d , m, · · · , m + 2 n + 1 , z ) . (4.13) Using the Pfaffian identitiesPf ( d , d , ∗ , m + 2 n, z ) Pf ( ∗ ) = Pf ( d , d , ∗ ) Pf ( ∗ , m + 2 n, z ) − Pf ( d , ∗ , m + 2 n ) Pf ( d , ∗ , z ) + Pf ( d , ∗ , z ) Pf ( d , ∗ , m + 2 n ) with {∗} = { m, · · · , m + 2 n − } andPf ( d , d , ∗ , m + 2 n + 1) Pf ( ∗ , z ) = Pf ( d , ∗ ) Pf ( d , ∗ , m + 2 n + 1 , z ) − Pf ( d , ∗ ) Pf ( d , ∗ , m + 2 n + 1 , z ) + Pf ( ∗ , m + 2 n + 1) Pf ( d , d , ∗ , z ) with {∗} = { m, · · · , m + 2 n } , one can find the following time evolutions ∂ t Q ( m ) n ( z ) + I mn ∂ t Q ( m ) n − ( z ) = I mn ( K mn + K mn − ) Q ( m ) n − ( z ) , I mn = τ ( m ) n +1 τ ( m ) n − ( τ ( m ) n ) , (4.14)where K mn is defined in (4.7). The compatibility condition of (4.6) and (4.14) gives us the B-Todalattice D t τ ( m ) n · τ ( m ) n = 2 D t τ ( m ) n +1 · τ ( m ) n − . (4.15)Since we focus on the adjacent PSOPs in this paper, in what follows, we show how to apply theChristoffel transformation to this kind of moment constraint.By using the derivative formula (4.13) and Pfaffian identitiesPf ( d , d , m, ∗ , z ) Pf ( ∗ ) = Pf ( d , d , ∗ ) Pf ( m, ∗ , z ) − Pf ( d , m, ∗ ) Pf ( d , ∗ , z ) + Pf ( d , ∗ , z ) Pf ( d , m, ∗ ) with {∗} = { m + 1 , · · · , m + 2 n } andPf ( d , d , m, ∗ ) Pf ( ∗ , z ) = Pf ( d , ∗ ) Pf ( d , m, ∗ , z ) − Pf ( d , ∗ ) Pf ( d , m, ∗ , z ) + Pf ( d , d , ∗ , z ) Pf ( m, ∗ ) with {∗} = { m + 1 , · · · , m + 2 n + 1 } , we find the following formula ∂ t Q ( m ) n ( z ) + ∂ t log τ ( m ) n τ ( m +1) n Q ( m ) n ( z )= z τ ( m ) n +1 τ ( m +1) n − τ ( m ) n τ ( m +1) n ∂ t Q ( m +1) n − ( z ) + D t τ ( m ) n +1 · τ ( m +1) n − τ ( m ) n τ ( m +1) n Q ( m +1) n − ( z ) ! , (4.16)which is of degree n on both sides, and therefore, it must be D t τ ( m ) n · τ ( m +1) n = D t τ ( m ) n +1 · τ ( m +1) n − . This integrable lattice is the Bäcklund transformation of the B-Toda lattice (4.15). We now proceedto the Lax-type equation. Noting that the equation (4.16) can be rewritten as Q ( m ) n ( z ) + A mn ∂ t Q ( m ) n ( z ) = z ( Q ( m +1) n − ( z ) + B mn Q ( m +1) n − ( z )) , where the coefficients A mn = ( ∂ t log τ ( m ) n τ ( m +1) n ) − , B mn = τ ( m ) n +1 τ ( m +1) n − D t τ ( m ) n +1 · τ ( m +1) n − . HRISTOFFEL TRANSFORMATIONS FOR SOPS AND PSOPS 15
Combining the Christoffel transformation (2.5) and the formulas above, one can find z∂ t Φ ( m +1) = L ( m )1 ∂ t Φ ( m ) + L ( m )2 Φ ( m ) , Φ ( m ) = (cid:16) Q ( m )0 ( z ) , Q ( m )1 ( z ) , · · · (cid:17) ⊤ , (4.17)where the coefficients matrices L ( m )1 = ( a + Λ a ) − ( a + a Λ ⊤ ) , L ( m )2 = ( a + Λ a ) − a with a = diag ( B m , B m , · · · ) , a = diag ( η m B m , η m B m , · · · ) , a = diag ( η m A m , η m A m , · · · ) , a = diag ( A m , A m , · · · ) and a = diag ( η m − ξ m , η m − ξ m , · · · ) . Moreover, with the notation b = diag ( I m , I m , · · · ) , b = diag ( I m ( K m + K m ) , · · · ) , the time evolution (4.14) can be rewritten as ∂ t Φ ( m ) = M ( m ) Φ ( m ) , M ( m ) = ( b + Λ ⊤ ) − b . (4.18)and by denoting b = diag ( ξ m , ξ m , · · · ) and b = diag ( η m , η m , · · · ) , so that the Christoffel trans-formation (2.5) has the form Φ ( m ) = zN ( m ) Φ ( m +1) , N ( m ) = ( b + Λ ⊤ ) − ( I + Λ b ) . (4.19)The compatibility conditions of (4.17), (4.18) and (4.19) give two alternative forms M ( m +1) = ( L ( m )1 M ( m ) + L ( m )2 ) N ( m ) , ∂ t N ( m ) = (cid:16) M ( m ) − N ( m ) ( L ( m )1 M ( m ) + L ( m )2 ) (cid:17) N ( m ) . Moment constraint III.
In this section, we consider the moment constraint µ i,j +1 − µ i +1 ,j = 2 β i β j or U Λ − Λ ⊤ U = 2 ββ ⊤ , (4.20)where U = ( u i,j ) i,j ∈ N , Λ is the shift operator, and β = ( β , β , · · · ) ⊤ . From this constraint, we canexpress the bi-moments { µ i,j } i,j ∈ N in terms of single moments { β i } i ∈ N as follows µ i,i +2 k +1 = 2 k − X s =0 β i + s β i +2 k − s + β i + k , µ i,i +2 k = 2 k − X s =0 β i + s β i +2 k − − s . This kind of constraint is amount to τ ( m )2 n τ ( m )2 n +2 = ( τ ( m )2 n +1 ) , (4.21)followed by the observationPf ( d, m, · · · , n + m ) Pf ( d, m, · · · , n + m ) = Pf ( m, · · · , n − m ) Pf ( m, · · · , n + 1 + m ) . Moreover, the time evolutions on { µ i,j } i,j ∈ N is dependent on { β i } i ∈ N , so we can state thefollowing proposition. Proposition 4.3.
If the time evolutions on { β i } i ∈ N satisfy ∂ t n β i = β i + n , then the evolutions of { µ i,j } i,j ∈ N satisfy ∂ t n µ i,j = µ i,j + n + µ i + n,j . Proof.
Here we prove the time evolution on µ i,i +2 k , while for µ i,i +2 k +1 the statement of timeevolution could be verified similarly. For odd-order time flow t n +1 , we have ∂ t n +1 µ i,i +2 k = 2 k − X s =0 ( β i +2 n +1+ s β i +2 k − − s + β i + s β i +2 k +2 n − s ) and k − X s =0 β i + s β i +2( k + n ) − s = n + k − X s =0 β i + s β i +2( k + n ) − s − n − X s =0 β i + k + s β i +2 n + k − s , k − X s =0 β i +2 n +1+ s β i +2 k − − s = k − X s = n +1 β i + n + s β i +2 k + n − s + n X s =0 β i + k + s β i +2 n + k − s , So one can immediately get the result. For the even-order flow t n , we have ∂ t n µ i,i +2 k = 2 k − X s =0 ( β i +2 n + s β i +2 k − s − + β i + s β i +2 k +2 n − − s ) . Since the following identities hold k − X s =0 β i +2 n + s β i +2 k − s − = k − X s = n β i + n + s β i +2 k + n − s − + n − X s =0 β i + m + s β i +2 n + m − s − , k − X s =0 β i + s β i +2 k +2 n − s − = k + n − X s =0 β i + s β i +2 n +2 k − s − − n − X s =0 β i + k + s β i +2 n + k − s − , the derivatives for even flows could be verified. (cid:3) Therefore, the evolutions on the moments satisfy the equation (4.1), and we can get the one-component integrable system τ ( m )2 n +2 τ ( m +1)2 n − = D t τ ( m +1)2 n · τ ( m )2 n +1 + τ ( m )2 n τ ( m +1)2 n +1 ,τ ( m )2 n +1 τ ( m +1)2 n − = D t τ ( m +1)2 n − · τ ( m )2 n + τ ( m )2 n − τ ( m +1)2 n ,τ ( m )2 n τ ( m )2 n +2 = ( τ ( m )2 n +1 ) , τ ( m +1)2 n τ ( m +1)2 n +2 = ( τ ( m +1)2 n +1 ) . If we rewrite τ ( m )2 n = f n , τ ( m +1)2 n = f n +1 , τ ( m )2 n +1 = g n +1 , τ ( m +1)2 n +1 = g n +2 , then the equation above can be written in a unified form D t g n · f n − g n +1 f n − + g n − f n +1 = 0 , f n +1 f n − = g n , which is the so-called modified KdV equation [15]. Proposition 4.4.
The polynomials { Q ( m )2 n ( z ) } n ∈ N satisfy the following three term recurrence re-lation zQ ( m )2 n ( z ) = Q ( m )2 n +1 ( z ) + K m n Q ( m )2 n ( z ) + J m n Q ( m )2 n − ( z ) , (4.22) where K m n and J m n are given in (4.7) .Proof. The proof is based on the expansion of the polynomial Q ( m )2 n +1 , that is, the expansion ofPf ( d, m, · · · , m + 2 n + 1 , z ) Pf ( d, m, · · · , m + 2 n ) . Similar to (4.21), one can get Q ( m )2 n +1 ( z ) = 12 zQ ( m )2 n ( z ) + 12 1 τ ( m )2 n z m Pf ( m, · · · , m + 2 n − , m + 2 n + 1 , z ) − I m n +2 Q ( m )2 n ( z ) . HRISTOFFEL TRANSFORMATIONS FOR SOPS AND PSOPS 17
Now we need to deal with the mid term on the right hand side. By using the Pfaffian identityPf ( d, ∗ ,m + 2 n, m + 2 n + 1 , z ) Pf ( ∗ ) = Pf ( d, ∗ , m + 2 n ) Pf ( ∗ , m + 2 n + 1 , z ) − Pf ( d, ∗ , m + 2 n + 1) Pf ( ∗ , m + 2 n, z ) + Pf ( d, ∗ , z ) Pf ( ∗ , m + 2 n, m + 2 n + 1) with {∗} = { m, · · · , m + 2 n − } and equation (4.21), one could find that it is equal to Q ( m )2 n +1 ( z ) + I m n +1 Q ( m )2 n ( z ) − J mn Q ( m )2 n − ( z ) . Noting K m n = K m n +1 and combining these gives the result. (cid:3) Moreover, the even-order polynomials satisfy the following time evolutions.
Corollary 4.5.
The following time evolutions for the specific { Q ( m )2 n ( z ) } n ∈ N hold ∂ t Q ( m )2 n ( z ) = − J m n Q ( m )2 n − ( z ) . (4.23) Proof.
As was shown in Proposition 3.1, the following identity is true, ( z + ∂ t )( τ ( m )2 n Q ( m )2 n ( z )) = Pf ( m, · · · , m + 2 n − , m + 2 n + 1 , z ) . According to the above Pfaffian identity, one has that the right-hand side is τ ( m )2 n (cid:16) Q ( m )2 n +1 ( z ) + ∂ t τ ( m )2 n +1 Q ( m )2 n ( z ) − J m n Q ( m )2 n − ( z ) (cid:17) . By dividing τ ( m )2 n on both sides and using (4.21), we get (4.23). (cid:3) The spectral problem for the odd-order polynomials is more difficult. The observation h z m +1 Q ( m )2 n +1 ( z ) , z m + j i = Pf ( d, m, · · · , m + 2 n + 1 , m + j + 1) τ ( m )2 n +1 − Pf ( d, m + j + 1) τ ( m )2 n +2 τ ( m )2 n +1 , (4.24)implies the following proposition. Proposition 4.6.
The following spectral problem for the specific odd order PSOPs holds z (cid:16) Q ( m )2 n +1 ( z ) − J m n Q ( m )2 n − ( z ) (cid:17) = Q ( m )2 n +2 + K m n Q ( m )2 n +1 ( z ) + (cid:16) α mn + K m n ∂ t log τ ( m )2 n +1 (cid:17) Q ( m )2 n ( z ) − K m n J m n Q ( m )2 n − ( z ) − J m n − J m n Q ( m )2 n − ( z ) , (4.25) where α mn is given in (4.27) and J mn and K mn are given in (4.7) .Proof. From (4.24) we obtain D z m +1 (cid:16) Q ( m )2 n +1 ( z ) − J m n Q ( m )2 n − ( z ) (cid:17) , z m + j E = Pf ( d, m, · · · , m + 2 n + 1 , m + j + 1) τ ( m )2 n +1 − τ ( m )2 n +2 Pf ( d, m, · · · , m + 2 n − , m + j + 1) τ ( m )2 n τ ( m )2 n +1 , which is equal to when j = 0 , · · · , n − . At this step, we need to consider a set of basis for theSOPs and then transform the set into the basis for the PSOPs. Expanding the left hand of (4.25)in terms of SOPs, we get z (cid:16) Q ( m )2 n +1 ( z ) − J m n Q ( m )2 n − ( z ) (cid:17) = Q ( m )2 n +2 ( z )+ z − m n X i =0 α mi τ ( m )2 i Pf ( m, · · · , m + 2 i, z ) + n X i =0 β mi τ ( m )2 i Pf ( m, · · · , m + 2 i − , m + 2 i + 1 , z ) ! . (4.26) Taking the skew inner product with h z m · , z m + j i for j = 0 , · · · , n − , we obtain that α mi = 0 for i = 0 , · · · , n − and β mj = 0 whenever j = 0 , · · · , n − . For j = 2 n − , j = 2 n and j = 2 n + 1 ,we have α mn − = −J m n − J m n , β mn = K m n , α mn = J ( m )2 n +1 − s ( − ˜ ∂ t ) τ ( m )2 n +1 τ ( m )2 n +1 + s ( − ˜ ∂ t ) τ ( m )2 n τ ( m )2 n , (4.27)where s ( − ˜ ∂ t ) is the Schur function mentioned before. From Prop. 3.1, we get z (cid:16) Q ( m )2 n +1 ( z ) − J m n Q ( m )2 n − ( z ) (cid:17) = Q ( m )2 n +2 ( z ) + α mn Q ( m )2 n ( z ) + α ( m ) n − Q ( m )2 n − ( z ) + β mn τ ( m )2 n ( z + ∂ t )( τ ( m )2 n Q ( m )2 n ( z )) , Substituting zQ ( m )2 n ( z ) and ∂ t Q ( m )2 n ( z ) into this identity, we obtain (4.25). (cid:3) Analogously to the even-order polynomials, the odd-order polynomials also have the time evo-lutions.
Corollary 4.7.
The specific odd order polynomials { Q ( m )2 n +1 ( z ) } n ∈ N admit the following time evo-lutions ∂ t Q ( m )2 n +1 ( z ) − J m n ∂ t Q ( m )2 n − ( z ) = (cid:16) J m n +1 + J m n − α mn − K mn ∂ t log τ ( m )2 n +1 (cid:17) Q ( m )2 n ( z )+ J m n (cid:0) K m n − K m n − (cid:1) Q ( m )2 n − ( z ) − J m n − J m n Q ( m )2 n − ( z ) . (4.28) Proof.
This proof using the following identity τ ( m )2 n +1 ( z + ∂ t )( τ ( m )2 n +1 Q ( m )2 n +1 ) = z − m Pf ( d, m, · · · , m + 2 n, m + 2 n + 2 , z ) . From the Pfaffian identityPf ( d, ∗ ,m + 2 n + 1 , m + 2 n + 2) Pf ( ∗ , z ) = Pf ( d, ∗ ) Pf ( ∗ , m + 2 n + 1 , m + 2 n + 2 , z ) − Pf ( ∗ , m + 2 n + 1) Pf ( d, ∗ , m + 2 n + 2 , z ) + Pf ( ∗ , m + 2 n + 2) Pf ( d, ∗ , m + 2 n + 1 , z ) , with {∗} = { m, · · · , m + 2 n } , we obtain ( z + ∂ t ) Q ( m )2 n +1 ( z ) = Q ( m )2 n +2 ( z ) + K ( m )2 n +1 Q ( m )2 n +1 ( z ) − J m n +1 Q ( m )2 n ( z ) . Now Proposition 4.6 implies (4.28). (cid:3)
Thus, denoting
Φ = (cid:16) Q ( m )0 ( z ) , Q ( m )1 ( z ) , · · · (cid:17) ⊤ , we get the Lax pair z Φ = L Φ , ∂ t Φ = M Φ , where L and M are constructed with (4.22), (4.23), (4.25) and (4.28). Another possible way tofind the Lax pair is due to the idea in [10]; one can make the use of the SOPs as the eigenfunctionsand regard the PSOPs as the auxiliary polynomials to obtain the Lax pair. Remark 4.8.
The essence of this reduction is a dimension integrable lattice, and thereforethe eigenfunction of this Lax pair involves { Q ( m ) n ( z ) } n ∈ N only, which requires a higher-order timeflow t to take place of τ ( m +1) n . HRISTOFFEL TRANSFORMATIONS FOR SOPS AND PSOPS 19
Multi-component case.
From [15] it follows that the above reduction has a multi-componentextension. If we extend the constraint (4.20) to µ i,j +1 − µ i +1 ,j = 2 N X a,b =1 β ai β bj , then we can express the bi-moments { µ i,j } i,j ∈ N in terms of single moments { β aj } j ∈ N for a =1 , · · · , N , and µ i,i +2 k +1 = N X a,b =1 k − X s =0 β ai + s β bi +2 k − s + β ai + k β bi + k ! , µ i,i +2 k = N X a,b =1 k − X s =0 β ai + k β bi +2 k − − s ! . Similar to Proposition 4.3, one can show that if ∂ t n β aj = β aj + n for all a = 1 , · · · , N , then µ i,j satisfies ∂ t n µ i,j = µ i + n,j + µ i,j + n . Moreover, the even and odd tau functions are connected witheach other by the formula τ ( m )2 n τ ( m )2 n +2 = N X a,b =1 τ ( m )2 n +1 ,a τ ( m )2 n +1 ,b . Thus we get the coupled modified KdV (cmKdV) hierarchy τ ( m +1)2 n s n +1 − ℓ ( − ˜ ∂ t ) τ ( m )2 n +1 ,k + τ ( m +1)2 n +1 ,k s n − ℓ ( − ˜ ∂ t ) τ ( m )2 n = τ ( m )2 n +1 ,k s n +1 − ℓ ( − ˜ ∂ t ) τ ( m +1)2 n + τ ( m )2 n +2 s n − ℓ ( − ˜ ∂ t ) τ ( m +1)2 n − ,k ,τ ( m +1)2 n +1 ,k s n +2 − ℓ ( − ˜ ∂ t ) τ ( m )2 n +2 + τ ( m +1)2 n +2 s n +1 − ℓ ( − ˜ ∂ t ) τ ( m )2 n +1 ,k = τ ( m )2 n +2 s n +2 − ℓ ( − ˜ ∂ t ) τ ( m +1)2 n +1 ,k + τ ( m )2 n +3 ,k s n +1 − ℓ ( − ˜ ∂ t ) τ ( m +1)2 n ,τ ( m )2 n τ ( m )2 n +2 = N X a,b =1 τ ( m )2 n +1 ,a τ ( m )2 n +1 ,b , τ ( m +1)2 n τ ( m +1)2 n +2 = N X a,b =1 τ ( m +1)2 n +1 ,a τ ( m +1)2 n +1 ,b . The first example of this hierarchy is the cmKdV equation.4.4.2.
Complex multi-component case.
The multi-component case admits a complex version as well.If the single moments { β aj } j ∈ N ,a ∈{ , ··· ,N } are complex and satisfy the time evolutions ∂ t n β aj = β aj + n , ∂ t n ¯ β aj = ¯ β aj + n then the bi-moments { µ i,j } i,j ∈ N can be expressed in terms of single moments as follows. µ i,j +1 − µ i +1 ,j = 2 N X a,b =1 β ai ¯ β bj , and satisfy the identity ∂ t n µ i,j = µ i + n,j + µ i,j + n . Thus, we have τ ( m )2 n τ ( m )2 n +2 = N X a,b =1 τ ( m )2 n +1 ,a ¯ τ ( m )2 n +1 ,b . Such kind of constraint may lead us to the discrete vector NLS hierarchy [22] τ ( m +1)2 n s n +1 − ℓ ( − ˜ ∂ t )˜ τ ( m )2 n +1 ,k + ˜ τ ( m +1)2 n +1 ,k s n − ℓ ( − ˜ ∂ t ) τ ( m )2 n = ˜ τ ( m )2 n +1 ,k s n +1 − ℓ ( − ˜ ∂ t ) τ ( m +1)2 n + τ ( m )2 n +2 s n − ℓ ( − ˜ ∂ t )˜ τ ( m +1)2 n − ,k , ˜ τ ( m +1)2 n +1 ,k s n +2 − ℓ ( − ˜ ∂ t ) τ ( m )2 n +2 + τ ( m +1)2 n +2 s n +1 − ℓ ( − ˜ ∂ t )˜ τ ( m )2 n +1 ,k = τ ( m )2 n +2 s n +2 − ℓ ( − ˜ ∂ t )˜ τ ( m +1)2 n +1 ,k + ˜ τ ( m )2 n +3 ,k s n +1 − ℓ ( − ˜ ∂ t ) τ ( m +1)2 n ,τ ( m )2 n τ ( m )2 n +2 = N X a,b =1 τ ( m )2 n +1 ,a ¯ τ ( m )2 n +1 ,b , τ ( m +1)2 n τ ( m +1)2 n +2 = N X a,b =1 τ ( m +1)2 n +1 ,a ¯ τ ( m +1)2 n +1 ,b , where ˜ τ means that τ and ¯ τ both satisfy those equations.5. Concluding remarks
In this article, we consider the Christoffel transformation for SOPs and PSOPs along with theirapplications in integrable systems. The eigenfunctions of the Lax pair are given in terms of SOPsor PSOPs and integrable hierarchies are expressed in terms of the coefficients of polynomials. Theadvantage of SOPs lies in the fact that the basis is skew orthogonal and therefore, it’s better forus to choose the SOPs as the basis to expand some polynomials, see for example, equation (4.26).On the other hand, the advantage of PSOPs is that we can naturally introduce the odd-order taufunctions that may shed lights into novel integrable hierarchies as well as iterative algorithms, forexample, in the design of Grave-Morris’ vector Padé approximation. Thus, both of the polynomialshave their own strengths and should be properly chosen while using.
Acknowledgement
The authors would like to thank Dr. Hiroshi Miki and Prof. Xing-Biao Hu for helpful discussionsand comments. G. Yu is supported by National Natural Science Foundation of China (Grant no.11871336).
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