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Two-Parameter Generalizations of Cauchy Bi-Orthogonal Polynomials and Integrable Lattices


In this article, we consider the generalised two-parameter Cauchy two-matrix model and the corresponding integrable lattice equation. It is shown that with parameters chosen as \(1/k_i\), \(k_i\in {\mathbb {Z}}_{>0}\) (\(i=1,\,2\)), the average characteristic polynomials admit \((k_1+k_2+2)\)-term recurrence relations, which can be interpreted as spectral problems for integrable lattices. The tau function is then given by the partition function of the generalised Cauchy two-matrix model as well as Gram determinant. The simplest solvable example is given.

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Fig. 1
Fig. 2


  1. One can follow the procedure exhibited in Wang et al. (2010) and take the Gram determinant into the first equation of CKP equation (i.e. Wang et al. 2010, Eq. (3)). It is not difficult to see the asymmetric Gram determinant doesn’t satisfy the bilinear equation of CKP equation.


  • Adler, M., van Moerbeke, P.: Group factorization, moment matrices, and Toda lattices. Int. Math Res. Not. 12, 555–572 (1997)

    MathSciNet  Article  Google Scholar 

  • Andréief, C.: Note sur une relation entre les intégrales définies desproduits des fonctions. Mém. Soc. Sci. Phys. Nat. Bordeaux 2, 1–14 (1886)

    MATH  Google Scholar 

  • Bertola, M., Gekhtman, M., Szmigieski, J.: The Cauchy two-matrix model. Commun. Math. Phys. 287, 983–1014 (2009)

    MathSciNet  Article  Google Scholar 

  • Bertola, M., Gekhtman, M., Szmigieski, J.: Cauchy biorthogonal polynomials. J. Approx. Theory 162, 832–867 (2010)

    MathSciNet  Article  Google Scholar 

  • Bertola, M., Gekhtman, M., Szmigielski, J.: Cauchy-Laguerre two-matrix model and the Meijer G-random point field. Commun. Math. Phys. 326, 111–144 (2014)

    MathSciNet  Article  Google Scholar 

  • Bertola, M., Harnad, J.: Rationally weighted Hurwitz numbers, Meijer G-functions and matrix integrals. J. Math. Phys. 60, 103504 (2019)

    MathSciNet  Article  Google Scholar 

  • Borodin, A.: Biorthogonal ensembles. Nucl. Phys. B 536, 704–732 (1999)

    MathSciNet  Article  Google Scholar 

  • Carlet, G.: The extended bigraded Toda hierarchy. J. Phys. A 39, 9411–9435 (2006)

    MathSciNet  Article  Google Scholar 

  • Chang, X., Hu, X., Li, S.: Degasperis-Procesi peakon dynamical system and finite Toda lattice of CKP type. Nonlinearity 31, 4746–4775 (2018)

    MathSciNet  Article  Google Scholar 

  • Chang, X., He, Y., Hu, X., Li, S.: Partial-skew-orthogonal polynomials and related integrable lattice with Pfaffian tau-functions. Commun. Math. Phys. 364, 1069–1119 (2018)

    MathSciNet  Article  Google Scholar 

  • Dubrovin, B., Zhang, Y.: Extended affine Weyl groups and Frobenius manifolds. Comp. Math. 111, 167–219 (1998)

    MathSciNet  Article  Google Scholar 

  • Forrester, P.: Log-Gases and Random Matrices. London Mathematical Society Monograph, vol. 34. Princeton University Press, New Jersey (2010)

    Google Scholar 

  • Forrester, P., Kieburg, M.: Relating the Bures measure to the Cauchy two-matrix model. Commun. Math. Phys. 342, 151–187 (2016)

    MathSciNet  Article  Google Scholar 

  • Forrester, P.: Meet Andréief, Bordeaux 1886, and Andreev, Kharkov 1882–83. Random Matrices Theory Appl. 08, 1930001 (2019)

    Article  Google Scholar 

  • Forrester, P., Li, S.: Fox H-kernel and \(\theta \)-deformation of the Cauchy two-matrix model and Bures ensemble. Int. Math Res. Not., rnz028, (2019)

  • Forrester, P., Li, S.: Rate of convergence at the hard edge for various Pólya ensembles of positive definite matrices. arXiv: 2008.01319

  • Gerasimov, A., Marshakov, A., Mironov, A., Morozov, A., Orlov, A.: Matrix models of two-dimensional gravity and Toda theory. Nucl. Phys. B 357, 565–618 (1991)

    MathSciNet  Article  Google Scholar 

  • Harnad, J., Yu Orlov, A.: Fermionic construction of partition function for two-matrix models and perturbative Schur function expansions. J. Phys. A 39(28), 8783–8809 (2006)

    MathSciNet  Article  Google Scholar 

  • Hirota, R.: (translated by Atsushi Nagai, Jon Nimmo and Claire Gilson). The direct method in soliton theory Cambridge Tracts in Mathematics, vol. 155. Cambridge University Press, Cambridge (2004)

  • Li, C., Li, S.: The Cauchy two-matrix model, C-Toda lattice and CKP hierarchy. J. Nonlinear Sci. 29, 3–27 (2019)

    MathSciNet  Article  Google Scholar 

  • Li, S., Yu, G.: Rank shift conditions and reductions of 2d-Toda theory. arXiv: 1908.08725

  • Lundmark, H., Szmigielski, J.: Multi-peakon solutions of the Degasperis-Procesi equation. Inverse Prob. 19, 1241–1245 (2003)

    MathSciNet  Article  Google Scholar 

  • Madhekar, H., Thakare, N.: Biorthogonal polynomials suggested by Jacobi polynomials. Pacific J. Math. 100, 417 (1982)

    MathSciNet  Article  Google Scholar 

  • Miki, H., Tsujimoto, S.: Cauchy biorthogonal polynomials and discrete integrable systems. J. Nonlinear Syst. Appl. 2, 195–199 (2001)

    Google Scholar 

  • Tsujimoto, S., Kondo, K.: The molecule solutions of discrete integrable systems and orthogonal polynomials (in Japanese). RIMS Kôkyûroku Bessatsu 1170, 1–8 (2000)

    MATH  Google Scholar 

  • Wang, H., Tam, H., Hu, X.: The 2+1 dimensional Kaup-Kuperschmidt equation with self-consistent sources and its exact solutions. In: AIP Conference Proceedings 1212, 273 (2010);

  • Zuo, D.: Frobenius manifolds and a new class of extended affine Weyl group of A-type. Lett. Math. Phys. 110, 1903–1940 (2020)

    MathSciNet  Article  Google Scholar 

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The authors would also like to thank Profs. Xingbiao Hu and Dafeng Zuo for helpful discussions, and referees for helpful comments which improve the article a lot. X. Chang is partially supported by National Natural Science Foundation of China (Grant Nos. 11688101, 11731014 and 11701550) and the Youth Innovation Promotion Association CAS. S. Li was supported by the ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS). S. Tsujimoto is partially supported by JSPS KAKENHI (Grant Nos. 19H01792, 17K18725) and G. Yu is partially supported by National Natural Science Foundation of China (Grant No. 11871336).

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Appendix A

Appendix A

This appendix is devoted to a statement about how the superpotential (1.3) relates to the extended affine group \({\tilde{W}}^{(k,k+1)}(A_\ell )\). To this end, firstly, we give some basic concepts about the extended affine Weyl group.

Let V be an \(\ell \)-dimensional Euclidean space with inner product \((\cdot ,\cdot )\) and \({\mathcal {R}}\) be an irreducible root system defined in V with simple roots \(\alpha _1,\cdots ,\alpha _\ell \) and coroots \(\alpha _1^\vee ,\cdots ,\alpha _\ell ^\vee \). Then, the Weyl group \(W({\mathcal {R}})\) can be generated by the reflection

$$\begin{aligned} x\mapsto x-(\alpha _j^\vee ,x)\alpha _j,\quad \forall x\in V,\quad j=1,\cdots ,\ell . \end{aligned}$$

The affine Weyl group \(W_a({\mathcal {R}})\) which acts on the Euclidean space V can be realised as the semi-product of \(W({\mathcal {R}})\) by the lattice of coroots via the map

$$\begin{aligned} {x}\mapsto w(x)+\sum _{j=1}^\ell m_j\alpha _j^\vee , \quad w\in W({\mathcal {R}}),\quad m_j\in {\mathbb {Z}}. \end{aligned}$$

Dubrovin and Zhang then proposed the extended affine Weyl group \({\tilde{W}}^{(k)}({\mathcal {R}})\) acting on \(V\oplus {\mathbb {R}}\) in the study of Frobenius manifold. It is generated by the transformations

$$\begin{aligned}&(x,x_{\ell +1})\mapsto (w(x)+\sum _{j=1}^\ell m_j\alpha _j^\vee , x_{\ell +1}),\quad w\in W({\mathcal {R}}), \quad m_j\in {\mathbb {Z}},\\&(x,x_{\ell +1})\mapsto (x+\gamma w_k, x_{\ell +1}-\gamma ),\quad 1\le k\le \ell \end{aligned}$$

with \(w_k\) being the fundamental weights defined by the relations \((w_k,\alpha _j^\vee )=\delta _{k,j}\) for \(k,j=1,\cdots ,\ell \) and \(\gamma \) is a constant related the root system. With \({\mathcal {R}}\) chosen as \(A_\ell \), a construction of Frobenius manifold structure on the orbit of \({\tilde{W}}^{(k)}(A_\ell )\) was given in Dubrovin and Zhang (1998). It was shown that \({\tilde{W}}^{(k)}(A_\ell )\) describes the monodromy of roots of trigonometric polynomials admits the form

$$\begin{aligned} \lambda (\phi )=e^{ik\phi }+\mathbf {a_1}e^{i(k-1)\phi }+\cdots +a_{\ell +1} e^{i(k-\ell -1)\phi },\quad a_{\ell +1}\ne 0. \end{aligned}$$

If one sets \(e^{i\phi }\) as the shift operator \(\Lambda \), then the above polynomials will become

$$\begin{aligned} L=\Lambda ^k+\mathbf {a_1}\Lambda ^{k-1}+\cdots +a_{\ell +1}\Lambda ^{k-\ell -1},\quad a_{\ell +1}\ne 0 \end{aligned}$$

which is regarded as the spectral operator of the bigraded Toda hierarchy (c.f. Eq. (1.2)).

In the recent work (Zuo 2020), the author studied another extended affine Weyl group \({\tilde{W}}^{(k,k+1)}(A_\ell )\) acting on the space \(V\oplus {\mathbb {R}}^2\), generated by the transformation

$$\begin{aligned}&(x,x_{\ell +1},x_{\ell +2})\mapsto (w(x)+\sum _{j=1}^\ell m_j\alpha _j^\vee ,x_{\ell +1},x_{\ell +2}),\quad w\in W(A_\ell ),\quad m_j\in {\mathbb {Z}},\\&(x,x_{\ell +1},x_{\ell +2})\mapsto (x+\gamma w_k,x_{\ell +1}-\gamma ,x_{\ell +2}),\quad (x,x_{\ell +1},x_{\ell +2})\\&\quad \mapsto (x+\gamma w_{k+1},x_{\ell +1},x_{\ell +2}-\gamma ) \end{aligned}$$

for \(1\le k\le \ell -1\). It was proven by Zuo (2020) that the orbit space of the extended affine Weyl group is locally isomorphic to a simple Hurwitz space \({\mathbb {M}}_{k,\ell -k+1,1}\) which contains a natural structure of Frobenius manifold. Moreover, this space consists of trigonometric Laurent series admitting the form

$$\begin{aligned} \lambda (\phi )=(e^{i\phi }-a_{\ell +2})^{-1}(e^{i(k+1)\phi }+\mathbf {a_1}e^{ik\phi } +\cdots +a_{\ell +1} e^{i(k-\ell )}\phi ),\quad a_{\ell +1}a_{\ell +2}\ne 0. \end{aligned}$$

Similarly, by setting \(e^{i\phi }=\Lambda \), one gets the spectral operator (1.3)

$$\begin{aligned} L\!=\!(\Lambda \!-\!a_{\ell +2})^{-1}(\Lambda ^{k+1}\!+\!\mathbf {a_1}\Lambda ^{k}\!+\cdots +a_{\ell +1}\Lambda ^{k-\ell }), \quad 1\!\le \! k\!\le \! \ell -1,\, a_{\ell +1}a_{\ell +2}\ne 0. \end{aligned}$$

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Chang, XK., Li, SH., Tsujimoto, S. et al. Two-Parameter Generalizations of Cauchy Bi-Orthogonal Polynomials and Integrable Lattices. J Nonlinear Sci 31, 30 (2021).

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  • Two-parameter Cauchy two-matrix model
  • Toda-type lattice
  • Gram determinant technique

Mathematics Subject Classification

  • 37K10
  • 37K20
  • 15A15