Adler, M., Horozov, E., van Moerbeke, P.: The Pfaff lattice and skew-orthogonal polynomials. Int. Math. Res. Not. 11, 569–588 (1999)
Article MathSciNet MATH Google Scholar
Adler, M., Shiota, T., van Moerbeke, P.: Pfaff \(\tau \)-functions. Math. Ann. 322, 423–476 (2002)
Article MathSciNet MATH Google Scholar
Adler, M., van Moerbeke, P.: Matrix integrals, Toda symmetries, Virasoro constraints and orthogonal polynomials. Duke Math J. 80, 863–911 (1995)
Article MathSciNet MATH Google Scholar
Adler, M., van Moerbeke, P.: The spectrum of coupled random matrices. Ann. Math. 149, 921–976 (1999)
Article MathSciNet MATH Google Scholar
Adler, M., van Moerbeke, P.: Toda versus Pfaff lattice and related polynomials. Duke Math J. 112, 1–58 (2002)
Article MathSciNet MATH Google Scholar
Adler, M., van Moerbeke, P., Vanhaecke, P.: Moment matrices and multi-component KP, with applications to random matrix theory. Comm. Math. Phys. 286, 1–38 (2009)
Article ADS MathSciNet MATH Google Scholar
Álvarez-Femández, C., Prieto, U., Mañas, M.: Multiple orthogonal polynomials of mixed type: Gauss-Borel factorization and the multi-component 2D Toda hierarchy. Adv. Math. 227, 1451–1525 (2011)
Article MathSciNet MATH Google Scholar
Álvarez-Fernández, C., Ariznabarreta, G., García-Ardila, J., Mañas, M., Marcellán, F.: Christoffel transformations for matrix orthogonal polynomials in the real line and the non-abelian 2D Toda lattice hierarchy. Int. Math. Res. Not. 2017, 1285–1341 (2017)
MathSciNet MATH Google Scholar
Aptekarev, A., Bleher, P., Kuijlaars, A.: Large n limit of Gaussian random matrices with external source, part II. Comm. Math. Phys. 259, 367–389 (2005)
Article ADS MathSciNet MATH Google Scholar
Aptekarev, A., Branquinho, A., Marcellán, F.: Toda-type differential equations for the recurrence coefficients of orthogonal polynomials and Freud transformation. J. Comp. Appl. Math. 78, 139–160 (1997)
Article MathSciNet MATH Google Scholar
Aptekarev, A., Derevyagin, M., Miki, H., van Assche, W.: Multidimensional Toda lattices: Continuous and discrete time. SIGMA, 12, 054, 30 pages (2016)
Bertola, M., Eynard, B., Harnad, J.: Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions. Comm. Math. Phys. 263, 401–437 (2006)
Article ADS MathSciNet MATH Google Scholar
Bertola, M., Gekhtman, M., Szmigieski, J.: The Cauchy two-matrix model. Comm. Math. Phys. 287, 983–1014 (2009)
Article ADS MathSciNet MATH Google Scholar
Bertola, M., Gekhtman, M., Szmigieski, J.: Cauchy biorthogonal polynomials. J. Approx. Theory 162, 832–867 (2010)
Article MathSciNet MATH Google Scholar
Bleher, P., Kuijlaars, A.: Random matrices with external source and multiple orthogonal polynomials. Int. Math. Res. Not. 3, 109–129 (2004)
Article MathSciNet MATH Google Scholar
Borodin, A.: Biorthogonal ensembles. Nucl. Phys. B 536, 704 (1998)
Article ADS MathSciNet MATH Google Scholar
Branquinho, A., Foulquié-Moreno, A., Mañas, M.: Multiple orthogonal polynomials and random walks. arXiv:2103.13715
Brézin, E., Hikami, S.: Level spacing of random matrices in an external source. Phys. Rev. E 58, 7176 (1998)
Article ADS MathSciNet Google Scholar
Carrozza, S., Tanasa, A.: Pfaffians and nonintersecting paths in graphs with cycles: Grassmann algebra methods. Adv. Appl. Math. 93, 108–120 (2018)
Article MathSciNet MATH Google Scholar
Chang, X., He, Y., Hu, X., Li, S.: Partial-skew-orthogonal polynomials and related integrable lattices with Pfaffian tau-functions. Comm. Math. Phys. 364, 1069–1119 (2018)
Article ADS MathSciNet MATH Google Scholar
Chihara, T.: An Introduction to Orthogonal Polynomials. Gordon and Breach, New York (1978)
Daems, E., Kuijlaars, A.: Multiple orthogonal polynomials of mixed type and non-intersecting Brownian motions. J. Approx. Theory 146, 91–114 (2007)
Article MathSciNet MATH Google Scholar
Deift, P.: Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes 3, American Mathematical Society, (2000)
Desrosiers, P., Forrester, P.: A note on biorthogonal ensembles. J. Approx. Theory 152, 167–187 (2008)
Article MathSciNet MATH Google Scholar
Eynard, B., Mehta, M.: Matrices coupled in a chain: I Eigenvalue correlations. J. Phys. A 31, 4449 (1998)
Article ADS MathSciNet MATH Google Scholar
Favard, J.: Sur les polyn\(}\)mes de Tchebicheff. C.R. Acad. Sci. Paris, 200, 2052-2053 (1935)
Fidalgo, P., López, L.: Nikishin systems are perfect. Constr. Approx. 34, 297–356 (2011)
Article MathSciNet MATH Google Scholar
Forrester, P.J.: Log-Gases and Random Matrices. Princeton University Press, Princeton, NJ (2010)
Freud, G.: On the coefficients in the recursion formulae of orthogonal polynomials. Proc. Roy. Irish Acad. Sect. A 76, 1–6 (1976)
MathSciNet MATH Google Scholar
Gilson, C., Nimmo, J.: The relation between a 2D Lotka-Volterra equation and a 2D Toda lattice. J. Nonlinear Math. Phys. 12(Supplement 2), 169–179 (2005)
Article ADS MathSciNet MATH Google Scholar
Hirota, R.: The direct method in soliton theory. (Edited and translated by A. Nagai, J. Nimmo and C. Gilson), Cambridge Tracts in Mathematics 155, Cambridge University Press, (2004)
Hu, X., Li, C., Nimmo, J., Yu, G.: An integrable symmetric \((2+1)\)-dimension Lotka-Volterra equation and a family of its solutions. J. Phys. A 38, 195–204 (2005)
Article ADS MathSciNet MATH Google Scholar
Hu, X., Li, S.: The partition function of the Bures ensemble as the \(\tau \)-function of BKP and DKP hierarchies: continuous and discrete. J. Phys. A 50, 285201 (2017)
Article MathSciNet MATH Google Scholar
Hu, X., Zhao, J.: Commutativity of Pfaffianization and Bäcklund transformation: the KP equation. Inverse Problem 21, 1461–1472 (2005)
Article ADS MATH Google Scholar
Ismail, M.: Classical and quantum orthogonal polynomials in one variable. Cambridge University Press, (2009)
Jimbo, M., Miwa, T.: Solitons and infinite dimensional Lie algebras. Publ. RIMS, Kyoto Univ., 19: 943-1001, (1983)
Kac, V., van de Leur, J.: The geometry of spinors and the multicomponent BKP and DKP hierarchies, in The bispectral problem (Montreal, PQ, 1997), 159–202, CRM Proc. Lecture Notes, 14, Amer. Math. Soc., Providence, RI, (1998)
Kakei, S.: Orthogonal and symplectic matrix integrals and coupled KP hierarchy. J. Phys. Soc. Jpn. 68, 2875–2879 (1999)
Article ADS MathSciNet MATH Google Scholar
Kodama, Y., Pierce, V.: The Pfaff lattice on symplectic matrices. J. Phys. A 43, 055206 (2010)
Article ADS MathSciNet MATH Google Scholar
Li, S.: Matrix orthogonal polynomials, non-abelian Toda lattice and Bäcklund transformation. https://doi.org/10.48550/arXiv.2109.00671
Li, C., Li, S.: The Cauchy two-matrix model, C-Toda lattice and CKP hierarchy. J. Nonlinear Sci. 29, 3–27 (2019)
Article ADS MathSciNet MATH Google Scholar
Li, Shi-Hao.: Yu, Guo-Fu: Integrable lattice hierarchies behind Cauchy two-matrix model and Bures ensemble. Nonlinearity 35, 5109–5149 (2022)
Article ADS MathSciNet MATH Google Scholar
Lundmark, H., Szmigieski, J.: Degasperis-Procesi peakons and the discrete cubic string. Int. Math. Res. Papers 2005, 53–116 (2005)
Article MathSciNet MATH Google Scholar
Martínez-Finkelshtein, A., van Assche, W.: What is ... a multiple orthogonal polynomials. Notices AMS, 63, 1029-1031 (2016)
Mehta, M.: Zeros of some bi-orthogonal polynomials. J. Phys. A 35, 517 (2002)
Article ADS MathSciNet MATH Google Scholar
Muttalib, K.: Random matrix models with additional interactions. J. Phys. A 28, L159 (1995)
Article ADS MathSciNet Google Scholar
Nikishin, E., Sorokin, V.: Rational approximations and orthogonality. Translations of Mathematical Monographs, Vol. 92, Amer. Math. Soc., Providence, RI, (1991)
Ohta, Y.: Special solutions of discrete integrable systems. Dis. Integrable Syst. Lecture Notes in Phys. 644, 57–83 (2004)
Comments (0)