Multiple Skew-Orthogonal Polynomials and 2-Component Pfaff Lattice Hierarchy

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)

MATH  Google Scholar 

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)

Book  MATH  Google Scholar 

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)

Article  ADS  MathSciNet  MATH 

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