Bilinear semiclassical moment functionals and their integral representation

Journal of Approximation Theory

Volumn 121, Issue 1, 2003, Pages 71-99

  Bertola, Marco ^a,b  

^a UNIVERSITÉ DE MONTRÉAL   (Canada)

^b Gina Cody School of Engineering and Computer Science   (Canada)

Author keywords

Biorthogonal polynomials; Moment functionals; Semiclassical functionals

Indexed keywords

EID: 0037356057     PISSN: 00219045     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0021-9045(02)00044-8     Document Type: Review

References (20)

1. M. Adler, P. Van Moerbeke, The spectrum of coupled random matrices, Ann. Math. 149 (1999) 921-976.
2. M. Bertola, B. Eynard, J. Harnad, Duality, biorthogonal polynomials and multi-matrix models, Comm. Math. Phys. 229 (2002) 73-120.
3. T.S. Chihara, An introduction to orthogonal polynomials, in: Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York, London, Paris, 1978.
4. P. Di Francesco, P. Ginsparg, J. Zinn-Justin, 2D gravity and random matrices, Phys. Rep. 254 (1995) 1.
5. N.M. Ercolani, K.T.-R. McLaughlin, Asymptotics and integrable structures for biorthogonal polynomials associated to a random two-matrix model, Advances in nonlinear mathematics and science, Phys. D 152/153 (2001) 232-268.
6. B. Eynard, M.L. Mehta, Matrices coupled in a chain: eigenvalue correlations, J. Phys. A: Math. Gen. 31 (1998) 4449 cond-mat/9710230.
7. M. Ismail, D. Masson, M. Rahman, Complex weight functions for classical orthogonal polynomials, Canad. J. Math. 43 (1991) 1294-1308.
8. E.N. Laguerre, Sur la réduction en fractions continues d'une fraction qui satisfait une équation différentielle linéaire du premier ordre dont les coefficients sont rationnels, J. Math. Pures Appl. 1 (1885) 135-165.
9. F. Marcellán, I.A. Rocha, On semiclassical linear functionals: Integral representations, J. Comput. Appl. Math. 57 (1995) 239-249.
10. F. Marcellán, I.A. Rocha, Complex path integral representation for semiclassical linear functionals, J. Appr. Theory 94 (1998) 107-127.
11. P. Maroni, Prolégomènes à l'étude des polynômes semiclassiques, Ann. Mat. Pura Appl. 149 (1987) 165-184.
12. M.L. Mehta, A method of integration over matrix variables, Comm. Math. Phys. 79 (1981) 327.
13. M.L. Mehta, Random Matrices, Academic Press Inc., Boston, MA, 1991.
14. M.L. Mehta, P. Shukla, Two coupled matrices: Eigenvalue correlations and spacing functions, J. Phys. A: Math. Gen. 27 (1994) 7793-7803.
15. K.S. Miller, H.S. Shapiro, On the linear independence of Laplace integral solutions of certain differential equations, Comm. Pure Appl. Math. 14 (1961) 125-135.
16. H.S. Shapiro, private communication.
17. J. Shohat, A differential equation for orthogonal polynomials, Duke Math. J. 5 (1939) 401-417.
18. G. Szegö, Orthogonal Polynomials, 1st Edition, American Mathematical Society, Colloquium Publications, Vol. 23, Amer. Math. Soc., Providence, RI, 1939.
19. K. Ueno, K. Takasaki, Toda lattice hierarchy, Adv. Stud. Pure Math. 4 (1984) 1-95.
20. J.L. Walsh, Interpolation and Approximation, 2nd Edition, American Mathematical Society Colloquium Publications, Vol. 20, Amer. Math. Soc., Providence, RI, 1956.