Integral representations for multiple Hermite and multiple Laguerre polynomials
نویسنده
چکیده
converges. Random matrices with external source were introduced and studied by Brézin and Hikami [7, 8, 9, 10, 11], and P. Zinn-Justin [18, 19]. In what follows, we assume that A hasm distinct eigenvalues a1, . . . , am of multiplicities n1, . . . , nm. We consider m fixed and use multi-index notation ~n = (n1, . . . , nm) and |~n| = n1 + · · ·+ nm. The average characteristic polynomial P~n(x) = E [det(xI −M)] of the ensemble (1.1) is a monic polynomial of degree |~n| which satisfies for k = 1, . . . ,m,
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