Generalized ChristoffelDarboux formula for classical skeworthogonal polynomials
Abstract
We show that skeworthogonal functions, defined with respect to Jacobi weight $w_{a,b}(x)={(1x)}^a{(1+x)}^b$, $a$, $b>1$, including the limiting cases of Laguerre ($w_{a}(x)=x^{a}e^{x}$, $a > 1$) and Gaussian weight ($w(x)=e^{x^2}$), satisfy threeterm recursion relation in the quaternion space. From this, we derive generalized ChristoffelDarboux (GCD) formulæ for kernel functions arising in the study of the corresponding orthogonal and symplectic ensembles of random $2N\times 2N$ matrices. Using the GCD formulæwe calculate the leveldensities and prove that in the bulk of the spectrum, under appropriate scaling, the eigenvalue correlations are universal. We also provide evidence to show that there exists a mapping between skeworthogonal functions arising in the study of orthogonal and symplectic ensembles of random matrices.
 Publication:

arXiv eprints
 Pub Date:
 November 2007
 arXiv:
 arXiv:0711.4432
 Bibcode:
 2007arXiv0711.4432S
 Keywords:

 Mathematical Physics
 EPrint:
 29 pages