THE LAGUERRE-SOBOLEV-TYPE ORTHOGONAL POLYNOMIALS. HOLONOMIC EQUATION AND ELECTROSTATIC INTERPRETATION

In this paper we find the second order linear differential equation satisfied by orthogonal polynomials with respect to the inner product < p, q > = integral(infinity)(0) p(x)q(x)x(alpha)e(-x)dx+Np'(0)q'(0) where alpha > -1, N is an element of R+ and p, q are polynomials with real coefficients. We also find some numerical results concerning the distribution of their zeros and their electrostatic interpretation in terms of a logarithmic potential with an external field. We deduce the hypergeometric expression of these polynomials. Finally, the analysis of asymptotic behavior of such polynomials is presented.