Jacobi polynomials are solutions of the differential equation (z − 1)f ′′(z) + (az + b)f ′(z) + cf(z) = 0, (1) where a, b, c are constants satisfying a > b, a + b > 0 and c = n(1 − a − n) for some nonnegative integer n. It is a classical fact that the zeros of the Jacobi polynomials lie in the interval [−1, 1], and that their density in this interval is proportional to 1/ √ 1 − |z|2 in the limi...

We study the combinatorics of a continued fraction formula due to Wall. We also derive the orthogonality of little q-Jacobi polynomials from this formula, as Wall did for little q-Laguerre polynomials.

Classical Jacobi polynomials P (α,β) n , with α, β > −1, have a number of well-known properties, in particular the location of their zeros in the open interval (−1, 1). This property is no longer valid for other values of the parameters; in general, zeros are complex. In this paper we study the strong asymptotics of Jacobi polynomials where the real parameters αn, βn depend on n in such a way t...

We study the monotonicity with respect to a parameter of zeros of orthogonal polynomials. Our method uses the tridiagonal (Jacobi) matrices arising from the three-term recurrence relation for the polynomials. We obtain new results on monotonicity of zeros of associated Laguerre, Al-Salam-Carlitz, Meixner and PoJlaczek polynomials. We also derive inequalities for the zeros of the Al-Salam-Carlit...

We present results on co-recursive associated Laguerre and Jacobi polynomials which are of interest for the solution of the Chapman-Kolmogorov equations of some birth and death processes with or without absorption. Explicit forms, generating functions, and absolutely continuous part of the spectral measures are given. We derive fourth-order differential equations satisfied by the polynomials wi...