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...

For the generalized Jacobi, Laguerre and Hermite polynomials P (αn,βn) n (x), L (αn) n (x), H (γn) n (x) the limit distributions of the zeros are found, when the sequences αn or βn tend to infinity with a larger order than n. The derivation uses special properties of the sequences in the corresponding recurrence formulae. The results are used to give second order approximations for the largest ...

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 This property is no longer valid for other values of the parameters; in general, zeros are complex.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...

In this paper Refinement of Generalized Jacobi (RGJ) method for solving systems of linear algebraic equations is proposed and its convergence is discussed. Few numerical examples are considered to show the efficiency of the Refinement of Generalized Jacobi method over generalized Jacobi method. Mathematics Subject Classification: 65F10, 65F50

Abstract. We investigate the simplest class of hyperdeterminants defined by Cayley in the case of Hankel hypermatrices (tensors of the form Ai1i2...ik = f(i1+i2+· · ·+ik)). It is found that many classical properties of Hankel determinants can be generalized, and a connection with Selberg type integrals is established. In particular, Selberg’s original formula amounts to the evaluation of all Ha...

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...