On the Zeros of Jacobi Polynomials P ( | alpha n , | beta n ) n (X)

Abstract

If r/sub n/ and s/sub n/ denote, respectively, the smallest and largest zeros of the Jacobi polynomial P/sub n//sup (..cap alpha../sub n/,..beta../sub n/)/, where ..cap alpha../sub n/ > 1, ..beta../sub n/ - 1, and if lim/sub n ..-->.. infinity/ ..cap alpha../sub n//(2n + ..cap alpha../sub n/ + ..beta../sub n/ + 1) = ..cap alpha.. and if lim/sub n ..-->.. infinity/ ..beta../sub n//(2n+..cap alpha../sub n/ + ..beta../sub n/ + 1) = b, then the numbers r/sub a,b/ and s/sub a,b/ are determined. Furthermore, the zeros of )P/sub n//sup (..cap alpha../sub n/,..beta../sub n/)(x))/sub n = 0//sup infinity/ are dense in (r/sub a,b/, s/sub a,b/).