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We derive upper bounds for the smallest zero and lower bounds for the largestzero of Laguerre, Jacobi and Gegenbauer polynomials. Our approach uses mixedthree term recurrence relations satisfied by polynomials corresponding todifferent parameter(s) within the same classical family. We prove thatinterlacing properties of the zeros impose restrictions on the possiblelocation of common zeros of the polynomials involved and deduce strict boundsfor the extreme zeros of polynomials belonging to each of these three classicalfamilies. We show numerically that the bounds generated by our method improveknown lower (upper) bounds for the largest (smallest) zeros of polynomials inthese families, notably in the case of Jacobi and Gegenbauer polynomials.

Bounds for extreme zeros of some classical orthogonal polynomials