In a generalization that is complementary to that of Segura in [12], it was proved in [7] that the zeros of [P.sup.[alpha], [beta].sub.n] interlace with the zeros of polynomials from some different Jacobi sequences, including those of [P.sup.[alpha]-t,[beta]+k.sub.n] and [P.sup.[alpha]-t,[beta]+k.sub.n-1] for 0 [less than or equal to] t, k [less than or equal to] 2, thereby confirming and extending a conjecture made by Richard Askey in [2].
(a) the zeros of [P.sup.[alpha]+t,[beta].sub.n-1] and [[[beta].sup.2] - [[alpha].sup.2] + t([beta] - [alpha] + 2n (n + [beta] + 1))]/[(2n + [alpha] + [beta] + t)(2n + [alpha] + [beta] + 2)] interlace with the zeros of [P.sup.[alpha],[beta].sub.n+1] for fixed t [member of] {0, 1, 2};
(b) the zeros of [P.sup.[alpha]+3,[beta].sub.n-1] and [n(n + [alpha] + [beta] + 2) + ([alpha] + 2)(n - [alpha] + [beta])]/[(n + [alpha] + 2)(2 + [alpha] + [beta] + 2)] interlace with the zeros of [P.sup.[alpha],[beta].sub.n+1];
(c) the zeros of [P.sup.[alpha]+4,[beta].sub.n-1] and [2n(n + [alpha] + [beta] + 3) + ([alpha] + 3)([beta] - [alpha])]/[2n(n + [alpha] + [beta] + 3)([alpha] + 3)([alpha] + [beta] + 2)] interlace with the zeros of [P.sup.[alpha],[beta].sub.n+1].
(ii) If [P.sup.[alpha]+t,[beta].sub.n-1] and [P.sup.[alpha],[beta].sub.n+1] are not co-prime, they have one common zero located at the respective points identified in (i) (a) to (c) and the n - 1 zeros of [P.sup.[alpha]+t,[beta].sub.n-1] interlace with the remaining n (non-common) zeros of [P.sup.[alpha],[beta].sub.n+1].
(a) The zeros of [P.sup.[alpha],[beta]+t.sub.n-1] and [[[beta].sup.2] - [[alpha].sup.2] - t([alpha] - [beta] + 2n (n + [alpha] + 1))[/[(2n + [alpha] + [beta] + t)(2n + [alpha] + [beta] + 2)] interlace with the zeros of [P.sup.[alpha],[beta].sub.n+1] for fixed t [member of] {1, 2};
(b) The zeros of [P.sup.[alpha],[beta]+3.sub.n-1] and - [n(n + [alpha] + [beta] + 2) + ([beta] + 2)(n - [beta] + [alpha])]/[(n + [beta] + 2)(n + [alpha] + [beta] + 2)] interlace with the zeros of [P.sup.[alpha],[beta].sub.n+1];
(c) The zeros of [P.sup.[alpha],[beta]+4.sub.n-1] and - [2n(n + [alpha] + [beta] + 3) + ([beta] + 3)([alpha] - [beta])]/[2n(n + [alpha] + [beta] + 3)([beta] + 3)([alpha] + [beta] + 2)] interlace with the zeros of [P.sup.[alpha],[beta].sub.n+1].
(ii) If [P.sup.[alpha],[beta]+t.sub.n-1] and [P.sup.[alpha],[beta].sub.n+1] are not co-prime, they have one common zero located at the respective points identified in (i) (a) to (c) and the n - 1 zeros of [P.sup.[alpha],[beta]+t.sub.n-1] interlace with the remaining n (non-common) zeros of [P.sup.[alpha],[beta].sub.n+1].
For t [member of] (0,2), if [P.sup.[alpha]+t,[beta].sub.n-1] and [P.sup.[alpha],[beta].sub.n+1] are co-prime, the zeros of [P.sup.[alpha]+t,[beta].sub.n-1] and [[[beta].sup.2] - [[alpha].sup.2] + t([beta] - [alpha] + 2n (n + [beta] + 1))]/[(2n + [alpha] + [beta] + t)(2n + [alpha] + [beta] + 2)] interlace with the zeros of [P.sup.[alpha],[beta].sub.sub.n-1].
(a) are co-prime, then the zeros of [P.sup.[alpha]+j,[beta]+k.sub.n-1] and [[beta] - [alpha] - n(k - j)]/[[alpha] + [beta] + 2 + n(4 - j - k)] interlace with the zeros of [P.sup.[alpha],[beta].sub.n+1];
(b) are not co-prime, they have one common zero located at the point identified in (i) (a) and the n - 1 zeros of [P.sup.[alpha]+j,[beta]+k.sub.n-1] interlace with the n remaining (non-common) zeros of [P.sup.[alpha],[beta].sub.n+1].
(a) are co-prime, then the zero of [P.sup.[alpha]+3,[beta]+1.sub.n-1] and [[n.sup.2] + n([alpha] + [beta] + 3) - ([alpha] + 2)([alpha] - [beta])]/[[n.sup.2] + n([alpha] + [beta] + 3) + ([alpha] + 2)([alpha] + [beta])] interlace with the zeros of [P.sup.[alpha],[beta].sub.n+1];
(b) are not co-prime, then they have one common zero located at the point identified in (ii) (a) and the n - 1 zeros of [P.sup.[alpha]+j,[beta]+k.sub.n-1] interlace with the n remaining (non-common) zeros of [P.sup.[alpha],[beta].sub.n+1].