Inequalities for Extreme Zeros of Some Classical Orthogonal and q-orthogonal Polynomials

Abstract

Abstract. Let {pn} ∞ n=0 be a sequence of orthogonal polynomials. We briefly review properties of pn that have been used to derive upper and lower bounds for the largest and smallest zero of pn. Bounds for the extreme zeros of Laguerre, Jacobi and Gegenbauer polynomials that have been obtained using different approaches are numerically compared and new bounds for extreme zeros of q-Laguerre and little q-Jacobi polynomials are proved.

The sum-of-squares hierarchy on the sphere and applications in quantum information theory

“…Here, the minimization is over polynomials q(t) of degree , and λ 2k is the 2k'th coefficient of φ(t) = (q(t)) 2 in its Gegenbauer expansion, see Equation (12). In words, ρ 2n (d, ) quantifies how close we can get the Gegenbauer coefficients of φ(t) = (q(t)) 2 to 1 (note however that the distance to 1 is measured by |λ −1 2k − 1| and not linearly).…”

Interlacing properties and bounds for zeros of $${}_2\phi _1$$ 2 ϕ 1 hypergeometric and little q-Jacobi polynomials

“…7.1.1]) and the connection of little q-Jacobi polynomials with 2 φ 1 hypergeometric polynomials. The results obtained are applied to obtain inequalities satisfied by the extreme zeros of little q-Jacobi polynomials which improve inequalities obtained in [10] and, in some cases, also those in [17].…”

“…This implies that the little q-Jacobi weight (2) is defined for general x and in particular for x ∈ (0, ∞). We analyse inequalities satisfied by the zeros of different sequences of 2 φ 1 hypergeometric polynomials and little q-Jacobi polynomials in order to obtain an inner bound for the extreme zeros that improves the upper bound for the smallest zero of little qJacobi polynomials in [10] and, for certain parameter values, the bound due to Gupta and Muldoon (cf. [17]).…”

New bounds for the extreme zeros of Jacobi polynomials

“…The extreme zeros of the classical orthogonal polynomials of Jacobi, Laguerre and Hermite have been a subject of intensive study. We refer to Szegő's monograph [2] for earlier results, and to [1,[3][4][5][6][7][8][9][10][11][12] for some recent developments.…”

“…Of course, having found the power sums p i (P ), 1 ≤ i ≤ 4, one could apply Proposition 6 for derivation of lower bounds for 1 − x n,n (α, β) as well. For instance, the first inequality in (8) with p 4 (P ) given by (11) and a = α + 1, b = β + 1, t = n(n + α + β + 1). However, the expression on the right-hand side looks rather complicated to be of any use.…”