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

Mixed recurrence equations and interlacing properties for zeros of sequences of classical q -orthogonal polynomials

“…Corollary 2 (cf. [2,7,12,13]). Let (c, d) be a finite or infinite interval and assume that p n and q n are monic polynomials (not necessarily orthogonal) of degree n, with zeros c <…”

“…Interlacing results for the zeros of different sequences of q-orthogonal sequences with shifted parameters are given for q-Laguerre polynomials in [12,21], for Al-Salam-Chihara, q-Meixner-Pollaczek and q-ultraspherical polynomials in [12] and for 2 φ 1 hypergeometric polynomials, associated with the little q-Jacobi polynomials, in [7]. The recurrence equations necessary to prove these results were obtained respectively from relationships between polynomials orthogonal to a positive measure dΨ(x) and those orthogonal to xdΨ(x) (cf.…”

“…The necessary equations are obtained using an algorithmic approach, whereas one may also use contiguous function relations for the basic hypergeometric series (see e.g. [7,9,23]) to get some of these recurrence equations. We use an adaption of the q-version of Zeilberger's algorithm which is an extension of Gosper's algorithm.…”

Quasi-orthogonality and zeros of some 2ϕ2 and 3<

“…Clearly, the coefficients of L (z, q) as ν 1,n < κ 1,n < ν 2,n < κ 2,n < · · · < ν n,n < κ n,n . Again from [29, Further applications of [17,Lemma 4(b)] shows that the real, simple zeros of L (δ+1) n−1 (z, q) interlace with those L (δ) n (z, q) as ν 1,n < κ 1,n−1 < ν 2,n < κ 2,n−1 < · · · < ν n−1,n < κ n−1,n−1 < ν n,n .…”

Solutions of convex Bethe Ansatz equations and the zeros of (basic) hypergeometric orthogonal polynomials

“…In this work we are mainly concerned with estimates for the locations of the zeros of some well-studied (basic) hypergeometric orthogonal polynomial families belonging to the (q−)Askey scheme [AW85, KLS10], while other relevant issues concerning these zeros, such as e.g. their dependence on the parameters [S75, M93, I05], interlacing properties [S75,D09,HV12,GJRS16], or their asymptotical behavior [S75,S10,DHK10] will not be addressed.…”