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

Abstract

Let \(\displaystyle \{p_n\}_{n=0}^{\infty }\), where \(p_n\) is a polynomial of degree n, be a sequence of polynomials orthogonal with respect to a positive probability measure. If \(x_{1,n} < \cdots < x_{n,n}\) denotes the zeros of \(p_n\) while \(x_{1,n-1} < \cdots < x_{n-1,n-1}\) are the zeros of \(p_{n-1}\), the inequality $$\begin{aligned} x_{1,n} < x_{1,n-1} < x_{2,n} < \cdots < x_{n-1,n}< x_{n-1,n-1}< x_{n,n}, \end{aligned}$$ known as the interlacing property, is satisfied. We use a consequence of a generalised version of Markov’s monotonicity results to investigate interlacing properties of zeros of contiguous basic hypergeometric polynomials associated with little q-Jacobi polynomials and determine inequalities for extreme zeros of the above two polynomials. It is observed that the new bounds which are obtained in this paper give more precise upper bounds for the smallest zero of little q-Jacobi polynomials, improving previously known results by Driver and Jordaan (Math Model Nat Phenom 8(1):48–59, 2013), and in some cases, those by Gupta and Muldoon (J Inequal Pure Appl Math 8(1):7, 2007). Numerical examples are given in order to illustrate the accuracy of our bounds.