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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 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. [ABSTRACT FROM AUTHOR]

The zeros of quasi-orthogonal polynomials play a key role in applications in areas such as interpolation theory, Gauss-type quadrature formulas, rational approximation and electrostatics. We extend previous results on the quasi-orthogonality of Jacobi polynomials and discuss the quasi-orthogonality of Meixner–Pollaczek, Hahn, Dual-Hahn and Continuous Dual-Hahn polynomials using a characterization of quasi-orthogonality due to Shohat. Of particular interest are the Meixner–Pollaczek polynomials whose linear combinations only exhibit quasi-orthogonality of even order. In some cases, we also investigate the location of the zeros of these polynomials for quasi-orthogonality of order 1 and 2 with respect to the end points of the interval of orthogonality, as well as with respect to the zeros of different polynomials in the same orthogonal sequence. [ABSTRACT FROM AUTHOR]

In this paper, we prove the quasi-orthogonality of a family of F 2 2 polynomials and several classes of F 2 3 polynomials that do not appear in the Askey scheme for hypergeometric orthogonal polynomials. Our results include, as a special case, two F 2 3 polynomials considered by Dickinson in 1961. We also discuss the location and interlacing of the real zeros of our polynomials. [ABSTRACT FROM AUTHOR]

The monotonicity properties of all the zeros with respect to a parameter of orthogonal polynomials associated with an even weight function are studied. The results we obtain extend the work of A. Markoff. The monotonicity of the zeros of Gegenbauer, Freud-type and symmetric Meixner–Pollaczek orthogonal polynomials as well as Al-Salam–Chiharaq-orthogonal polynomials are investigated. For the Meixner–Pollaczek polynomials, a special case of a conjecture by Jordaan and Toókos which concerns the interlacing of their zeros between two different sequences of Meixner–Pollaczek polynomials is proved. [ABSTRACT FROM AUTHOR]

Abstract: The family of general Jacobi polynomials where can be characterised by complex (non-Hermitian) orthogonality relations (cf. Kuijlaars et al. (2005)). The special subclass of Jacobi polynomials where are classical and the real orthogonality, quasi-orthogonality as well as related properties, such as the behaviour of the real zeros, have been well studied. There is another special subclass of Jacobi polynomials with , which are known as Pseudo-Jacobi polynomials. The sequence of Pseudo-Jacobi polynomials is the only other subclass in the general Jacobi family (beside the classical Jacobi polynomials) that has real zeros for every for certain values of . For some parameter ranges Pseudo-Jacobi polynomials are fully orthogonal, for others there is only complex (non-Hermitian) orthogonality. We summarise the orthogonality and quasi-orthogonality properties and study the zeros of Pseudo-Jacobi polynomials, providing asymptotics, bounds and results on the monotonicity and convexity of the zeros. [Copyright &y& Elsevier]

Abstract: We use a method based on the division algorithm to determine all the values of the real parameters and for which the hypergeometric polynomials have real, simple zeros. Furthermore, we use the quasi-orthogonality of Jacobi polynomials to determine the intervals on the real line where the zeros are located. [Copyright &y& Elsevier]