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In this paper we consider connection formulae for orthogonal polynomials in the context of Christoffel transformations for the case where a weight function, not necessarily even, is multiplied by an even function $c_{2k}(x),k\in N_0$, to determine new lower bounds for the largest zero and upper bounds for the smallest zero of a Meixner-Pollaczek polynomial. When $p_n$ is orthogonal with respect to a weight $w(x)$ and $g_{n-m}$ is orthogonal with respect to the weight $c_{2k}(x)w(x)$, we show that $k\in\{0,1,\dots,m\}$ is a necessary and sufficient condition for existence of a connection formula involving a polynomial $G_{m-1}$ of degree $(m-1)$, such that the $(n-1)$ zeros of $G_{m-1}g_{n-m}$ and the $n$ zeros of $p_n$ interlace. We analyse the new inner bounds for the extreme zeros of Meixner-Pollaczek polynomials to determine which bounds are the sharpest. We also briefly discuss bounds for the zeros of Pseudo-Jacobi polynomials., Comment: 12 pages, 0 figures

For each fixed value of $\beta$ in the range $-2<\beta<-1$ and $0

We state and prove the $q$-extension of a result due to Johnston and Jordaan (cf. \cite{Johnston-2015}) and make use of this result, the orthogonality of $q$-Laguerre, little $q$-Jacobi, $q$-Meixner and Al-Salam-Carlitz I polynomials as well as contiguous relations satisfied by the polynomials, to establish the quasi-orthogonality of certain $_2\phi_2$ and $_3\phi_2$ polynomials. The location and interlacing properties of the real zeros of these quasi-orthogonal polynomials are studied. Interlacing properties of the zeros of $q$-Laguerre quasi-orthogonal polynomials $L_n^{(\delta)}(z;q)$ when $-2<\delta<-1$ with those of $L_{n-1}^{(\delta+1)}(z;q)$ and $L_n^{(\delta+1)}(z;q)$ are also considered.

We use a method based on the division algorithm to determine all the values of the real parameters $b$ and $c$ for which the hypergeometric polynomials $_2F_1(-n, b; c; z)$ have $n$ 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.

We derive upper bounds for the smallest zero and lower bounds for the largest zero of Laguerre, Jacobi and Gegenbauer polynomials. Our approach uses mixed three term recurrence relations satisfied by polynomials corresponding to different parameter(s) within the same classical family. We prove that interlacing properties of the zeros impose restrictions on the possible location of common zeros of the polynomials involved and deduce strict bounds for the extreme zeros of polynomials belonging to each of these three classical families. We show numerically that the bounds generated by our method improve known lower (upper) bounds for the largest (smallest) zeros of polynomials in these families, notably in the case of Jacobi and Gegenbauer polynomials.

We investigate the zeros of a family of hypergeometric polynomials $_2F_1(-n,-x;a;t)$, $n\in\nn$ that are known as the Meixner polynomials for certain values of the parameters $a$ and $t$. When $a=-N$, $N\in\nn$ and $t=\frac1{p}$, the polynomials $K_n(x;p,N)=(-N)_n\phantom{}_2F_1(-n,-x;-N;\frac1{p})$, $n=0,1,...N$, $0n-1$, the quasi-orthogonal polynomials $K_{n}(x;p,a)$, $k-11$ or $p<0$ as well as the non-orthogonal polynomials $K_{n}(x;p,N)$, $0

The Pad\'e table of $\phantom{}_2F_1(a,1;c;z)$ is normal for $c>a>0$ (cf. \cite{3}). For $m \geq n-1$ and $c \notin {\zz}^{\phantom{}^-}$, the denominator polynomial $Q_{mn}(z)$ in the $[m/n]$ Pad\'e approximant $P_{mn}(z)/Q_{mn}(z)$ for $\phantom{}_2F_1(a,1;c;z)$ and the remainder term $Q_{mn}(z)\phantom{}_2F_1(a,1;c;z)-P_{mn}(z)$ were explicitly evaluated by Pad\'e (cf. \cite{2}, \cite{5} or \cite{7}). We show that for $c>a>0$ and $m\geq n-1$, the poles of $P_{mn}(z)/Q_{mn}(z)$ lie on the cut $(1,\infty)$. We deduce that the sequence of approximants $P_{mn}(z)/Q_{mn}(z)$ converges to $\phantom{}_2F_1(a,1;c;z)$ as $m \to \infty$, $ n/m \to \rho$ with $0<\rho \leq 1$, uniformly on compact subsets of the unit disc $|z|<1$ for $c>a>0$

Our interest lies in describing the zero behaviour of Gauss hypergeometric polynomials $F(-n,b; c; z)$ where $b$ and $c$ are arbitrary parameters. In general, this problem has not been solved and even when $b$ and $c$ are both real, the only cases that have been fully analyzed impose additional restrictions on $b$ and $c$. We review recent results that have been proved for the zeros of several classes of hypergeometric polynomials $F(-n,b; c; z)$ where $b$ and $c$ are real. We show that the number of real zeros of $F(-n,b; c; z)$ for arbitrary real values of the parameters $b$ and $c$, as well as the intervals in which these zeros (if any) lie, can be deduced from corresponding results for Jacobi polynomials.

We study convexity properties of the zeros of some special functions that follow from the convexity theorem of Sturm. We prove results on the intervals of convexity for the zeros of Laguerre, Jacobi and ultraspherical polynomials, as well as functions related to them, using transformations under which the zeros remain unchanged. We give upper as well as lower bounds for the distance between consecutive zeros in several cases.

We study interlacing properties of the zeros of two types of linear combinations of Laguerre polynomials with different parameters, namely $R_n=L_n^{\alpha}+aL_{n}^{\alpha'}$ and $S_n=L_n^{\alpha}+bL_{n-1}^{\alpha'}$. Proofs and numerical counterexamples are given in situations where the zeros of $R_n$, and $S_n$, respectively, interlace (or do not in general) with the zeros of $L_k^{\alpha}$, $L_k^{\alpha'}$, $k=n$ or $n-1$. The results we prove hold for continuous, as well as integral, shifts of the parameter $\alpha$.