Turán's inequality, nonnegative linearization and amenability properties for associated symmetric Pollaczek polynomials (Q2037070)

The main results can be found in Theorem 4.1. Let \(\alpha >-1/2,\,\lambda\geq 0\) and \(\nu\geq 0\) be arbitrary, and let \(P_n(x)=Q_n^{(\alpha,\lambda,\nu)}\ (n\in\mathbb{N}_0)\). Then \((c_n)_{n\in\mathbb{N}}\) is strictly increasing and \((P_n(x))_{n\in\mathbb{N})_0}\) satisfies both the nonnegative linearization property \((P)\) and Turán's inequality. Theorem 5.1. Let \(\alpha >-1/2,\,\lambda\geq 0\) and \(\nu\geq 0\) be arbitrary, and let \(P_n(x)=Q_n^{(\alpha,\lambda,\nu)}\ (n\in\mathbb{N}_0)\). Then \(\ell^1(h)\) is \begin{itemize} \item[{(i)}] point amenable if and only iif \(\alpha<1/2\) and \(\lambda=0\), \item[ (ii)] weakly amenable if and only if \(\alpha<0\) and \(\lambda=\nu=0\), \item[ (iii)] never right character amenable, \item[ (iv)] never amenable. \end{itemize} The underlying setting and the notions used are: \begin{itemize} \item \(\{P_n(x)\}_{n\in\mathbb{N}_0} \subset\mathbb{R}[x]\) with deg\(P_n(x)=n\) a sequence of polynomials with recurrence relation \(P_0(x)=1,\,P_1(x)=x\) and \[xP_n(x)=a_nP_{n+1}(x)+c_nP_{n-1}(x)\ (n\in\mathbb{N}),\] with \(a_n,c_n \subseteq (0,1)\) and \(a_n+c_n=1\ (n\in\mathbb{N})\). \item `Property \((P)\)' indicates that the polynomials satisfy the condition that all linearization coefficients \(g(m,n;k)\), given below, are nonnegatrive: \[P_n(x)p_m(x)=\sum_{k=0}^{+n}\,g(m,n;k)P_k(x)\ (m,n\in\mathbb{N}_0).\] \item The \(Q_n^{(\alpha,\lambda,\nu)}(x)\ (n\in\mathbb{N}_0)\) are the associated symmetric Pollaczek polynomials given by \(\alpha>-1/2,\lambda\geq 0,\nu\geq 0\) that satisfy the recurrence relation with \[c_n=\frac{n+\nu+2\alpha}{2n+2\nu+2\alpha+2\lambda+1}\,\frac{L_{n-1}^{(2\alpha,\nu)}(-2\lambda)}{L_{n}^{(2\alpha,\nu)}(-2\lambda)}\in (0,1)\ (n\in\mathbb{N})\] and \(a_n=1-c_n\). Here \((L_n^{(2\alpha,\nu)}(x))_{n\in\mathbb{N}_0}\) denotes the associated Laguerre polynomials corresponding to \(2\alpha\) and \(\nu\). \item The Haar function used can be given by the underlying measure of the polynomials \[h(n)=\frac{1}{g(n,n;0)}=\frac{1}{\int_{\mathbb{R}}\,P_n^2(x)d\mu(x)},\] or by the recurrence relation \[h(0)=1,\ h(n+1)=\frac{a_n}{c_{n+1}}\,h(n)\ (n\in\mathbb{N}_0).\] \item For every \(p\in[1,\infty)\), let \(\ell^p(h)=\{f:\mathbb{N}_0\,|\,\|f\|_p <\infty\}\) with \[\|f\|_p=\left(\sum_{k=0}^{\infty}\,|f(k)|^p h(k)\right)^{1/p}.\] \item Given a Banach algebra \(A\), a linear mapping \(D\) from \(A\) into a Banach \(A\)-bimodule \(X\) is called a derivation if \(D(ab)=a\cdot D(b)+D(a)\cdot b\ (a,b\in A)\), an inner derivation if \(D(a)=a\cdot x-x\cdot a\ (a\in A)\) for some \(x\in X\) and apoint derivation at \(\varphi\in\Delta(A)\) (structure space) if \(X=\mathbb{C}\) and \(D(ab)=\varphi(a)D(b)+\varphi(b)D(a)\ (a,b\in A)\). \item \(A\) is called amenable if for every Banach \(A\)-bimodule \(X\) every bounded derivation into the dual module \(X^{\ast}\) is an inner derivation, weakly amenable if every bounded bounded derivation into \(A^{\ast}\) is an inner derivation, \(\varphi\)-amenable w.r.t. \(\varphi\in\Delta(A)\) if for every Banach A-bimodule \(X\) such that \(a\cdot x=\varphi(a)x\ (a\in A, x\in X)\) every bounded derivation from A into the dual module \(X^{\ast}\) is an inner derivation, and right character amenable if A is \(\varphi\)-amenable for every \(\varphi\in\Delta(A)\) and A has a bounded right approximate identity. \end{itemize} The paper starts with an extensive Introduction and a section on Preliminaries, together 9 pages, followed by a section on `Transformation into random walk polynomials' (\(3\,1/2\) pages). In the sections 4 and 5 the main results are given. The paper concludes with an Appendix given an alternative proof of Theorem 4.4 (`via Turán's inequality for Laguerre polynomials') and a list of references containing \(59\) items.