In this paper we show that the only sequences of orthogonal polynomials $$(P_n)_{n\ge 0}$$ satisfying $$\begin{aligned} \phi (x){\mathcal {D}}_q P_{n}(x)=a_n{\mathcal {S}}_q P_{n+1}(x) +b_n{\mathcal P_n(x) +c_n{\mathcal P_{n-1}(x), \end{aligned}$$ ( $$c_n\ne 0$$ ) where $$\phi $$ is a well chosen polynomial degree at most two, $${\mathcal {D}}_q$$ Askey-Wilson operator and {S}}_q$$ averaging op...

Abstract This is the first of a series two papers dealing with local limit theorems in relatively hyperbolic groups. In this paper, we prove rough estimates for Green function. Along way, introduce notion relative automaticity which will be useful both and show that groups are automatic. We also define spectral positive recurrence random walks on then use our function to $p_n\asymp R^{-n}n^{-3/...

We study a new kind of symmetric polynomials P_n(x_1,...,x_m) degree n in m real variables, which have arisen the theory numerical semigroups. establish their basic properties and find representation through power sums E_k=\sum_{j=1}^m x_j^k. observe visual similarity between normalized P_n(x_1,...,x_m)/\chi_m, where \chi_m=\prod_{j=1}^m x_j, polynomial part partition function W(s,{d_1,...,d_m}...

The QE constant of a finite connected graph $G$, denoted by $\mathrm{QEC}(G)$, is definition the maximum quadratic function associated to distance matrix on certain sphere codimension two. We prove that constants paths $P_n$ form strictly increasing sequence converging $-1/2$. Then we formulate problem determining all graphs $G$ satisfying $\mathrm{QEC}(P_n)\le\mathrm{QEC}(G)<\mathrm{QEC}(P_{n+...

Abstract Let and $f_\infty$?> be formal power series at the origin infinity, $P_n/Q_n$?> , $\deg(P_n),\deg(Q_n)\leq n$?> rational function that simultaneously interpolates with order $n$?> infinity ${n+1}$?> . When germs represent multi-valued functions finitely many branch points, it was shown by Buslaev there exists a unique compact set $F$?> in complement of which approximant...

Deficiency zero is an important network structure and has been the focus of many celebrated results within reaction theory. In our previous paper Prevalence deficiency networks in Erdős-Rényi framework, we provided a framework to quantify prevalence among randomly generated networks. Specifically, given binary with n species, edge between two arbitrary vertices occurring independently probabili...

We study the numerical evaluation of Legendre polynomials $P_n$ onthe interval $[-1,1]$ via a three-term recurrence. prove that ina neighborhood an endpoint, computed approximation exactlyagrees with line tangent to at this endpoint. As aconsequence, we obtain sharp error bounds for