In this contribution we deal with pairs of symmetric linear functionals \(({\textbf {u}} ,{\textbf {v}} )\) such that the corresponding sequences of monic orthogonal polynomials \(\{P_n(x;{\textbf {u}} )\}_{n\ge 0}\) and \(\{ P_n(x;{\textbf {v}} )\}_{n\ge 0}\) are related by \( \frac{T_{\mu }P_{n+1}(x;{\textbf {u}} )}{\mu _{n+1}}=P_n(x;{\textbf {v}} )-\tau _{n-1}P_{n-2}(x;{\textbf {v}} ),\quad \tau _{n-1}\ne 0,\quad n\ge 2. \) Here \(T_{\mu }\) is the standard Dunkl operator in one variable. Such a pair of linear functionals is said to be a symmetric \(T_{\mu }\) -coherent pair of the second kind. We give a necessary and sufficient condition in order for a pair of symmetric linear functionals to be a \(T_{\mu }\) -coherent pair of the second kind. We describe all of such linear functionals. In particular, we focus our attention on the companions of \(T_{\mu }\) -classical linear functionals (Generalized Hermite and Generalized Gegenbauer).
