Suppose $p(x_1, x_2, \cdots, x_n)$ is a symmetric polynomial. Given any univariate polynomial $u$, we can define a new polynomial $q(x_1, x_2, \cdots, x_{n+1})$ as

$q(x_1, x_2, \cdots, x_{n+1}) = u(x_1)p(x_2, x_3, \cdots, x_{n+1}) + u(x_2)p(x_1, x_3, \cdots, x_{n+1}) + \cdots \\ \phantom{q(x_1, x_2, \cdots, x_{n+1}) = } \qquad + u(x_{n+1})p(x_1, x_2, \cdots, x_n).$

It is easy to verify that $q$ is a symmetric polynomial. My question is: Is there a name already defined for such a mapping from $(p, u)$ to $q$? Thanks.