On the analytic conjugacy of Touchard polynomials and the polynomials quasi-orthogonal to them

We consider the Touchard polynomials, which are extensions of the Bell polynomials, and the $P$-polynomials which, together with the Touchard polynomials, form a quasi-orthogonal system. We establish analytical conjugacy of the Touchard polynomials and the $P$-polynomials, which allows us, in particular, to find a series of new recurrence relations for the $P$-polynomials.