Polynomials of binomial type from truncated delta series (Q1176033)

The author calls a sequence of polynomials \((b_ n(x))_ {n\geq0}\) of binomial type if there exists a formal power series \(\beta (t)=\sum _ {k\geq1} ^ {}\beta _ kt^ k\) with \(\beta _ 1\neq0\) such that \[ \sum _ {n\geq0} ^ {}b_ n(x)t^ n=e^ {x\beta (t)}. \] (Classically the sequence \((b_ n(x)n!)_ {n\geq0}\) would be called of binomial type.) The series \(\beta (t)\) is called the delta series conjugate to \((b_ n(x))\). This paper is devoted to investigate the effect of ``truncation'' of the delta series. Namely, define the \(k\)-th level sequence \((b_ n^ {(k)}(x))\) to be the binomial sequence conjugate to \(\beta ^ {(k)}(t):=\beta _ {k + 1}t + \beta _ {k + 2}t^ 2 + \dots\) (provided \(\beta _ {k + 1}\neq0\)). It is shown that the umbral composition of these level sequences with \(\left(\binom xn\right)\) has interesting properties. Besides, it is exposed what it means, in terms of species, that a binomial sequence coincides with its \(k\)-th umbral level.