On a Pair of Operator Series Expansions Implying a Variety of Summation Formulas

Abstract

With the aid of Mullin-Rota's substitution rule, we show that the Sheffer-type differential operators together with the delta operators $\Delta$ and $D$ could be used to construct a pair of expansion formulas that imply a wide variety of summation formulas in the discrete analysis and combinatorics. A convergence theorem is established for a fruitful source formula that implies more than 20 noted classical formulas and identities as consequences. Numerous new formulas are also presented as illustrative examples. Finally, it is shown that a kind of lifting process can be used to produce certain chains of $(\infty^m)$ degree formulas for $m\geq 3$ with $m\equiv 1$ (mod 2) and $m\equiv 1$ (mod 3), respectively.