TY - JOUR
T1 - On a Pair of Operator Series Expansions Implying a Variety of Summation Formulas
JO - Analysis in Theory and Applications
VL - 3
SP - 260
EP - 282
PY - 2017
DA - 2017/07
SN - 31
DO - http://doi.org/10.4208/ata.2015.v31.n3.5
UR - https://global-sci.org/intro/article_detail/ata/4639.html
KW - Delta operator, Sheffer-type operator, $(\infty^m)$ degree formula, triplet, lifting process.
AB - <p style="text-align: justify;">With the aid of Mullin-Rota&#39;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.</p>