Volume 31, Issue 3
On a Pair of Operator Series Expansions Implying a Variety of Summation Formulas

Anal. Theory Appl., 31 (2015), pp. 260-282

Published online: 2017-07

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• Abstract
With the aid of Mullin-Rota's substitution rule,we show that the Sheffer-type differential operators together withthe delta operators $\Delta$ and $D$ could be used to construct apair of expansion formulas that imply a wide variety of summationformulas in the discrete analysis and combinatorics. A convergencetheorem is established for a fruitful source formula that impliesmore than 20 noted classical fomulas 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 toproduce certain chains of $(\infty^m)$ degree formulas for $m\geq 3$with $m\equiv 1$ (mod 2) and $m\equiv 1$ (mod 3), respectively.
• Keywords

Delta operator Sheffer-type operator $(\infty^m)$ degree formula triplet lifting process

12E10 13F25 16S32 65B10

@Article{ATA-31-260, author = {L. C. Hsu}, title = {On a Pair of Operator Series Expansions Implying a Variety of Summation Formulas}, journal = {Analysis in Theory and Applications}, year = {2017}, volume = {31}, number = {3}, pages = {260--282}, abstract = { With the aid of Mullin-Rota's substitution rule,we show that the Sheffer-type differential operators together withthe delta operators $\Delta$ and $D$ could be used to construct apair of expansion formulas that imply a wide variety of summationformulas in the discrete analysis and combinatorics. A convergencetheorem is established for a fruitful source formula that impliesmore than 20 noted classical fomulas 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 toproduce certain chains of $(\infty^m)$ degree formulas for $m\geq 3$with $m\equiv 1$ (mod 2) and $m\equiv 1$ (mod 3), respectively.}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2015.v31.n3.5}, url = {http://global-sci.org/intro/article_detail/ata/4639.html} }
TY - JOUR T1 - On a Pair of Operator Series Expansions Implying a Variety of Summation Formulas AU - L. C. Hsu 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 KW - Sheffer-type operator KW - $(\infty^m)$ degree formula KW - triplet KW - lifting process AB - With the aid of Mullin-Rota's substitution rule,we show that the Sheffer-type differential operators together withthe delta operators $\Delta$ and $D$ could be used to construct apair of expansion formulas that imply a wide variety of summationformulas in the discrete analysis and combinatorics. A convergencetheorem is established for a fruitful source formula that impliesmore than 20 noted classical fomulas 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 toproduce certain chains of $(\infty^m)$ degree formulas for $m\geq 3$with $m\equiv 1$ (mod 2) and $m\equiv 1$ (mod 3), respectively.