On a Pair of Operator Series Expansions Implying a Variety of Summation Formulas
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@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.
L. C. Hsu. (1970). On a Pair of Operator Series Expansions Implying a Variety of Summation Formulas.
Analysis in Theory and Applications. 31 (3).
260-282.
doi:10.4208/ata.2015.v31.n3.5
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