With the aid of Mullin-Rota's substitution rule, we show that the Sheffertype differential operators together with the delta operators ? and D could be used to construct a pair of expansion formulas that imply a w...With the aid of Mullin-Rota's substitution rule, we show that the Sheffertype differential operators together with the delta operators ? 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 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 to produce certain chains of(∞~m) degree formulas for m ≥ 3 with m ≡ 1(mod 2) and m ≡ 1(mod 3), respectively.展开更多
Here presented is a further investigation on a general source formula（GSF） that has been proved capable of deducing more than 30 special formulas for series expansions and summations in the author＇s recent paper [O...Here presented is a further investigation on a general source formula（GSF） that has been proved capable of deducing more than 30 special formulas for series expansions and summations in the author＇s recent paper [On a pair of operator series expansions implying a variety of summation formulas.Anal.Theory Appl.,2015,31（3）：260–282].It is shown that the pair of series transformation formulas found and utilized by He,Hsu and Shiue [cf.Disc.Math.,2008,308：3427–3440] is also deducible from the GSF as consequences.Thus it is found that the GSF actually implies more than 50 special series expansions and summation formulas.Finally,several expository remarks relating to the（Σ△D） formula class are given in the closing section.展开更多
文摘With the aid of Mullin-Rota's substitution rule, we show that the Sheffertype differential operators together with the delta operators ? 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 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 to produce certain chains of(∞~m) degree formulas for m ≥ 3 with m ≡ 1(mod 2) and m ≡ 1(mod 3), respectively.
文摘Here presented is a further investigation on a general source formula（GSF） that has been proved capable of deducing more than 30 special formulas for series expansions and summations in the author＇s recent paper [On a pair of operator series expansions implying a variety of summation formulas.Anal.Theory Appl.,2015,31（3）：260–282].It is shown that the pair of series transformation formulas found and utilized by He,Hsu and Shiue [cf.Disc.Math.,2008,308：3427–3440] is also deducible from the GSF as consequences.Thus it is found that the GSF actually implies more than 50 special series expansions and summation formulas.Finally,several expository remarks relating to the（Σ△D） formula class are given in the closing section.
