Here introduced and studied are two formulaic classes consisting of various combinatorial algebraic identities and series summation formulas. The basic ideas include utilizing properly the △-operator and Stirling num...Here introduced and studied are two formulaic classes consisting of various combinatorial algebraic identities and series summation formulas. The basic ideas include utilizing properly the △-operator and Stirling numbers for some series transformations. A variety of classic formulas and remarkable identities are shown to be the members of the classes.展开更多
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
This paper provides a pair of summation formulas for a kind of combinatorial series involvingak＋b m as a factor of the summand. The construction of formulas is based on a certain series transformation formula [2, 7, ...This paper provides a pair of summation formulas for a kind of combinatorial series involvingak＋b m as a factor of the summand. The construction of formulas is based on a certain series transformation formula [2, 7, 9] and by making use of the C-numbers [3]. Various consequences and examples including several remarkable classic identities are presented to illustrate some applications of the formulas obtained.展开更多
The method of non-standard analysis （NSA） is used to construct a pair of hy-perstandard reciprocal formulas involving certain non-standard difference operators with real-number orders. Our main result consists of so...The method of non-standard analysis （NSA） is used to construct a pair of hy-perstandard reciprocal formulas involving certain non-standard difference operators with real-number orders. Our main result consists of some extensions of earlier results appearing previ-ously [5]. An essential meaning of the paper is to indicate the fact that only the basic idea of NSA is applicable to the construction of a unified pattern that may have certain applications to both the analysis and the number theory.展开更多
Here concerned and further investigated is a certain operator method for the computation of convolutions of polynomials.We provide a general formulation of the method with a refinement for certain old results,and also...Here concerned and further investigated is a certain operator method for the computation of convolutions of polynomials.We provide a general formulation of the method with a refinement for certain old results,and also give some new applications to convolved sums involving several noted special polynomials.The advantage of the method using operators is illustrated with concrete examples.Finally,also presented is a brief investigation on convolution polynomials having two types of summations.展开更多
文摘Here introduced and studied are two formulaic classes consisting of various combinatorial algebraic identities and series summation formulas. The basic ideas include utilizing properly the △-operator and Stirling numbers for some series transformations. A variety of classic formulas and remarkable identities are shown to be the members of the classes.
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.
基金Supported by the National Natural Science Foundation of China (Grant No.11071183)
文摘This paper provides a pair of summation formulas for a kind of combinatorial series involvingak＋b m as a factor of the summand. The construction of formulas is based on a certain series transformation formula [2, 7, 9] and by making use of the C-numbers [3]. Various consequences and examples including several remarkable classic identities are presented to illustrate some applications of the formulas obtained.
文摘The method of non-standard analysis （NSA） is used to construct a pair of hy-perstandard reciprocal formulas involving certain non-standard difference operators with real-number orders. Our main result consists of some extensions of earlier results appearing previ-ously [5]. An essential meaning of the paper is to indicate the fact that only the basic idea of NSA is applicable to the construction of a unified pattern that may have certain applications to both the analysis and the number theory.
文摘Here concerned and further investigated is a certain operator method for the computation of convolutions of polynomials.We provide a general formulation of the method with a refinement for certain old results,and also give some new applications to convolved sums involving several noted special polynomials.The advantage of the method using operators is illustrated with concrete examples.Finally,also presented is a brief investigation on convolution polynomials having two types of summations.
