An operator on formal power series of the form S μS , where μ is an invertible power series, and σ is a series of the form?t+(t2)?is called a unipotent substitution with pre-function. Such operators, denoted by a p...An operator on formal power series of the form S μS , where μ is an invertible power series, and σ is a series of the form?t+(t2)?is called a unipotent substitution with pre-function. Such operators, denoted by a pair (μ ,σ )? , form a group. The objective of this contribution is to show that it is possible to define a generalized powers for such operators, as for instance fractional powers σ for every .展开更多
In two centuries ago, Ming Autu discovered the famous Catalan numbers while he tried to expand the function sin(2px) as power series of sin(x) for the case p = 1, 2, 3. Very recently, P. J. Larcombe shows that for any...In two centuries ago, Ming Autu discovered the famous Catalan numbers while he tried to expand the function sin(2px) as power series of sin(x) for the case p = 1, 2, 3. Very recently, P. J. Larcombe shows that for any p, sin(2px) can always be expressed as an infinite power series of sin(x) involving precise combinations of Catalan numbers as part of all but the initial p terms and gave all expansions for the case p = 4, 5. The present paper presents the desired expansion for arbitrary integer p.展开更多
We investigate Bell polynomials, also called Touchard polynomials or exponential polynomials, by using and without using umbral calculus. We use three different formulas in order to express various known families of p...We investigate Bell polynomials, also called Touchard polynomials or exponential polynomials, by using and without using umbral calculus. We use three different formulas in order to express various known families of polynomials such as Bernoulli polynomials, poly-Bernoulli polynomials, Cauchy polynomials and falling factorials in terms of Bell polynomials and vice versa. In addition, we derive several properties of Bell polynomials along the way.展开更多
文摘An operator on formal power series of the form S μS , where μ is an invertible power series, and σ is a series of the form?t+(t2)?is called a unipotent substitution with pre-function. Such operators, denoted by a pair (μ ,σ )? , form a group. The objective of this contribution is to show that it is possible to define a generalized powers for such operators, as for instance fractional powers σ for every .
文摘In two centuries ago, Ming Autu discovered the famous Catalan numbers while he tried to expand the function sin(2px) as power series of sin(x) for the case p = 1, 2, 3. Very recently, P. J. Larcombe shows that for any p, sin(2px) can always be expressed as an infinite power series of sin(x) involving precise combinations of Catalan numbers as part of all but the initial p terms and gave all expansions for the case p = 4, 5. The present paper presents the desired expansion for arbitrary integer p.
文摘We investigate Bell polynomials, also called Touchard polynomials or exponential polynomials, by using and without using umbral calculus. We use three different formulas in order to express various known families of polynomials such as Bernoulli polynomials, poly-Bernoulli polynomials, Cauchy polynomials and falling factorials in terms of Bell polynomials and vice versa. In addition, we derive several properties of Bell polynomials along the way.
