定义了Norland Bernoulli多项式和Norland Eurler多项式,证明了恒等式: B_(m_1·m_2·…·m_p)^((k))(x_1,x_2,…x_p,y_1,y_2,…y_k)=(1/2^(sum from i to p(m_i)))((sum from s_1=0 to m_1)(sum from s_2=0 to m_2)…(sum ...定义了Norland Bernoulli多项式和Norland Eurler多项式,证明了恒等式: B_(m_1·m_2·…·m_p)^((k))(x_1,x_2,…x_p,y_1,y_2,…y_k)=(1/2^(sum from i to p(m_i)))((sum from s_1=0 to m_1)(sum from s_2=0 to m_2)…(sum from s_p=0 to m_p)(m_1 s_1)…(m_p s_p))E_(s_1·s_2·…·s_p)^((k))(x_1,x_2,…x_p,y_1,y_2,…y_k) B_(m_1-s_1,m_2-s_2,…,m_p-s_p)^((k))(x_1,x_2,…,x_p,y_1,y_2,…,y_k)展开更多
Utilization of the shift operator to represent Euler polynomials as polynomials of Appell type leads directly to its algebraic properties, its relations with powers sums;may be all its relations with Bernoulli polynom...Utilization of the shift operator to represent Euler polynomials as polynomials of Appell type leads directly to its algebraic properties, its relations with powers sums;may be all its relations with Bernoulli polynomials, Bernoulli numbers;its recurrence formulae and a very simple formula for calculating simultaneously Euler numbers and Euler polynomials. The expansions of Euler polynomials into Fourier series are also obtained;the formulae for obtaining all π<sup>m</sup> as series on k<sup>-m</sup> and for expanding functions into series of Euler polynomials.展开更多
For any Pisot number β it is known that the set F（β） ={t ： limn→∞‖tβn‖ = 0} is countable, where ‖α‖ is the distance between a real number a and the set of integers. In this paper it is proved that every m...For any Pisot number β it is known that the set F（β） ={t ： limn→∞‖tβn‖ = 0} is countable, where ‖α‖ is the distance between a real number a and the set of integers. In this paper it is proved that every member in this set is of the form cβn, where n is a nonnegative integer and e is determined by a linear system of equations. Furthermore, for some self-similar measures μ associated with β, the limit at infinity of the Fourier transforms limn→μ（tβn）≠0 if and only if t is in a certain subset of F（β）. This generalizes a similar result of Huang and Strichartz.展开更多
文摘定义了Norland Bernoulli多项式和Norland Eurler多项式,证明了恒等式: B_(m_1·m_2·…·m_p)^((k))(x_1,x_2,…x_p,y_1,y_2,…y_k)=(1/2^(sum from i to p(m_i)))((sum from s_1=0 to m_1)(sum from s_2=0 to m_2)…(sum from s_p=0 to m_p)(m_1 s_1)…(m_p s_p))E_(s_1·s_2·…·s_p)^((k))(x_1,x_2,…x_p,y_1,y_2,…y_k) B_(m_1-s_1,m_2-s_2,…,m_p-s_p)^((k))(x_1,x_2,…,x_p,y_1,y_2,…,y_k)
文摘Utilization of the shift operator to represent Euler polynomials as polynomials of Appell type leads directly to its algebraic properties, its relations with powers sums;may be all its relations with Bernoulli polynomials, Bernoulli numbers;its recurrence formulae and a very simple formula for calculating simultaneously Euler numbers and Euler polynomials. The expansions of Euler polynomials into Fourier series are also obtained;the formulae for obtaining all π<sup>m</sup> as series on k<sup>-m</sup> and for expanding functions into series of Euler polynomials.
文摘For any Pisot number β it is known that the set F（β） ={t ： limn→∞‖tβn‖ = 0} is countable, where ‖α‖ is the distance between a real number a and the set of integers. In this paper it is proved that every member in this set is of the form cβn, where n is a nonnegative integer and e is determined by a linear system of equations. Furthermore, for some self-similar measures μ associated with β, the limit at infinity of the Fourier transforms limn→μ（tβn）≠0 if and only if t is in a certain subset of F（β）. This generalizes a similar result of Huang and Strichartz.
