提出研究形如fp,r,s,x(n)=∑ from k=0 to rn 〔pn k〕s xk的组合和的闭形式问题的Z-P方法,并利用此方法得到了如下结果:1)当s=1,p=2r时证明了对未定元x,∑ from k=0 to rn 〔2rn k〕xk无闭形式表示;2)对p、r、s及x取特定的值,借助计算...提出研究形如fp,r,s,x(n)=∑ from k=0 to rn 〔pn k〕s xk的组合和的闭形式问题的Z-P方法,并利用此方法得到了如下结果:1)当s=1,p=2r时证明了对未定元x,∑ from k=0 to rn 〔2rn k〕xk无闭形式表示;2)对p、r、s及x取特定的值,借助计算机归纳出几个值得探索的猜测.展开更多
With the help of the classical Abel’s lemma on summation by parts and algorithm of q-hypergeometric summations, we deal with the summation, which can be written as multiplication of a q-hypergeometric term and q-harm...With the help of the classical Abel’s lemma on summation by parts and algorithm of q-hypergeometric summations, we deal with the summation, which can be written as multiplication of a q-hypergeometric term and q-harmonic numbers. This enables us to construct and prove identities on q-harmonic numbers. Several examples are also given.展开更多
In this paper,we prove two supercongruences conjectured by Z.-W.Sun via the Wilf-Zeilberger method.One of them is,for any prime p>3,4F3[7/6 1/2 1/2 1/2 1/6 1 1]-1/8]-1/2≡p(-2/p)+p^(3)/4(2/p)Ep-3 (mod p^(4))where(&...In this paper,we prove two supercongruences conjectured by Z.-W.Sun via the Wilf-Zeilberger method.One of them is,for any prime p>3,4F3[7/6 1/2 1/2 1/2 1/6 1 1]-1/8]-1/2≡p(-2/p)+p^(3)/4(2/p)Ep-3 (mod p^(4))where(·/p)stands for the Legendre symbol,and E_(n)is the n-th Euler number.展开更多
The m-th order detrended Brownian motion is defined as the orthogonal component of projection of the standard Brownian motion onto the subspace spanned by polynomials of degree up to m. We obtain the Karhunen-Loeve ex...The m-th order detrended Brownian motion is defined as the orthogonal component of projection of the standard Brownian motion onto the subspace spanned by polynomials of degree up to m. We obtain the Karhunen-Loeve expansion for the process and establish a connection with the generalized （m-th order） Brownian bridge developed by MacNeill （1978） in the study of distributions of polynomial regression. The resulting distribution identity is also verified by a stochastic Fubini approach. As applications, large and small deviation asymptotic behaviors for the L2 norm are given.展开更多
文摘提出研究形如fp,r,s,x(n)=∑ from k=0 to rn 〔pn k〕s xk的组合和的闭形式问题的Z-P方法,并利用此方法得到了如下结果:1)当s=1,p=2r时证明了对未定元x,∑ from k=0 to rn 〔2rn k〕xk无闭形式表示;2)对p、r、s及x取特定的值,借助计算机归纳出几个值得探索的猜测.
文摘With the help of the classical Abel’s lemma on summation by parts and algorithm of q-hypergeometric summations, we deal with the summation, which can be written as multiplication of a q-hypergeometric term and q-harmonic numbers. This enables us to construct and prove identities on q-harmonic numbers. Several examples are also given.
基金Supported by the National Natural Science Foundation of China(Grant No.12001288)。
文摘In this paper,we prove two supercongruences conjectured by Z.-W.Sun via the Wilf-Zeilberger method.One of them is,for any prime p>3,4F3[7/6 1/2 1/2 1/2 1/6 1 1]-1/8]-1/2≡p(-2/p)+p^(3)/4(2/p)Ep-3 (mod p^(4))where(·/p)stands for the Legendre symbol,and E_(n)is the n-th Euler number.
基金supported by the Fundamental Research Funds for the Central Universities(Grant No.DL13BBX10)
文摘The m-th order detrended Brownian motion is defined as the orthogonal component of projection of the standard Brownian motion onto the subspace spanned by polynomials of degree up to m. We obtain the Karhunen-Loeve expansion for the process and establish a connection with the generalized （m-th order） Brownian bridge developed by MacNeill （1978） in the study of distributions of polynomial regression. The resulting distribution identity is also verified by a stochastic Fubini approach. As applications, large and small deviation asymptotic behaviors for the L2 norm are given.
