Utilizing Gamma-Beta function, we can build one series involving reciprocal of non-central binomial coefficients, then We can structure several new series of reciprocals of non-central binomial coefficients by item sp...Utilizing Gamma-Beta function, we can build one series involving reciprocal of non-central binomial coefficients, then We can structure several new series of reciprocals of non-central binomial coefficients by item splitting, these new created denominator of series contain 1 to 4 odd factors of binomial coefficients. As the result of splitting items, some identities of series of numbers values of reciprocals of binomial coefficients are given. The method of splitting terms offered in this paper is a new combinatorial analysis way and elementary method to create new series.展开更多
Let p be an odd prime and let a,m ∈ Z with a 】 0 and p ︱ m.In this paper we determinep ∑k=0 pa-1(2k k=d)/mk mod p2 for d=0,1;for example,where(-) is the Jacobi symbol and {un}n≥0 is the Lucas sequence given by u0...Let p be an odd prime and let a,m ∈ Z with a 】 0 and p ︱ m.In this paper we determinep ∑k=0 pa-1(2k k=d)/mk mod p2 for d=0,1;for example,where(-) is the Jacobi symbol and {un}n≥0 is the Lucas sequence given by u0 = 0,u1 = 1 and un+1 =(m-2)un-un-1(n = 1,2,3,...).As an application,we determine ∑0【k【pa,k≡r(mod p-1) Ck modulo p2 for any integer r,where Ck denotes the Catalan number 2kk /(k + 1).We also pose some related conjectures.展开更多
设p>3为素数.对任何p-adic整数a,我们决定出p−1 X k=0 −k a a−1 k Hk,p−1 X k=0 −k a a−1 k Hk(2),p−1 X k=0 −k a a−1 k H(2)k 2k+1模p 2,其中Hk=P 0<j<k 1/j且Hk(2)=P 0<j>k 1/j2.特别地,我们证明了p−1 X k=0 −k a a−1...设p>3为素数.对任何p-adic整数a,我们决定出p−1 X k=0 −k a a−1 k Hk,p−1 X k=0 −k a a−1 k Hk(2),p−1 X k=0 −k a a−1 k H(2)k 2k+1模p 2,其中Hk=P 0<j<k 1/j且Hk(2)=P 0<j>k 1/j2.特别地,我们证明了p−1 X k=0 −k a a−1 k Hk≡(−1)p 2(Bp−1(a)−Bp−1)(mod p),p−1 X k=0 −k a a−1 k Hk(2)≡−Ep−3(a)(mod p),(2a−1)p−1 X k=0 −k a a−1 k H(2)k 2k+1≡Bp−2(a)(mod p)其中p表示满足a≤r(mod p)的最小非负整数r,Bn(x)与En(x)分别表示次数为n的伯努利多项式与欧拉多项式.展开更多
In this paper,we show several arithmetic properties on the residues of binomial coefficients and their products modulo prime powers,e.g.. for any distinct odd primes p and q.Meanwhile.we discuss the connections with ...In this paper,we show several arithmetic properties on the residues of binomial coefficients and their products modulo prime powers,e.g.. for any distinct odd primes p and q.Meanwhile.we discuss the connections with the prime recognitions.展开更多
An efficient method for the analytic evaluation of the plasma dispersion function for the Fermi-Dirac distribution is proposed.The new method has been developed using the binomial expansion theorem and the Gamma funct...An efficient method for the analytic evaluation of the plasma dispersion function for the Fermi-Dirac distribution is proposed.The new method has been developed using the binomial expansion theorem and the Gamma functions.The general formulas obtained for the plasma dispersion function are utilized for the evaluation of the response function.The resulting series present better convergence rates.Several acceleration techniques are combined to further improve the efficiency.The obtained results for the plasma dispersion function are in good agreement with the known numerical data.展开更多
Summetor is an operator used in the mathematics to calculate the special numbers like binomial coefficients and combinations of group elements. It has many applications in algebra, matrices like calculation of pascal ...Summetor is an operator used in the mathematics to calculate the special numbers like binomial coefficients and combinations of group elements. It has many applications in algebra, matrices like calculation of pascal triangle elements and pascal matrix formation, etc. This paper explains about its functions and properties of N-Summet-k. The result of variation between N and k is shown in tabulation.展开更多
文摘Utilizing Gamma-Beta function, we can build one series involving reciprocal of non-central binomial coefficients, then We can structure several new series of reciprocals of non-central binomial coefficients by item splitting, these new created denominator of series contain 1 to 4 odd factors of binomial coefficients. As the result of splitting items, some identities of series of numbers values of reciprocals of binomial coefficients are given. The method of splitting terms offered in this paper is a new combinatorial analysis way and elementary method to create new series.
基金supported by National Natural Science Foundation of China (Grant No.10871087)the Overseas Cooperation Fund of China (Grant No.10928101)
文摘Let p be an odd prime and let a,m ∈ Z with a 】 0 and p ︱ m.In this paper we determinep ∑k=0 pa-1(2k k=d)/mk mod p2 for d=0,1;for example,where(-) is the Jacobi symbol and {un}n≥0 is the Lucas sequence given by u0 = 0,u1 = 1 and un+1 =(m-2)un-un-1(n = 1,2,3,...).As an application,we determine ∑0【k【pa,k≡r(mod p-1) Ck modulo p2 for any integer r,where Ck denotes the Catalan number 2kk /(k + 1).We also pose some related conjectures.
基金Supported by the National Natural Science Foundation of China(11971222)the initial version was posted to arXiv in 2014 with the ID arXiv:1407.8465.
文摘设p>3为素数.对任何p-adic整数a,我们决定出p−1 X k=0 −k a a−1 k Hk,p−1 X k=0 −k a a−1 k Hk(2),p−1 X k=0 −k a a−1 k H(2)k 2k+1模p 2,其中Hk=P 0<j<k 1/j且Hk(2)=P 0<j>k 1/j2.特别地,我们证明了p−1 X k=0 −k a a−1 k Hk≡(−1)p 2(Bp−1(a)−Bp−1)(mod p),p−1 X k=0 −k a a−1 k Hk(2)≡−Ep−3(a)(mod p),(2a−1)p−1 X k=0 −k a a−1 k H(2)k 2k+1≡Bp−2(a)(mod p)其中p表示满足a≤r(mod p)的最小非负整数r,Bn(x)与En(x)分别表示次数为n的伯努利多项式与欧拉多项式.
基金Supported partly by NNSFCA Presidential Faculty Fellowsupported in part by the NSF of the United States
文摘In this paper,we show several arithmetic properties on the residues of binomial coefficients and their products modulo prime powers,e.g.. for any distinct odd primes p and q.Meanwhile.we discuss the connections with the prime recognitions.
文摘An efficient method for the analytic evaluation of the plasma dispersion function for the Fermi-Dirac distribution is proposed.The new method has been developed using the binomial expansion theorem and the Gamma functions.The general formulas obtained for the plasma dispersion function are utilized for the evaluation of the response function.The resulting series present better convergence rates.Several acceleration techniques are combined to further improve the efficiency.The obtained results for the plasma dispersion function are in good agreement with the known numerical data.
文摘Summetor is an operator used in the mathematics to calculate the special numbers like binomial coefficients and combinations of group elements. It has many applications in algebra, matrices like calculation of pascal triangle elements and pascal matrix formation, etc. This paper explains about its functions and properties of N-Summet-k. The result of variation between N and k is shown in tabulation.
