Two Kinds of Series Involving the Reciprocals of Binomial Coefficients

By applying the theory of formal power series,the author obtains the closed forms for two kinds of infinite series involving the reciprocals of binomial coefficients,and the author gets another closed form for the infinite series Σr≥m tn＋r/（n＋rr）.

On the Residues of Binomial Coefficients and Their Products Modulo Prime Powers

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.

Quasi-binomial coefficients stemming from Nakayama algebras

It is known that there is a very closed connection between the set of non-isomorphic indecomposable basic Nakayama algebras and the set of admissible sequences.To determine the cardinal number of all nonisomorphic indecomposable basic Nakayama algebras,we describe the cardinal number of the set of all t-length admissible sequences using a new type of integers called quasi-binomial coefficients.Furthermore,we find some intrinsic relations among binomial coefficients and quasi-binomial coefficients.

Identities Involving Powers and Inverse of Binomial Coefficients

In this paper,we give several identities of finite sums and some infinite series involving powers and inverse of binomial coefficients,which extends the results of T.Trif.

A Journey into Fermat＇s Equation

As expounded in some recent mathematical conferences, this research on that amazing source of algebraic ideas known as Fermat＇s equation is aimed to prove how Fermat triples can be limited until the impossible existence through a criterion of incompatible parities related to unexplored properties of the binomial coefficients. In this paper, the authors use a technique based on the analysis of four numbers and their internal relations with three basic compulsory factors. It leads to the practical impossibility to find any triple of natural numbers candidate to satisfy Fermat＇s equation, because when the authors try to meet a condition between parity and range the authors are compelled to violate the other one, so that they are irreducibly alternative. In particular, there is a parity violation when the authors choose all the basic factors in the allowed range and the authors obtain exceeding values of one of the involved variables when the authors try to restore the parity. Since Fermat＇s last theorem would consequently be demonstrated, many readers could recall the never found elementary proof of FLT （Fermat＇s last theorem） claimed by Pierre de Fermat. The authors are not encouraging such an interpretation because this paper is intended as a journey into Fermat＇s equation and the reader＇s attitude should be towards the algebraic achievements here proposed, with their possible hidden flaws and future developments, rather than to legendary problems like Fermat＇s riddle.

Analytical evaluation of the plasma dispersion function for a Fermi-Dirac distribution

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.

N-Summet-k and Its Application in the Construction of Pascal Triangle and Pascal Matrix

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.

CONVERGENCE OF AN IMMERSED INTERFACE UPWIND SCHEME FOR LINEAR ADVECTION EQUATIONS WITH PIECEWISE CONSTANT COEFFICIENTS Ⅱ: SOME RELATED BINOMIAL COEFFICIENT INEQUALITIES

In this paper we give proof of three binomial coefficient inequalities. These inequalities are key ingredients in [Wen and Jin, J. Comput. Math. 26, （2008）, 1-22] to establish the L^1-error estimates for the upwind difference scheme to the linear advection equations with a piecewise constant wave speed and a general interface condition, which were further used to establish the L^1-error estimates for a Hamiltonian-preserving scheme developed in [Jin and Wen, Commun. Math. Sci. 3, （2005）, 285-315] to the Liouville equation with piecewise constant potentials [Wen and Jin, SIAM J. Numer. Anal. 46, （2008）, 2688-2714].

Partial Bell Polynomials, Falling and Rising Factorials, Stirling Numbers, and Combinatorial Identities

In the paper,the authors collect,discuss,and find out several connections,equivalences,closed-form formulas,and combinatorial identities concerning partial Bell polynomials,falling factorials,rising factorials,extended binomial coefficients,and the Stirling numbers of the first and second kinds.These results are new,interesting,important,useful,and applicable in combinatorial number theory.

Generalized Trigonometric Power Sums Covering the Full Circle

The analytical calculation of the area moments of inertia used for special mechanical tests in materials science and further generalizations for moments of different orders and broader symmetry properties has led to a new type of trigonometric power sums. The corresponding generalized equations are presented, proven, and their characteristics discussed. Although the power sums have a basic form, their results have quite different properties, dependent on the values of the free parameters used. From these equations, a large variety of power reduction formulas can be derived. This is shown by some examples.

Super congruences and Euler numbers

Let p 〉 3 be a prime. A p-adic congruence is called a super congruence if it happens to hold modulo some higher power of p. The topic of super congruences is related to many fields including Gauss and Jacobi sums and hypergeometric series. We prove that ∑k=0^p-1（k^2k/2k）≡（-1）^（p-1）/2-p^2Ep-3（modp^3） ∑k=1^（p-1）/2（k^2k）/k≡（-1）^（p＋1）/2 8/3pEp-3（mod p^2）,∑k=0^（p-1）/2（k^2k）^2/16k≡（-1）^（p-1）/2＋p^2Ep-3（mod p^3）,where E0, E1, E2,... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for π2, π-2 and the constant K ：= ∑k=1^∞（k/3）/k^2 （with （-） the Jacobi symbol）, two of which are ∑k=1^∞（10k-3）8k/k2（k^2k）^2（k^3k）=π^2/2and ∑k=1^∞（15k-4）（-27）^k-1/k^3（k^2k）^2（k^3k）=K.

On a Supercongruence Conjecture of Z.-W.Sun

In this paper,the author partly proves a supercongruence conjectured by Z.-W.Sun in 2013.Let p be an odd prime and let a∈Z^(+).Then,if p≡1(mod 3),[5/6p^(a)]∑k=0(2kk)/16^(k)≡(3/p^(a))(mod p^(2))is obtained,where(■)is the Jacobi symbol.