Powers of the Catalan Generating Function and Lagrange''s 1770 Trinomial Equation Series

The Catalan numbers $1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862,\ldots$ are given by $C(n)=\frac{1}{n+1}\binom{2n}{n}$ for $n\geq 0$. They are named for Eugene Catalan who studied them as early as 1838. They were also found by Leonhard Euler (1758), Nicholas von Fuss (1795), and Andreas von Segner (1758). The Catalan numbers have the binomial generating function $$\mathbf{C}(z) = \sum_{n=0}^{\infty}C(n)z^n = \frac{1 - \sqrt{1-4z}}{2z}$$ It is known that powers of the generating function $\mathbf{C}(z)$ are given by $$\mathbf{C}^a(z) = \sum_{n=0}^{\infty}\frac{a}{a+2n}\binom{a+2n}{n}z^n.$$ The above formula is not as widely known as it should be. We observe that it is an immediate, simple consequence of expansions first studied by J. L. Lagrange. Such series were used later by Heinrich August Rothe in 1793 to find remarkable generalizations of the Vandermonde convolution. For the equation $x^3 - 3x + 1 =0$, the numbers $\frac{1}{2k+1}\binom{3k}{k}$ analogous to Catalan numbers occur of course. Here we discuss the history of these expansions. and formulas due to L. C. Hsu and the author.     

A Remark on Jesmanowicz' Conjecture for the Non-coprimality Case

Let a, b, c be relatively prime positive integers such that a2+ b2= c2. Je′smanowicz'conjecture on Pythagorean numbers states that for any positive integer N, the Diophantine equation(aN)x+(b N)y=(cN)zhas no positive solution(x, y, z) other than x = y = z = 2. In this paper, we prove this conjecture for the case that a or b is a power of 2.     

New series of odd non-congruent numbers Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

We determine all square-free odd positive integers n such that the 2-Selmer groups Sn and Sn of the elliptic curve En: y2 = x(x -n)(x - 2n) and its dual curve En: y2 = x3 6nx2 n2x have the smallest size: Sn = {1}, Sn = {1,2,n,2n}. It is well known that for such integer n, the rank of group En(Q) of the rational points on En is zero so that n is a non-congruent number. In this way we obtain many new series of elliptic curves En with rank zero and such series of integers n are non-congruent numbers.     

Blow-up of p-Laplacian evolution equations with variable source power

We study the blow-up and/or global existence of the following p-Laplacian evolution equation with variable source power ut(x,t)=div(|?u|~(p-2)?u)+u~(q(x)) in?×(0,T),where ? is either a bounded domain or the whole space R~N,and q(x) is a positive and continuous function defined in ? with 0q_-=inf q(x)=q(x)=sup q(x)=q_+∞.It is demonstrated that the equation with variable source power has much richer dynamics with interesting phenomena which depends on the interplay of q(x) and the structure of spatial domain ?,compared with the case of constant source power.For the case that ? is a bounded domain,the exponent p-1 plays a crucial role.If q_+p-1,there exist blow-up solutions,while if q_+p-1,all the solutions are global.If q_-p-1,there exist global solutions,while for given q_-p-1q_+,there exist some function q(x) and ? such that all nontrivial solutions will blow up,which is called the Fujita phenomenon.For the case ?=R~N,the Fujita phenomenon occurs if 1q_-=q_+=p-1+p/N,while if q_-p-1+p/N,there exist global solutions.     

OSCILLATION AND ASYMPTOTIC BEHAVIOR OF SOME SECOND-ORDER RETARDED DIFFERENTIAL EQUATIONS

In this paper, we consider the following second order retarded differential equations x″(t)+cx′(t)=qx(t-σ)-lx(t-δ) (1) x″(t)+p(t)x(t-τ)=0 (2) We give some sufficient conditions for the oscillation of all solutions of Eq. (1) in the case where q, ι, σ, δ are positive numbers and c is a real number. And also, we study the asymptotic behavior of the nonoscillatory solutions. If necessary, we give some examples to illustrate our results. At last, we study Eq. (2) with some conditions on p(t).     

COMPARISONS OF ABSOLUTELY CONVERGENT TRIGONOMETRIC SERIES

In this paper, we first discuss the methods of comparing two special absolutely convergentsine series, sinnx and sinnx. We state the theorem in.one dimensional case as follows; Theorem. Let be convergent series with nonnegative terms. SupposeThen for all x∈[0,π]If, in addition, then     

NOTES ON ERDOS' CONJECTURE

Let Xn,n 1,be a sequence of independent random variables satisfying P(Xn = O) = 1 － P(Xn = an) = 1 － 1/pn,where an,n 1,is a sequence of real numbers,andpn isthenthprime,setFN(x)=P(N∑Xnn=1 x).The authors investigate a conjecture of Erdos in probabilistic number theory and show that in order for the sequence FN to be weakly convergent,it is both sufficient and necessary that there exist three numbers x0 and x1 ＜ x2 such that limsup(FN(x2) －FN(x1)) ＞ 0 holds,and Lo =N→∞ lim FN(x0) exists.Moreover,the authors point ont that they can also obtain the same N→∞ result in the weakened case of liminfP(Xn = 0) ＞ 0.n→∞     

Romanoff theorem in a sparse set

Let A be any subset of positive integers,and P the set of all positive primes.Two of our results are:(a) the number of positive integers which are less than x and can be represented as 2k + p(resp.p-2k) with k ∈ A and p ∈ P is more than 0.03A(log x/log 2)π(x) for all sufficiently large x;(b) the number of positive integers which are less than x and can be represented as 2q + p with p,q ∈ P is(1 + o(1))π(log x/log 2)π(x).Four related open problems and one conjecture are posed.     

Building New Functions from Old

The functions we have considered so far, such as x ,x10, 10x, or logx, can be thought of as building blocks out of which we construct other,more complicated,functions. In a simple case,we can take the power functions x and x2 and the constants 3,4, and -5 to create a new function f(x) = 3x2 4x-5. Thus we have produced a quadratic function as a linear combination of power functions. In fact,we have already seen that any polynomial can be thought of as a linear combination of power functions. In this section,we investigate in more de-     

Estimates for wave and Klein-Gordon equations on modulation spaces

We prove that the fundamental semi-group eit(m 2I+|Δ|)1/2(m = 0) of the Klein-Gordon equation is bounded on the modulation space M ps,q(Rn) for all 0 < p,q ∞ and s ∈ R.Similarly,we prove that the wave semi-group eit|Δ|1/2 is bounded on the Hardy type modulation spaces μsp,q(Rn) for all 0 < p,q ∞,and s ∈ R.All the bounds have an asymptotic factor tn|1/p 1/2| as t goes to the infinity.These results extend some known results for the case of p 1.Also,some applications for the Cauchy problems related to the semi-group eit(m2I+|Δ|)1/2 are obtained.Finally we discuss the optimum of the factor tn|1/p 1/2| and raise some unsolved problems.     

Sharp power mean bounds for Sei?ert mean简

In this paper,we find the greatest value p = log2/(log π. log 2) = 1.53 ··· and the least value q = 5/3 = 1.66 ··· such that the double inequality Mp(a,b) T(a,b) Mq(a,b) holds for all a,b 0 with a = b. Here,Mp(a,b) and T(a,b) are the p-th power and Seiffert means of two positive numbers a and b,respectively.     

EXISTENCE AND NONEXISTENCE OF ENTIRE LARGE SOLUTIONS FOR SOME SEMILINEAR ELLIPTIC EQUATIONS

In this note, we consider positive entire large solutions for semilinear elliptic equations Au = p（x）f（u） in R^N with N ≥ 3. More precisely, we are interested in the link between the existence of entire large solution with the behavior of solution for --△u = p（x） in R^N. Especially for the radial case, we try to give a survey of all possible situations under Keller-Osserman type conditions.     

避免$2$字母符号模式的具有偶数个符号的带符号排列

Let Dn be the set of all signed permutations on [n] = {1,... ,n} with even signs, and let ：Dn（T） be the set of all signed permutations in Dn which avoids a set T of signed patterns. In this paper, we find all the cardinalities of the sets Dn（T） where T B2. Some of the cardinalities encountered involve inverse binomial coefficients, binomial coefficients, Catalan numbers, and Fibonacci numbers.     

ON A SECOND ORDER DISSIPATIVE ODE IN HILBERT SPACE WITH AN INTEGRABLE SOURCE TERM Dedicated to Professor Constantine M. Dafermos on the occasion of his 70th birthday

Asymptotic behaviour of solutions is studied for some second order equations including the model case x(t) + γ x˙(t) + ■Φ(x(t)) = h(t) with γ > 0 and h ∈ L 1(0,+∞;H),Φ being continuouly differentiable with locally Lipschitz continuous gradient and bounded from below.In particular when Φ is convex,all solutions tend to minimize the potential Φ as time tends to infinity and the existence of one bounded trajectory implies the weak convergence of all solutions to equilibrium points.     

Sharp constants in the doubly weighted Hardy-Littlewood-Sobolev inequality

It is well known that the doubly weighted Hardy-Littlewood-Sobolev inequality is as follows,Z Rn Z Rn f(x)g(y)|x||x.y||y|dxdy6 B(p,q,,,,n)kfkLp(Rn)kgkLq(Rn).The main purpose of this paper is to give the sharp constants B(p,q,,,,n)for the above inequality for three cases:(i)p=1 and q=1;(ii)p=1 and 1q 6∞,or 1p 6∞and q=1;(iii)1p,q∞and 1p+1q=1.In addition,the explicit bounds can be obtained for the case 1p,q∞and 1p+1q1.     

NONEXISTENCE OF POSITIVE SOLUTIONS FOR A SEMI-LINEAR EQUATION INVOLVING THE FRACTIONAL LAPLACIAN IN R~N

In this paper,we consider the semilinear equation involving the fractional Laplacian in the Euclidian space R~n:(-△)~(α/2)u(x) = f(x_n)u~p(x),x ∈ R~n(0.1)in the subcritical case with 1 p (n+α)/(n-α).Instead of carrying out direct investigations on pseudo-differential equation(0.1),we first seek its equivalent form in an integral equation as below:u(x) = ∫R~n G∞(x,y) f(y_n) u~p(y)dy,(0.2)where G∞(x,y) is the Green's function associated with the fractional Laplacian in R~n.Employing the method of moving planes in integral forms,we are able to derive the nonexistence of positive solutions for(0.2) in the subcritical case.Thanks to the equivalence,same conclusion is true for(0.1).     

Notes on the Borwein-Choi conjecture of Littlewood cyclotomic polynomials

Borwein and Choi conjectured that a polynomial P（x） with coefficients ±1 of degree N - 1 is cyclotomic iff
P（x）=±Φp1（±x）ΦP2（±x^p1）…Φpr（±x^p1p2…pr-1）,
where N = P1P2 … pτ and the pi are primes, not necessarily distinct. Here Φ（x） ：= （x^p - 1）/（x - 1） is the p-th cyclotomic polynomial. They also proved the conjecture for N odd or a power of 2. In this paper we introduce a so-called E-transformation, by which we prove the conjecture for a wider variety of cases and present the key as well as a new approach to investigate the coniecture.     

OSCILLATIONS OF NEUTRAL DIFFERENTIAL DIFFERENCE EQUATIONS

Recently the authors have studied the oscillations of some neutral differential difference equations and obtained very good results (see [1—4]). In this paper we consider the oscillations of the neutral differential difference equationd/dt[x(t)+sum from i=1 px(t-τ)+sum form j=1 to n qx(t-σ)=0, t≥t,] (*)where p, τ, q and σ (i=1, 2, …, m; j=1, 2, …, n) are positive constants. Some sufficient conditions for all solutions of (*) to oscillate are obtained. And in some ease we give neeessaxy and sufficient conditions for (*) to oscillate.     

PERTURBED PERIODIC SOLUTIONFOR BOUSSINESQ EQUATION

We consider the solution of the good Boussinesq equation Utt -Uxx ＋ Uxxxx = （U2）xx, -∞ 〈 x 〈 ∞, t ≥ 0, with periodic initial value U（x, 0） = ε（μ ＋ φ（x））, Ut（x, 0） = εψ（x）, -∞ 〈 x 〈 ∞, where μ = 0, φ（x） and ψ（x） are 2π-periodic functions with 0-average value in [0, 2π], and ε is small. A two parameter Bcklund transformation is found and provide infinite conservation laws for the good Boussinesq equation. The periodic solution is then shown to be uniformly bounded for all small ε, and the H1-norm is uniformly bounded and thus guarantees the global existence. In the case when the initial data is in the simplest form φ（x） = μ＋a sin kx, ψ（x） = b cos kx, an approximation to the solution containing two terms is constructed via the method of multiple scales. By using the energy method, we show that for any given number T 〉 0, the difference between the true solution u（x, t; ε） and the N-th partial sum of the asymptotic series is bounded by εN＋1 multiplied by a constant depending on T and N, for all -∞ 〈 x 〈 ∞, 0 ≤ ｜ε｜t ≤ T and 0 ≤ ｜ε｜≤ε0.     

LARGE DEVIATIONS AND MODERATE DEVIATIONS FOR m-NEGATIVELY ASSOCIATED RANDOM VARIABLES

M-negatively associated random variables,which generalizes the classical one of negatively associated random variables and includes m-dependent sequences as its par- ticular case,are introduced and studied.Large deviation principles and moderate devi- ation upper bounds for stationary m-negatively associated random variables are proved. Kolmogorov-type and Marcinkiewicz-type strong laws of large numbers as well as the three series theorem for m-negatively associated random variables are also given.