Zeros of polynomials embedded in an orthogonal sequence

Let $\{x_{k,n}\}_{k=1}^n$ and $\{x_{k,n+1}\}_{k=1}^{n+1}$ , n?????, be two given sets of real distinct points with x _1,n?+?1?<?x _1,n ?<?x _2,n?+?1?<?...?<?x _n,n ?<?x _n?+?1,n?+?1. Wendroff (cf. Proc Am Math Soc 12:554?C555, 1961) proved that if $p_n(x)=\displaystyle{\prod\limits_{k=1}^n(x-x_{k,n})}$ and $p_{n+1}(x)=\displaystyle \prod\limits_{k=1}^{n+1}(x-x_{k,n+1})$ then p _n and p _n?+?1 can be embedded in a non-unique infinite monic orthogonal sequence $\{p_n\}_{n=0}^{\infty}$ . We investigate the connection between the zeros of p _n?+?2 and the two coefficients b _n?+?1????? and ?? _n?+?1?>?0, which are chosen arbitrarily, that define p _n?+?2 via the three term recurrence relation $$ p_{n+2}(x)=(x-b_{n+1})p_{n+1}(x)-\lambda_{n+1}p_n(x). $$      

Identities for Bernoulli polynomials and Bernoulli numbers

We prove that if m and \({\nu}\) are integers with \({0 \leq \nu \leq m}\) and x is a real number, then

1. $$\sum_{k=0 \atop k+m \, \, odd}^{m-1} {m \choose k}{k+m \choose \nu} B_{k+m-\nu}(x) = \frac{1}{2} \sum_{j=0}^m (-1)^{j+m} {m \choose j}{j+m-1 \choose \nu} (j+m) x^{j+m-\nu-1},$$ where B _n (x) denotes the Bernoulli polynomial of degree n. An application of (1) leads to new identities for Bernoulli numbers B _n . Among others, we obtain
2. $$\sum_{k=0 \atop k+m \, \, odd}^{m -1} {m \choose k}{k+m \choose \nu} {k+m-\nu \choose j}B_{k+m-\nu-j} =0 \quad{(0 \leq j \leq m-2-\nu)}. $$ This formula extends two results obtained by Kaneko and Chen-Sun, who proved (2) for the special cases j = 1, \({\nu=0}\) and j = 3, \({\nu=0}\) , respectively.

     

On multiplicative congruences

Let ${\varepsilon}$ be a fixed positive quantity, m be a large integer, x _j denote integer variables. We prove that for any positive integers N ₁, N ₂, N ₃ with ${N_1N_2N_3 > m^{1+\varepsilon}, }$ the set $$\{x_1x_2x_3 \quad ({\rm mod}\,m): \quad x_j\in [1,N_j]\}$$ contains almost all the residue classes modulo m (i.e., its cardinality is equal to m + o(m)). We further show that if m is cubefree, then for any positive integers N ₁, N ₂, N ₃, N ₄ with ${ N_1N_2N_3N_4 > m^{1+\varepsilon}, }$ the set $$\{x_1x_2x_3x_4 \quad ({\rm mod}\,m): \quad x_j\in [1,N_j]\}$$ also contains almost all the residue classes modulo m. Let p be a large prime parameter and let ${p > N > p^{63/76+\varepsilon}.}$ We prove that for any nonzero integer constant k and any integer ${\lambda\not\equiv 0 \,\, ({\rm mod}\,p)}$ the congruence $$p_1p_2(p_3+k)\equiv \lambda \quad ({\rm mod}\, p) $$ admits (1 + o(1))π(N)³/p solutions in prime numbers p ₁, p ₂, p ₃ ≤ N.     

Traces of Permuting n-Additive Maps and Permuting n-Derivations of Rings

Let R be a prime ring with center Z(R). For a fixed positive integer n, a permuting n-additive map ${\Delta : R^n \to R}$ is known to be permuting n-derivation if ${\Delta(x_1, x_2, \ldots, x_i x'_{i},\ldots, x_n) = \Delta(x_1, x_2, \ldots, x_i, \ldots, x_n)x'_i + x_i \Delta(x_1, x_2, \ldots, x'_i, \ldots, x_n)}$ holds for all ${x_i, x'_i \in R}$ . A mapping ${\delta : R \to R}$ defined by δ(x) = Δ(x, x, . . . ,x) for all ${x \in R}$ is said to be the trace of Δ. In the present paper, we have proved that a ring R is commutative if there exists a permuting n-additive map ${\Delta : R^n \to R}$ such that ${xy + \delta(xy) = yx + \delta(yx), xy- \delta(xy) = yx - \delta(yx), xy - yx = \delta(x) \pm \delta(y)}$ and ${xy + yx = \delta(x) \pm \delta(y)}$ holds for all ${x, y \in R}$ . Further, we have proved that if R is a prime ring with suitable torsion restriction then R is commutative if there exist non-zero permuting n-derivations Δ₁ and Δ₂ from ${R^n \to R}$ such that Δ₁(δ ₂(x), x, . . . ,x) =  0 for all ${x \in R,}$ where δ ₂ is the trace of Δ₂. Finally, it is shown that in a prime ring R of suitable torsion restriction, if ${\Delta_1, \Delta_2 : R^n \longrightarrow R}$ are non-zero permuting n-derivations with traces δ ₁, δ ₂, respectively, and ${B : R^n \longrightarrow R}$ is a permuting n-additive map with trace f such that δ ₁ δ ₂(x) =  f(x) holds for all ${x \in R}$ , then R is commutative.     

Convergence of weighted averages of noncommutative martingales

Let x = (x _n ) _n?1 be a martingale on a noncommutative probability space ( $\mathcal{M}$ , τ) and (w _n ) _n?1 a sequence of positive numbers such that $W_n = \sum\nolimits_{k = 1}^n {w_k \to \infty } $ as n → ∞. We prove that x = (x _n ) _n?1 converges bilaterally almost uniformly (b.a.u.) if and only if the weighted average (σ _n (x)) _n?1 of x converges b.a.u. to the same limit under some condition, where σ _n (x) is given by $\sigma _n (x) = \frac{1} {{W_n }}\sum\limits_{k = 1}^n {w_k x_k } ,n = 1,2,... $ Furthermore, we prove that x = (x _n ) _n?1 converges in L _p ( $\mathcal{M}$ ) if and only if (σ _n (x)) _n?1 converges in L _p ( $\mathcal{M}$ ), where 1 ? p < ∞. We also get a criterion of uniform integrability for a family in L ₁( $\mathcal{M}$ ).     

Limit ultraspherical series and their approximation properties

We study new series of the form $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ in which the general term $f_k^{ - 1} \hat P_k^{ - 1} (x)$ , k = 0, 1, …, is obtained by passing to the limit as α→?1 from the general term $\hat f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)$ of the Fourier series $\sum\nolimits_{k = 0}^\infty {f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)} $ in Jacobi ultraspherical polynomials $\hat P_k^{\alpha ,\alpha } (x)$ generating, for α> ?1, an orthonormal system with weight (1 ? x ₂)α on [?1, 1]. We study the properties of the partial sums $S_n^{ - 1} (f,x) = \sum\nolimits_{k = 0}^n {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ of the limit ultraspherical series $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ . In particular, it is shown that the operator S _n ^?1 (f) = S _n ^?1 (f, x) is the projection onto the subspace of algebraic polynomials p _n = p _n (x) of degree at most n, i.e., S _n (p _n ) = p _n ; in addition, S _n ^?1 (f, x) coincides with f(x) at the endpoints ±1, i.e., S _n ^?1 (f,±1) = f(±1). It is proved that the Lebesgue function Λ _n (x) of the partial sums S _n ^?1 (f, x) is of the order of growth equal to O(ln n), and, more precisely, it is proved that $\Lambda _n (x) \leqslant c(1 + \ln (1 + n\sqrt {1 - x^2 } )), - 1 \leqslant x \leqslant 1$ .     

Euler–Maclaurin Expansions for Integrals with Arbitrary Algebraic-Logarithmic Endpoint Singularities

In this paper, we provide the Euler?CMaclaurin expansions for (offset) trapezoidal rule approximations of the finite-range integrals $I[f]=\int^{b}_{a}f(x)\,dx$ , where f??C ^??(a,b) but can have general algebraic-logarithmic singularities at one or both endpoints. These integrals may exist either as ordinary integrals or as Hadamard finite part integrals. We assume that f(x) has asymptotic expansions of the general forms where $\widehat{P}(y),P_{s}(y)$ and $\widehat{Q}(y),Q_{s}(y)$ are polynomials in y. The ?? _s and ?? _s are distinct, complex in general, and different from ?1. They also satisfy The results we obtain in this work extend the results of a recent paper [A.?Sidi, Numer. Math. 98:371?C387, 2004], which pertain to the cases in which $\widehat{P}(y)\equiv0$ and $\widehat{Q}(y)\equiv0$ . They are expressed in very simple terms based only on the asymptotic expansions of f(x) as x??a+ and x??b?. The results we obtain in this work generalize, and include as special cases, all those that exist in the literature. Let $D_{\omega}=\frac{d}{d\omega}$ , h=(b?a)/n, where n is a positive integer, and define $\check{T}_{n}[f]=h\sum^{n-1}_{i=1}f(a+ih)$ . Then with $\widehat{P}(y)=\sum^{\hat{p}}_{i=0}{\hat{c}}_{i}y^{i}$ and $\widehat{Q}(y)=\sum^{\hat{q}}_{i=0}{\hat{d}}_{i}y^{i}$ , one of these results reads where ??(z) is the Riemann Zeta function and ?? _i are Stieltjes constants defined via $\sigma_{i}= \lim_{n\to\infty}[\sum^{n}_{k=1}\frac{(\log k)^{i}}{k}-\frac{(\log n)^{i+1}}{i+1}]$ , i=0,1,???.     

Combinational properties of sets of residues modulo a prime and the Erdős—Graham problem

Consider an arbitrary ε > 0 and a sufficiently large prime p > 2. It is proved that, for any integer a, there exist pairwise distinct integers x ₁, x ₂, ..., x _N , where N = 8([1/ε + 1/2] + 1)² such that 1 ≤ x _i ≤ p ^ε, i = 1, ..., N, and $$a \equiv x_1^{ - 1} + \cdots + x_N^{ - 1} (\bmod p)$$ , where x _i ^?1 is the least positive integer satisfying x _i ^?1 x _i ≡ 1 (modp). This improves a result of Sparlinski.     

Tail homogeneity of invariant measures of multidimensional stochastic recursions in a critical case

We consider the stochastic recursion ${X_{n+1} = M_{n+1}X_{n} + Q_{n+1}, (n \in \mathbb{N})}$ , where ${Q_n, X_n \in \mathbb{R}^d }$ , M _n are similarities of the Euclidean space ${ \mathbb{R}^d }$ and (Q _n , M _n ) are i.i.d. We study asymptotic properties at infinity of the invariant measure for the Markov chain X _n under assumption ${\mathbb{E}{[\log|M|]}=0}$ i.e. in the so called critical case.     

О приближении функци й средними Чезаро вто рого

Let σ _n ² (f, x) be the Cesàro means of second order of the Fourier expansion of the function f. Upper bounds of the deviationf(x)-σ _n ² (f, x) are studied in the metricC, while f runs over the class \(\bar W^1 C\) , i. e., of the deviation $$F_n^2 (\bar W^1 ,C) = \mathop {\sup }\limits_{f \in \bar W^1 C} \left\| {f(x) - \sigma _n^2 (f,x)} \right\|_c$$ . It is proved that the function $$g^* (x) = \frac{4}{\pi }\mathop \sum \limits_{v = 0}^\infty ( - 1)^v \frac{{\cos (2v + 1)x}}{{(2v + 1)^2 }}$$ , for whichg ^*′(x)=sign cosx, satisfies the following asymptotic relation: $$F_n^2 (\bar W^1 ,C) = g^* (0) - \sigma _n^2 (g^* ,0) + O\left( {\frac{1}{{n^4 }}} \right)$$ , i.e.g ^* is close to the extremal function. This makes it possible to find some of the first terms in the asymptotic formula for \(F_n^2 (\bar W^1 ,C)\) asn → ∞. The corresponding problem for approximation in the metricL is also considered.     

Bifurcation of the Point of Equilibrium in Systems with Zero Roots of the Characteristic Equation

We consider the following real autonomous system of 2d differential equations with a small positive parameter ε: $\dot x_i = x_{i + d} + X_i^{(n + 1)} (x,\varepsilon ), \dot x_{i + d} = - x_i^{2n - 1} + X_{i + d}^{(2n)} (x,\varepsilon ), i = 1,...,d,$ where d≥ 2, n≥2, and the $X_j^{(k)} $ are continuous functions continuously differentiable with respect to x and ε the required number of times in the neighborhood of zero; their expansion begins with order k if we assume that the variables x _i are of first order of smallness, ε is of second order, and the variables x _i+d are of order n. We write out a finite number of explicit conditions on the coefficients of the lower terms in the expansion of the right-hand side of this system guaranteeing that for any sufficiently small ε> 0 the system has one or several d-dimensional invariant tori with infinitely small frequencies of motions on them.     

A hybrid mean value involving two-term exponential sums and polynomial character sums

Let q ? 3 be a positive integer. For any integers m and n, the two-term exponential sum C(m, n, k; q) is defined by \(C(m,n,k;q) = \sum\limits_{a = 1}^q {e((ma^k + na)/q)} \) , where \(e(y) = e^{2\pi iy} \) . In this paper, we use the properties of Gauss sums and the estimate for Dirichlet character of polynomials to study the mean value problem involving two-term exponential sums and Dirichlet character of polynomials, and give an interesting asymptotic formula for it.     

Siegel’s Lemma and Sum-Distinct Sets

Let L(x)=a ₁ x ₁+a ₂ x ₂+???+a _n x _n , n≥2, be a linear form with integer coefficients a ₁,a ₂,…,a _n which are not all zero. A basic problem is to determine nonzero integer vectors x such that L(x)=0, and the maximum norm ‖x‖ is relatively small compared with the size of the coefficients a ₁,a ₂,…,a _n . The main result of this paper asserts that there exist linearly independent vectors x ₁,…,x _n?1∈? ⁿ such that L(x _i )=0, i=1,…,n?1, and $$\|{\mathbf{x}}_{1}\|\cdots\|{\mathbf{x}}_{n-1}\|<\frac{\|{\mathbf{a}}\|}{\sigma_{n}},$$ where a=(a ₁,a ₂,…,a _n ) and $$\sigma_{n}=\frac{2}{\pi}\int_{0}^{\infty}\left(\frac{\sin t}{t}\right)^{n}\,dt.$$ This result also implies a new lower bound on the greatest element of a sum-distinct set of positive integers (Erdös–Moser problem). The main tools are the Minkowski theorem on successive minima and the Busemann theorem from convex geometry.     

Freud's conjecture and approximation of reciprocals of weights by polynomials

LetW(x) be a function that is nonnegative inR, positive on a set of positive measure, and such that all power moments ofW ² (x) are finite. Let {p _n (W ²;x)} ₀ ^∞ denote the sequence of orthonormal polynomials with respect to the weightW ², and let {α _n } ₁ ^∞ and {β _n } ₁ ^∞ denote the coefficients in the recurrence relation $$xp_n (W^2 ,x) = \alpha _{n + 1} p_{n + 1} (W^2 ,x) + \beta _n p_n (W^2 ,x) + \alpha _n p_{n - 1} (W^2 ,x).$$ We obtain a sufficient condition, involving mean approximation ofW ^?1 by reciprocals of polynomials, for $$\mathop {\lim }\limits_{n \to \infty } {{\alpha _n } \mathord{\left/ {\vphantom {{\alpha _n } {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }} = \tfrac{1}{2}and\mathop {\lim }\limits_{n \to \infty } {{\beta _n } \mathord{\left/ {\vphantom {{\beta _n } {c_{n + 1} }}} \right. \kern-\nulldelimiterspace} {c_{n + 1} }} = 0,$$ wherec _{n 1} ^∞ is a certain increasing sequence of positive numbers. In particular, we obtain a sufficient condition for Freud's conjecture associated with weights onR.     

Die metrisierbaren linearen Teilräume des Raumes\mathfrak{O}_M von L. Schwartz

In Schwartz' terminology, a real or complex valued functionf, defined and infinitely differentiable on ? _n , belongs to \(\mathfrak{O}_M \) iff, as well as any of its derivatives, is at most of polynomial growth. The topology of \(\mathfrak{O}_M \) is defined by the seminorms sup{∣?(x)D ^p f(x)∣;x∈? ⁿ }, where ? belongs to \(\mathfrak{S}\) andD ^p is any derivative. It is well-known that \(\mathfrak{O}_M \) is non-metrisable. For any μ: ? ⁿ →?, let \(\mathfrak{B}_\mu \) be the space of all infinitely differentiable functionsf satisfying, for eachp, sup{∣(1+∣x∣²)^?μ(p) D ^p f(x)∣;x∈? ⁿ }<∞, with the obvious topology. These spaces, which are of very little use elsewhere in the theory of distributions, can be conveniently applied to characterise the metrisable linear subspaces of \(\mathfrak{O}_M \) : A linear subspace of \(\mathfrak{O}_M \) is metrisable if and only if it is, algebraically and topologically, a subspace of some \(\mathfrak{B}_\mu \) .     

Higher Order Coherent Pairs

In this paper, we study necessary and sufficient conditions for the relation $$\begin{array}{@{}l}P_n^{{[r]}}(x) + a_{n-1,r} P_{n-1}^{{[r]}}(x)= R_{n-r}(x) + b_{n-1,r} R_{n-r-1}(x),\\[5pt]\quad a_{n-1,r}\neq0,\ n\geq r+1,\end{array}$$ where {P _n (x)} _n??0 and {R _n (x)} _n??0 are two sequences of monic orthogonal polynomials with respect to the quasi-definite linear functionals $\mathcal{U},\mathcal{V}$ , respectively, or associated with two positive Borel measures ?? ₀,?? ₁ supported on the real line. We deduce the connection with Sobolev orthogonal polynomials, the relations between these functionals as well as their corresponding formal Stieltjes series. As sake of example, we find the coherent pairs when one of the linear functionals is classical.     

On the saturation of subfields of invariants of finite groups

Every subfield $ \mathbb{K} $ (φ) of the field of rational fractions $ \mathbb{K} $ (x ₁,..., x _n ) is contained in a unique maximal subfield of the form $ \mathbb{K} $ (ω). The element ω is said to be generating for the element φ. A subfield of $ \mathbb{K} $ (x ₁,..., x _n ) is said to be saturated if, together with every its element, the subfield also contains the generating element. In the paper, the saturation property is studied for the subfields of invariants $ \mathbb{K} $ (x ₁,..., x _n ) ^G of a finite group G of automorphisms of the field $ \mathbb{K} $ (x ₁..., x _n ).     

On the order of the error function of the square-full integers

LetL(x) denote the number of square full integers ≤x. By a square-full integer, we mean a positive integer all of whose prime factors have multiplicity at least two. It is well known that $$\left. {L(x)} \right| \sim \frac{{\zeta ({3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2})}}{{\zeta (3)}}x^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + \frac{{\zeta ({2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3})}}{{\zeta (2)}}x^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} ,$$ where ζ(s) denotes the Riemann Zeta function. Let Δ(x) denote the error function in the asymptotic formula forL(x). On the basis of the Riemann hypothesis (R.H.), it is known that \(\Delta (x) = O(x^{\tfrac{{13}}{{81}} + \varepsilon } )\) for every ε>0. In this paper, we prove the following results on the assumption of R.H.: (1) $$\frac{1}{x}\int\limits_1^x {\Delta (t)dt} = O(x^{\tfrac{1}{{12}} + \varepsilon } ),$$ (2) $$\int\limits_1^x {\frac{{\Delta (t)}}{t}\log } ^{v - 1} \left( {\frac{x}{t}} \right) = O(x^{\tfrac{1}{{12}} + \varepsilon } )$$ for any integer ν≥1. In fact, we prove some general results and deduce the above from them. On the basis of (1) and (2) above, we conjecture that \(\Delta (x) = O(x^{{1 \mathord{\left/ {\vphantom {1 {12}}} \right. \kern-0em} {12}} + \varepsilon } )\) under the assumption of R.H.     

On the distinctness of modular reductions of primitive sequences over Z/(232−1)

This paper studies the distinctness of modular reductions of primitive sequences over ${\mathbf{Z}/(2^{32}-1)}$ . Let f(x) be a primitive polynomial of degree n over ${\mathbf{Z}/(2^{32}-1)}$ and H a positive integer with a prime factor coprime with 2³²?1. Under the assumption that every element in ${\mathbf{Z}/(2^{32}-1)}$ occurs in a primitive sequence of order n over ${\mathbf{Z}/(2^{32}-1)}$ , it is proved that for two primitive sequences ${\underline{a}=(a(t))_{t\geq 0}}$ and ${\underline{b}=(b(t))_{t\geq 0}}$ generated by f(x) over ${\mathbf{Z}/(2^{32}-1), \underline{a}=\underline{b}}$ if and only if ${a\left( t\right) \equiv b\left( t\right) \bmod{H}}$ for all t ≥ 0. Furthermore, the assumption is known to be valid for n between 7 and 100, 000, the range of which is sufficient for practical applications.     

On definability in some lattices of semigroup varieties

An identity of the form x ₁?x _n ??x _1?? x _2?? ?x _n?? where ?? is a non-trivial permutation on the set {1,??,n} is called a permutation identity. If u??v is a permutation identity, then ?(u??v) [respectively r(u??v)] is the maximal length of the common prefix [suffix] of the words u and v. A variety that satisfies a permutation identity is called permutative. If $\mathcal{V}$ is a permutative variety, then $\ell=\ell(\mathcal{V})$ [respectively $r=r(\mathcal{V})$ ] is the least ? [respectively r] such that $\mathcal{V}$ satisfies a permutation identity ?? with ?(??)=? [respectively r(??)=r]. A?variety that consists of nil-semigroups is called a nil-variety. If ?? is a set of identities, then $\operatorname {var}\varSigma$ denotes the variety of semigroups defined by ??. If $\mathcal{V}$ is a variety, then $L (\mathcal{V})$ denotes the lattice of all subvarieties of $\mathcal{V}$ . For ?,r??0 and n>1 let $\mathfrak{B}_{\ell,r,n}$ denote the set that consists of n! identities of the form $$t_1\cdots t_\ell x_1x_2 \cdots x_n z_{1}\cdots z_{r}\approx t_1\cdots t_\ell x_{1\pi}x_{2\pi} \cdots x_{n\pi}z_{1}\cdots z_{r}, $$ where ?? is a permutation on the set {1,??,n}. We prove that for each permutative nil-variety $\mathcal{V}$ and each $\ell\ge\ell(\mathcal{V})$ and $r\ge r(\mathcal{V})$ there exists n>1 such that $\mathcal{V}$ is definable by a first-order formula in $L(\operatorname{var}{\mathfrak{B}}_{l,r,n})$ if ???r or $\mathcal{V}$ is definable up to duality in $L(\operatorname{var}{\mathfrak{B}}_{\ell,r,n})$ if ?=r.