On sums related to the numerator of generating functions for the <Emphasis Type="Italic">k</Emphasis>th power of Fibonacci numbers on sums related to the numerator of generating functions for the <Emphasis Type="Italic">k</Emphasis>th power of Fibonacci numbers

New results about some sums s _n (k, l) of products of the Lucas numbers, which are of similar type as the sums in [SEIBERT, J.—TROJOVSK Y, P.: On multiple sums of products of Lucas numbers, J. Integer Seq. 10 (2007), Article 07.4.5], and sums σ(k) = $ \sum\limits_{l = 0}^{\tfrac{{k - 1}} {2}} {(_l^k )F_k - 2l^S n(k,l)} $ \sum\limits_{l = 0}^{\tfrac{{k - 1}} {2}} {(_l^k )F_k - 2l^S n(k,l)} are derived. These sums are related to the numerator of generating function for the kth powers of the Fibonacci numbers. s _n (k, l) and σ(k) are expressed as the sum of the binomial and the Fibonomial coefficients. Proofs of these formulas are based on a special inverse formulas.     

On the Distribution in Residue Classes of Integers with a Fixed Sum of Digits

We give an asymptotic formula for the distribution of those integers n in a residue class, such that n has a fixed sum of base-g digits, with some uniformity over the choice of the modulus and g. We then use this formula to solve the problem of I. Niven of giving an asymptotic formula for the distribution of those integers n divisible by the sum of their base-g digits. Our results also allow us to give a stronger form of a result of M. Olivier dealing with the distribution of integers with a given gcd with their sum of base-g digits.For our friend Jean-Louis Nicolas on his sixtieth birthdayResearch partially supported by the Hungarian National Foundation for Scientific Research, Grant No. T029759, and “Balaton” French-Hungarian exchange program F-18/00.2000 Mathematics Subject Classification: Primary—11A63     

A shorter proof of Kanter’s Bessel function concentration bound

We give a shorter proof of Kanter’s (J. Multivariate Anal. 6, 222–236, 1976) sharp Bessel function bound for concentrations of sums of independent symmetric random vectors. We provide sharp upper bounds for the sum of modified Bessel functions I₀(x) + I₁(x), which might be of independent interest. Corollaries improve concentration or smoothness bounds for sums of independent random variables due to Čekanavičius & Roos (Lith. Math. J. 46, 54–91, 2006); Roos (Bernoulli, 11, 533–557, 2005), Barbour & Xia (ESAIM Probab. Stat. 3, 131–150, 1999), and Le Cam (Asymptotic Methods in Statistical Decision Theory. Springer, Berlin Heidelberg New York, 1986).      

Algebraic relations for reciprocal sums of odd terms in Fibonacci numbers

In this paper, we prove the algebraic independence of the reciprocal sums of odd terms in Fibonacci numbers ∑ _n=1^∞ F _2n−1⁻¹, ∑ _n=1^∞ F _2n−1⁻², ∑ _n=1^∞ F _2n−1⁻³ and write each ∑ _n=1^∞ F _2n−1^−s (s≥4) as an explicit rational function of these three numbers over ℚ. Similar results are obtained for various series including the reciprocal sums of odd terms in Lucas numbers.      

A Kuznetsov Formula for Kloosterman Sums on GLn

We prove a Kuznetsov sum formula for Kloosterman sums on GL _n corresponding to the big Bruhat cell. Using this formula, a weighted sum of Kloosterman sums can be expressed in spectral decomposition. A non-trivial estimate of the spectral side might lead to a proof of cancellations in sums of Kloosterman sums on GL _n.     

Une Remarque sur une Formule Sommatoire liée à la Somme des Chiffres

Let g ≥ 2 be an integer, and let s(n) be the sum of the digits of n in basis g. Let f(n) be a complex valued function defined on positive integers, such that ?_{n ￡ x} f(n)=o(x)\sum_{n\le x} f(n)=o(x) . We propose sufficient conditions on the function f to deduce the equality ?_{n ￡ x} f(s(n))=o(x)\sum_{n\le x} f(s(n))=o(x) . Applications are given, for instance, on the equidistribution mod 1 of the sequence (s(n))^α, where α is a positive real number.     

Extending analyticK-subanalytic functions

Letg:U→ℝ (U open in ℝⁿ) be an analytic and K-subanalytic (i. e. definable in ℝ _an ^K , whereK, the field of exponents, is any subfield ofℝ) function. Then the set of points, denoted Σ, whereg does not admit an analytic extension is K-subanalytic andg can be extended analytically to a neighbourhood of Ū\∑. Partially supported by the European RTN Network RAAG (contract no. HPRN-CT-00271)     

On some congruence with application to exponential sums

We will study the solution of a congruence,x ≡g ^1/2)ωg(2 ⁿ ) mod 2 ⁿ , depending on the integersg andn, where ω _g (2 ⁿ ) denotes the order ofg modulo 2 ⁿ . Moreover, we introduce an application of the above result to the study of an estimation of exponential sums.     

New multiple <Subscript>6</Subscript><Emphasis Type="Italic">ψ</Emphasis><Subscript>6</Subscript> summation formulas and related conjectures

Three new summation formulas for ₆ ψ ₆ bilateral basic hypergeometric series attached to root systems are presented. Remarkably, two of these formulae, labelled by the A_2n−1 and A_2n root systems, can be reduced to multiple ₆ φ ₅ sums generalising the well-known van Diejen sum. This latter sum serves as the weight-function normalisation for the BC _n q-Racah polynomials of van Diejen and Stokman. Two ₈ φ ₇-level extensions of the multiple ₆ φ ₅ sums, as well as their elliptic analogues, are conjectured. This opens up the prospect of constructing novel A-type extensions of the Koornwinder–Macdonald theory.     

Extremal Polynomials with Preassigned Zeros and Rational Approximants

Let p _n be the n th orthonormal polynomial with respect to a positive finite measure μ supported by Δ=[-1,1] . It is well known that, uniformly on compact subsets of C/Δ , and, for a large class of measures μ , where g _Ω (z) is Green's function of with pole at infinity. It is also well known that these limit relations give convergence of the diagonal Padé approximants of the Markov function to f on Ω with a certain geometric speed measured by g _Ω (z) . We prove corresponding results when we restrict the freedom of p _n by preassigning some of the zeros. This means that the Padé approximants are replaced by Padé-type approximants where some of the poles are preassigned. We also replace Δ by general compact subsets of C. July 12, 1995. Date revised: October 1, 1996.     

On the theorem of M Golomb

Let X ₁, …, X _n be compact spaces and X = X ₁ × … × X _n . Consider the approximation of a function ƒ ∈ C(X) by sums g ₁(x ₁)+…+g _n (x _n ), where g _i ∈ C(X _i ), i = 1, …, n. In [8], Golomb obtained a formula for the error of this approximation in terms of measures constructed on special points of X, called ‘projection cycles’. However, his proof had a gap, which was pointed out by Marshall and O’Farrell [15]. But the question if the formula was correct, remained open. The purpose of the paper is to prove that Golomb’s formula holds in a stronger form.     

Lie algebra of formal vector fields which are extended by formal g-valued functions

In this paper, we consider the infinite-dimensional Lie algebra W_n⋉g⊗O _n of formal vector fields on the n-dimensional plane which is extended by formal g-valued functions of n variables. Here g is an arbitrary Lie algebra. We show that the cochain complex of this Lie algebra is quasi-isomorphic to the quotient of the Weyl algebra of (gl _n ⊕ g) by the (2n+1)st term of the standard filtration. We consider separately the case of a reductive Lie algebra g. We show how one can use the methods of formal geometry to construct characteristic classes of bundles. For every G-bundle on an n-dimensional complex manifold, we construct a natural homomorphism from the ring A of relative cohomologies of the Lie algebra W_n⋉g⊗O _n to the ring of cohomologies of the manifold. We show that generators of the ring A are mapped under this homomorphism to characteristic classes of tangent and G-bundles. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 335, 2006, pp. 205–230.     

A Modification of the Lyons-Sullivan Discretization of Positive Harmonic Functions

A modification of the Lyons-Sullivan discretization of positive harmonic functions on a Riemannian manifold M is proposed. This modification, depending on a choice of constants C = {C _n :n = 1,2,..}, allows for constructing measures n_x^C, x ? M\nu_x^\mathbf{C},\ x\in M, supported on a discrete subset Γ of M such that for every positive harmonic function f on M

Bernoulli convolutions and an intermediate value theorem for entropies of<Emphasis Type="Italic">K</Emphasis>-partitions

We establish a strong regularity property for the distributions of the random sums Σ±λ ⁿ , known as “infinite Bernoulli convolutions”: For a.e. λ ∃ (1/2, 1) and any fixed ℓ, the conditional distribution of (w _n+1...,w _n+ℓ) given the sum Σ _n=0 ^∞ w _n λ ⁿ , tends to the uniform distribution on {±1}^ℓ asn → ∞. More precise results, where ℓ grows linearly inn, and extensions to other random sums are also obtained. As a corollary, we show that a Bernoulli measure-preserving system of entropyh hasK-partitions of any prescribed conditional entropy in [0,h]. This answers a question of Rokhlin and Sinai from the 1960’s, for the case of Bernoulli systems. The authors were partially supported by NSF grants DMS-9729992 (E. L.), DMS-9803597 (Y. P.) and DMS-0070538 (W. S.).     

Obstruction Theory and Coincidences in Positive Codimension

Let f, g ： X→Y be two maps between closed manifolds with dim X ≥ dim Y = n ≥ 3. We study the primary obstruction on（f, g） to deforming f and g to be coincidence free on the n-th skeleton of X. We give examples for which obstructions to deforming f and g to be coincidence free are detected by on （f, g）.     

Compactly Supported Distributional Solutions of Nonstationary Nonhomogeneous Refinement Equations

Let A be a matrix with the absolute values of all eigenvalues strictly larger than one, and let Z ₀ be a subset of Z such than n∈Z ₀ implies n + 1 ∈Z ₀. Denote the space of all compactly supported distributions by D′, and the usual convolution between two compactly supported distributions f and g by f*g. For any bounded sequence G _n and H _n , n∈Z ₀, in D′, define the corresponding nonstationary nonhomogeneous refinement equation Φ _n =H _n *Φ _n+1 (A·)+G _n for all n∈Z ₀ where Φ _n , n∈Z ₀, is in a bounded set of D′. The nonstationary nonhomogeneous refinement equation (*) arises in the construction of wavelets on bounded domain, multiwavelets, and of biorthogonal wavelets on nonuniform meshes. In this paper, we study the existence problem of compactly supported distributional solutions Φ _n , n∈Z ₀, of the equation (*). In fact, we reduce the existence problem to finding a bounded solution of the linear equations for all n∈Z ₀ where the matrices S _n and the vectors , n∈Z ₀, can be constructed explicitly from H _n and G _n respectively. The results above are still new even for stationary nonhomogeneous refinement equations. Received December 30, 1999, Accepted June 15, 2000     

Asymptotics of Huber-Dutter Estimators for Partial Linear Model with Nonstochastic Designs

For partial linear model Y=X~τβ_0 _(g0)(T) εwith unknown β_0∈R~d and an unknown smooth function go, this paper considers the Huber-Dutter estimators of β_0, scale σfor the errors and the function go respectively, in which the smoothing B-spline function is used. Under some regular conditions, it is shown that the Huber-Dutter estimators of β_0 and σare asymptotically normal with convergence rate n~((-1)/2) and the B-spline Huber-Dutter estimator of go achieves the optimal convergence rate in nonparametric regression. A simulation study demonstrates that the Huber-Dutter estimator of β_0 is competitive with its M-estimator without scale parameter and the ordinary least square estimator. An example is presented after the simulation study.     

Une Remarque sur une Formule Sommatoire liée à la Somme des Chiffres

Let g ≥ 2 be an integer, and let s(n) be the sum of the digits of n in basis g. Let f(n) be a complex valued function defined on positive integers, such that . We propose sufficient conditions on the function f to deduce the equality . Applications are given, for instance, on the equidistribution mod 1 of the sequence (s(n))^α, where α is a positive real number.     

The average of the values of a function at random points

The behavior of (1/N) asN→∞ is considered, wheref is a bounded measurable function on (−∞, ∞) and (S _n) _n ^=1/∞ are the partial sums of a sequence of independent and identically distributed rondom variables.     

Limsup results and LIL for partial sum processes of a Gaussian random field

Let {ξ _j ; j ∈ ℤ₊ ^d be a centered stationary Gaussian random field, where ℤ₊ ^d is the d-dimensional lattice of all points in d-dimensional Euclidean space ℝ^d, having nonnegative integer coordinates. For each j = (j ₁ , ..., jd) in ℤ₊ ^d , we denote |j| = j ₁ ... j _d and for m, n ∈ ℤ₊ ^d , define S(m, n] = Σ _m<j≤n ζ _j , σ²(|n−m|) = ES ² (m, n], S _n = S(0, n] and S ₀ = 0. Assume that σ(|n|) can be extended to a continuous function σ(t) of t > 0, which is nondecreasing and regularly varying with exponent α at b ≥ 0 for some 0 < α < 1. Under some additional conditions, we study limsup results for increments of partial sum processes and prove as well the law of the iterated logarithm for such partial sum processes. Research supported by NSERC Canada grants at Carleton University, Ottawa