Spectra of certain types of polynomials and tiling of integers with translates of finite sets

We investigate Fuglede's spectral set conjecture in the special case when the set in question is a union of finitely many unit intervals in dimension 1. In this case, the conjecture can be reformulated as a statement about multiplicative properties of roots of associated with the set polynomials with (0,1) coefficients. Let be an N-term polynomial. We say that {θ₁,θ₂,…,θ_N−1} is an N-spectrum for A(x) if the θ_j are all distinct and

     

Connection coefficients for Laguerre-Sobolev orthogonal polynomials

Laguerre-Sobolev polynomials are orthogonal with respect to an inner product of the form , where α>−1, λ?0, and , the linear space of polynomials with real coefficients. If dμ(x)=xαe−xdx, the corresponding sequence of monic orthogonal polynomials {Q_n^(α,λ)(x)} has been studied by Marcellán et al. (J. Comput. Appl. Math. 71 (1996) 245-265), while if dμ(x)=δ(x)dx the sequence of monic orthogonal polynomials {L_n^(α)(x;λ)} was introduced by Koekoek and Meijer (SIAM J. Math. Anal. 24 (1993) 768-782). For each of these two families of Laguerre-Sobolev polynomials, here we give the explicit expression of the connection coefficients in their expansion as a series of standard Laguerre polynomials. The inverse connection problem of expanding Laguerre polynomials in series of Laguerre-Sobolev polynomials, and the connection problem relating two families of Laguerre-Sobolev polynomials with different parameters, are also considered.     

On the Littlewood cyclotomic polynomials

In this article, we study the cyclotomic polynomials of degree N−1 with coefficients restricted to the set {+1,−1}. By a cyclotomic polynomial we mean any monic polynomial with integer coefficients and all roots of modulus 1. By a careful analysis of the effect of Graeffe's root squaring algorithm on cyclotomic polynomials, P. Borwein and K.K. Choi gave a complete characterization of all cyclotomic polynomials with odd coefficients. They also proved that a polynomial p(x) with coefficients ±1 of even degree N−1 is cyclotomic if and only if p(x)=±Φp₁(±x)Φp₂(±xp₁)?Φpr(±xp₁p₂?pr−1), where N=p₁p₂?pr and the pi are primes, not necessarily distinct. Here is the pth cyclotomic polynomial. Based on substantial computation, they also conjectured that this characterization also holds for polynomials of odd degree with ±1 coefficients. We consider the conjecture for odd degree here. Using Ramanujan's sums, we solve the problem for some special cases. We prove that the conjecture is true for polynomials of degree α²pβ−1 with odd prime p or separable polynomials of any odd degree.     

Diophantine equations for classical continuous orthogonal polynomials

Let A, B, C denote rational numbers with AB ≠ 0 and m > n ≥ 3 arbitrary rational integers. We study the Diophantine equation AP_m(x) + Bp_n(y) = C, in x, y ? , where {Pk(x)}I is one of the three classical continuous orthogonal polynomial families, i.e. Laguerre polynomials, Jacobi polynomials (including Gegenbauer, Legendre or Chebyshev polynomials) and Hermite polynomials. We prove that with exception of the Chebyshev polynomials for all such polynomial families there are at most finitely many solutions (x, y) ? ² provided n > 4. The tools are besides the criterion [3], a theorem of Szeg— [14] on monotonicity of stationary points of polynomials which satisfy a second order Sturm-Liouville differential equation,

     

Some identities involving theta functions

Let (|q|<1). For k∈N it is shown that there exist k rational numbers A(k,0),…,A(k,k−1) such that

     

Transcendence of the log gamma function and some discrete periods

We study transcendental values of the logarithm of the gamma function. For instance, we show that for any rational number x with 0<x<1, the number logΓ(x)+logΓ(1−x) is transcendental with at most one possible exception. Assuming Schanuel's conjecture, this possible exception can be ruled out. Further, we derive a variety of results on the Γ-function as well as the transcendence of certain series of the form , where P(x) and Q(x) are polynomials with algebraic coefficients.     

Generalized Haar integral

The aim of the paper is to generalize the notion of the Haar integral. For a compact semigroup S acting continuously on a Hausdorff compact space Ω, the algebra A(S)⊂C(Ω,R) of S-invariant functions and the linear space M(S) of S-invariant (real-valued) finite signed measures are considered. It is shown that if S has a left and right invariant measure, then the dual space of A(S) is isometrically lattice-isomorphic to M(S) and that there exists a unique linear operator (called the Haar integral) such that for each f∈A(S) and for any f∈C(Ω,R) and s∈S, , where .     

(DFS)-spaces of holomorphic functions invariant under differentiation

Let be a bounded convex domain, A^−∞(G) be the (DFS)-space of all holomorphic functions of polynomial growth on G and A^∞(G) be the Fréchet space of C^∞-functions on closure of G which are holomorphic on G. With the help of the Laplace transform we describe the strong dual of A^−∞(G) and prove that A^−∞(G) is the unique (DFS)-space H such that the space A^∞(G) is contained in H, H is embedded continuously in A^−∞(G) and H is invariant under differentiation.     

Ω-results for Beurling's zeta function and lower bounds for the generalised Dirichlet divisor problem

In this paper we study generalised prime systems for which the integer counting function NP(x) is asymptotically well behaved, in the sense that NP(x)=ρx+O(xβ), where ρ is a positive constant and . For such systems, the associated zeta function ζP(s) is holomorphic for . We prove that for , for any ε>0, and also for ε=0 for all such σ except possibly one value. The Dirichlet divisor problem for generalised integers concerns the size of the error term in NkP(x)−Ress=1(ζPk(s)xs/s), which is O(xθ) for some θ<1. Letting αk denote the infimum of such θ, we show that .     

Transcendence of certain infinite products

We prove the transcendence results for the infinite product , where Ek(x), Fk(x) are polynomials, α is an algebraic number, and r?2 is an integer. As applications, we give necessary and sufficient conditions for transcendence of and , where Fn and Ln are Fibonacci numbers and Lucas numbers respectively, and {ak}_k?0 is a sequence of algebraic numbers with log‖ak‖=o(rk).     

Divisibility properties of generalized Laguerre polynomials

In this paper we give effective upper bounds for the degree k of divisors (over ?) of generalized Laguerre polynomials L^α_n(x), i.e. of for α = −tn − s − 1 and α = tn + s with t,s ∈ ?, t = O(log k), s = O(k log k) and k sufficiently large.     

Bernoulli matrix and its algebraic properties

In this paper, we define the generalized Bernoulli polynomial matrix B^(α)(x) and the Bernoulli matrix B. Using some properties of Bernoulli polynomials and numbers, a product formula of B^(α)(x) and the inverse of B were given. It is shown that not only B(x)=P[x]B, where P[x] is the generalized Pascal matrix, but also B(x)=FM(x)=N(x)F, where F is the Fibonacci matrix, M(x) and N(x) are the (n+1)×(n+1) lower triangular matrices whose (i,j)-entries are and , respectively. From these formulas, several interesting identities involving the Fibonacci numbers and the Bernoulli polynomials and numbers are obtained. The relationships are established about Bernoulli, Fibonacci and Vandermonde matrices.     

Spectrum of a non-self-adjoint operator associated with the periodic heat equation

We study the spectrum of the linear operator L=−θ_∂−?θ_∂(sinθθ_∂) subject to the periodic boundary conditions on θ∈[−π,π]. We prove that the operator is closed in with the domain in for |?|<2, its spectrum consists of an infinite sequence of isolated eigenvalues and the set of corresponding eigenfunctions is complete. By using numerical approximations of eigenvalues and eigenfunctions, we show that all eigenvalues are simple, located on the imaginary axis and the angle between two subsequent eigenfunctions tends to zero for larger eigenvalues. As a result, the complete set of linearly independent eigenfunctions does not form a basis in .     

Cores for Feller semigroups with an invariant measure

Let be an elliptic differential operator with unbounded coefficients on RN and assume that the associated Feller semigroup (T(t))_t?0 has an invariant measure μ. Then (T(t))_t?0 extends to a strongly continuous semigroup (Tp(t))_t?0 on Lp(μ)=Lp(RN,μ) for every 1?p<∞. We prove that, under mild conditions on the coefficients of A, the space of test functions is a core for the generator (Ap,Dp) of (Tp(t))_t?0 in Lp(μ) for 1?p<∞.     

On the unitary equivalence of absolutely continuous parts of self-adjoint extensions

The classical Weyl-von Neumann theorem states that for any self-adjoint operator A₀ in a separable Hilbert space H there exists a (non-unique) Hilbert-Schmidt operator C=C^? such that the perturbed operator A₀+C has purely point spectrum. We are interesting whether this result remains valid for non-additive perturbations by considering the set ExtA of self-adjoint extensions of a given densely defined symmetric operator A in H and some fixed . We show that the ac-parts and of and A₀ are unitarily equivalent provided that the resolvent difference is compact and the Weyl function M(⋅) of the pair {A,A₀} admits weak boundary limits M(t):=w-lim_y→+0M(t+iy) for a.e. t∈R. This result generalizes the classical Kato-Rosenblum theorem. Moreover, it demonstrates that for such pairs {A,A₀} the Weyl-von Neumann theorem is in general not true in the class ExtA.     

Symmetry and specializability in the continued fraction expansions of some infinite products

Let f(x)∈Z[x]. Set f₀(x)=x and, for n?1, define fn(x)=f(fn−1(x)). We describe several infinite families of polynomials for which the infinite product

     

Congruences involving Bernoulli polynomials

Let {B_n(x)} be the Bernoulli polynomials. In the paper we establish some congruences for , where p is an odd prime and x is a rational p-integer. Such congruences are concerned with the properties of p-regular functions, the congruences for and the sum , where h(d) is the class number of the quadratic field of discriminant d and p-regular functions are those functions f such that are rational p-integers and for n=1,2,3,… . We also establish many congruences for Euler numbers.     

The index formula and the spectral shift function for relatively trace class perturbations

We compute the Fredholm index, index(DA), of the operator DA=(d/dt)+A on L²(R;H) associated with the operator path , where (Af)(t)=A(t)f(t) for a.e. t∈R, and appropriate f∈L²(R;H), via the spectral shift function ξ(⋅;A₊,A₋) associated with the pair (A₊,A₋) of asymptotic operators A_±=A(±∞) on the separable complex Hilbert space H in the case when A(t) is generally an unbounded (relatively trace class) perturbation of the unbounded self-adjoint operator A₋.We derive a formula (an extension of a formula due to Pushnitski) relating the spectral shift function ξ(⋅;A₊,A₋) for the pair (A₊,A₋), and the corresponding spectral shift function ξ(⋅;H₂,H₁) for the pair of operators in this relative trace class context,This formula is then used to identify the Fredholm index of DA with ξ(0;A₊,A₋). In addition, we prove that index(DA) coincides with the spectral flow of the family {A(t)}_t∈R and also relate it to the (Fredholm) perturbation determinant for the pair (A₊,A₋): with the choice of the branch of ln(detH(⋅)) on C₊ such thatWe also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta-invariant.