On the Rédei zeta function

Let L be a locally finite lattice. An order function ν on L is a function defined on pairs of elements x, y (with x ≤ y) in L such that ν(x, y) = ν(x, z) ν(z, y). The Rédei zeta function of L is given by ?(s; L) = Σ_x∈Lμ(Ô, x) ν(Ô, x)^?s. It generalizes the following functions: the chromatic polynomial of a graph, the characteristic polynomial of a lattice, the inverse of the Dedekind zeta function of a number field, the inverse of the Weil zeta function for a variety over a finite field, Philip Hall's φ-function for a group and Rédei's zeta function for an abelian group. Moreover, the paradigmatic problem in all these areas can be stated in terms of the location of the zeroes of the Rédei zeta function.     

Ramsey graphs and block designs

Symmetric (ν, κ, λ)-block designs admitting polarity maps are shown to be closely related to certain Ramsey numbers for bipartite graphs. In particular, if there exists a (ν, κ, λ)-difference set in an abelian group of order ν, then the Ramsey number R(K_2,λ+1, K_1,ν?k+1) is either 1 + ν or 2 + ν.     

Polynomial approximations in the complex plane

Good polynomial approximations for analytic functions are potentially useful but are in short supply. A new approach introduced here involves the Lanczos τ-method, with perturbations proportional to Faber or Chebyshev polynomials for specific regions of the complex plane. The results show that suitable forms of the τ-method, which are easy to use, can produce near-minimax polynomial approximations for functions which satisfy linear differential equations with polynomial coefficients. In particular, some accurate approximations of low degree for Bessel functions are presented. An appendix describes a simple algorithm which generates polynomial approximations for the Bessel function J_ν(z) of any given order ν.     

Some Indeterminate Moment Problems and Freud-Like Weights

We study two indeterminate Hamburger moment problems and the corresponding orthogonal polynomials. The coefficients in their recurrence relations are of exponential growth or are polynomials of degree 2. The entire functions in the Nevanlinna parametrization are found. The orthogonal polynomials with polynomial recurrence coefficients resemble the Freud polynomials with a = 1/2 . Inequalities are given for the largest zero and the asymptotic behavior of the largest zero is established. April 24, 1996. Date revised: March 3, 1997.     

Asymptotically extremal polynomials with respect to varying weights and application to Sobolev orthogonality

We study the asymptotic behavior of the zeros of a sequence of polynomials whose weighted norms, with respect to a sequence of weight functions, have the same nth root asymptotic behavior as the weighted norms of certain extremal polynomials. This result is applied to obtain the (contracted) weak zero distribution for orthogonal polynomials with respect to a Sobolev inner product with exponential weights of the form e−φ(x), giving a unified treatment for the so-called Freud (i.e., when φ has polynomial growth at infinity) and Erdös (when φ grows faster than any polynomial at infinity) cases. In addition, we provide a new proof for the bound of the distance of the zeros to the convex hull of the support for these Sobolev orthogonal polynomials.     

Une série génératrice des formes paraboliques de Maass

A certain periodic function F_μ,ν(z), an eigenfunction of the Laplacian on the upper half-plane with respect to z depending on some parameters μ, ν, is not automorphic, but the function μ → F_μ,ν(z)−F_μ,ν(−1/0 extends to a larger domain than the function F_μ,ν(z) itself. Consequently, at the poles of this latter function of μ, the coefficients of the polar parts provide non-analytic modular forms: all Maass cusp forms are finite linear combinations of forms obtained in this way, allowing the second parameter ν to vary     

Authentication perpendicular arrays APA1(2, 7, ν)

Stinson introduced authentication perpendicular arrays APA_λ (t, k,ν), as a special kind of perpendicular arrays, to construct authentication and secrecy codes. In this article, we generalize the known direct and recursive constructions and show that an APA₁(2, 7, ν) exists if νodd > 9384255. We also improve the known existence result for APA₁(2, 5, ν) by presenting five new orders so that such a design exists if and only if ν ≥ 5 is odd, except &ngr: = 7 and possibly excepting ν = 9, 13, 15, 17, 33, 39, 49, 57, 63, 69, 87, 97, and 113. © 1997 John Wiley & Sons, Inc.     

Some Orthogonal Polynomials Related to Elliptic Functions

We characterize the orthogonal polynomials in a class of polynomials defined through their generating functions. This led to three new systems of orthogonal polynomials whose generating functions and orthogonality relations involve elliptic functions. The Hamburger moment problems associated with these polynomials are indeterminate. We give infinite families of weight functions in each case. The different polynomials treated in this work are also polynomials in a parameter and as functions of this parameter they are orthogonal with respect to unique measures, which we find explicitly. Through a quadratic transformation we find a new exactly solvable birth and death process with quartic birth and death rates.     

Multiple Orthogonal Polynomials Associated with the Modified Bessel Functions of the First Kind

Abstract. We consider polynomials which are orthogonal with respect to weight functions, which are defined in terms of the modified Bessel function I _ν and which are related to the noncentral χ ² -distribution. It turns out that it is the most convenient to use two weight functions with indices ν and ν+1 and to study orthogonality with respect to these two weights simultaneously. We show that the corresponding multiple orthogonal polynomials of type I and type II exist and give several properties of these polynomials (differential properties, Rodrigues formula, explicit formulas, recurrence relation, differential equation, and generating functions).     

Interpolation transfinie sur le triangle,le tétraèdre et le pentaèdre. Application à la création de maillages et à la condition de Dirichlet

The Lagrange's interpolation L(ν) of a function ν on a polyhedron K is defined from the values of the function at points p_i of the boundary Γ(K) and ℒ(ν)(p_i) = ν(p_i) at these points. The transfinite interpolation is defined in the same way, but on the whole boundary Γ(K). Three new transfinite interpolations of degree one on the triangle, tetrahedron, and pentahedron are added to the Gordon and Hall's transfinite interpolations on the quadrangle and hexaedron. All these interpolations leave the polynomials of degree ≤1 invariant. The applications to the generation of regular meshes and the possibility of exactly taking into account the Dirichlet boundary condition, show the usefulness of these transfinite interpolations.     

Skew-orthogonal steiner triple systems

Two Steiner triple systems, S₁=(V,ℬ︁₁) and S₂=(V,ℬ︁₂), are orthogonal (S₁ ⟂ S₂) if ℬ︁₁ ∩ ℬ︁₂=∅︁ and if {u,ν} ≠ {x,y}, uνw,xyw ∈ ℬ︁₁, uνs, xyt ∈ ℬ︁₂ then s ≠ t. The solution to the existence problem for orthogonal Steiner triple systems, (OSTS) was a major accomplishment in design theory. Two orthogonal triple systems are skew-orthogonal (SOSTS, written S₁∼S₂) if, in addition, we require uνw, xys ∈ ℬ︁₁ and uνt, xyw∈ ℬ︁₂ implies s ≠ t. Orthogonal triple systems are associated with a class of Room squares, with the skew orthogonal triple systems corresponding to skew Room squares. Also, SOSTS are related to separable weakly union-free TTS. SOSTS are much rarer than OSTS; for example SOSTS(ν) do not exist for ν=3,9,15. Furthermore, a fundamental construction for the earlier OSTS proofs when ν ≡ 3 (mod 6) cannot exist. In the case ν≡ 1 ( mod 6) we are able to show existence except possibly for 22 values, the largest of which is 1315. There are at least two non-isomorphic OSTS(19)s one of which is SOSTS(19) and the other not. A SOSTS(27) was found, implying the existence of SOSTS(ν) for ν ≡ 3 (mod 6) with finitely many possible exceptions.     

On the power of the zeros of Bessel functions

For ν≥0 let c_νk be the k-th positive zero of the cylinder functionC _v(t)=J _v(t)cosα-Y _v(t)sinα, 0≤α>π whereJ _ν(t) andY _ν(t) denote the Bessel functions of the first and the second kind, respectively. We prove thatC _v,k ^1+H(x) is convex as a function of ν, ifc _νk≥x>0 and ν≥0, whereH(x) is specified in Theorem 1.1.     

A special class of polynomials orthogonal on the unit circle including the associated polynomials

Let (P _ν) be a sequence of monic polynomials orthogonal on the unit circle with respect to a nonnegative weight function, let (Ω_υ) the monic associated polynomials of (P _v), and letA andB be self-reciprocal polynomials. We show that the sequence of polynomials (AP_υλ+BΩ_υλ)/A^λ, λ stuitably determined, is a sequence of orthogonal polynomials having, up to a multiplicative complex constant, the same recurrence coefficients as theP _ν's from a certain index value onward, and determine the orthogonality measure explicity. Conversely, it is also shown that every sequence of orthogonal polynomials on the unit circle having the same recurrence coefficients from a certain index value onward is of the above form. With the help of these results an explicit representation of the associated polynomials of arbitrary order ofP _ν and of the corresponding orthogonality measure and Szegö function is obtained. The asymptotic behavior of the associated polynomials is also studied. Finally necessary and suficient conditions are given such that the measure to which the above introduced polynomials are orthogonal is positive.     

Linear birth and death models and associated Laguerre and Meixner polynomials

We study birth and death processes with linear rates λ_n = n + α + c + 1, μ_{n + 1} = n + c, n 0 and μ₀ is either zero or c. The spectral measures of both processes are found using generating functions and the integral transforms of Laplace and Stieltjes. The corresponding orthogonal polynomials generalize Laguerre polynomials and the choice μ₀ = c generates the associated Laguerre polynomials of Askey and Wimp. We investigate the orthogonal polynomials in both cases and give alternate proofs of some of the results of Askey and Wimp on the associated Laguerre polynomials. We also identify the spectra of the associated Charlier and Meixner polynomials as zeros of certain transcendental equations.     

Poisson algebras of polynomial growth

Consider the sequence c _n (V) of codimensions of a variety V of Poisson algebras. We show that the growth of every variety V of Poisson algebras over an arbitrary field is either bounded by a polynomial or at least exponential. Furthermore, if the growth of V is polynomial then there is a polynomial R(x) with rational coefficients such that c _n (V) = R(n) for all sufficiently large n. We present lower and upper bounds for the polynomials R(x) of an arbitrary fixed degree. We also show that the varieties of Poisson algebras of polynomial growth are finitely based in characteristic zero.     

Permutation Polynomials Modulo 2w

We give an exact characterization of permutation polynomials modulo n=2^w, w≥2: a polynomial P(x)=a₀+a₁x +···+a_dx^d with integral coefficients is a permutation polynomial modulo n if and only if a₁ is odd, (a₂+a₄+a₆+···) is even, and (a₃+a₅+a₇+···) is even. We also characterize polynomials defining latin squares modulo n=2^w, but prove that polynomial multipermutations (that is, a pair of polynomials defining a pair of orthogonal latin squares) modulo n=2^wdo not exist.     

Eigenfunction expansions for the Schrödinger equation with an inverse-square potential

We consider the one-dimensional Schrödinger equation -f″ + q_κf = Ef on the positive half-axis with the potential q_κ(r) = (κ² - 1/4)r^-2. For each complex number ν, we construct a solution u_ν^κ(E) of this equation that is analytic in κ in a complex neighborhood of the interval (-1, 1) and, in particular, at the “singular” point κ = 0. For -1 < κ < 1 and real ν, the solutions u_ν^κ(E) determine a unitary eigenfunction expansion operator U_κ,ν: L₂(0,∞) → L₂(R, Vκ,ν), where Vκ,ν is a positive measure on R. We show that every self-adjoint realization of the formal differential expression -?_r² + q_κ(r) for the Hamiltonian is diagonalized by the operator U_κ,ν for some ν ∈ R. Using suitable singular Titchmarsh–Weyl m-functions, we explicitly find the measures V_κ,ν and prove their continuity in κ and ν.     

A characterization of P2(μ) ≠ L2(μ)

It is shown that P²(μ) ≠ L²(μ) if and only if there exists a probability measure ν with ν ⊥ μ and 6pz.dfnc;_1,ν ? Cz.dfnc;pz.dfnc;_2,μ for all polynomials p and a fixed constant C < ∞. The relationship between this method and theorems of Berger and of Carey and Pincus concerning rationally cyclic vectors for powers of nonnormal subnormal Operators is examined.     

Convergence of solutions of a second order Cauchy problem

Let j_νk, y_νk and c_νk denote the kth positive zeros of the Bessel functions J_ν(x), Y_ν(x) and of the general cylinder function C_ν(x), respectively. We show, among other things, that, for k = 2, 3,… and 0 < ν < ∞, c_νk is a concave function of ν, c_νk > ν + c_0k and $\text{c}{}_{\mathrm{\nu k}}\text{[v + (}\text{2}\text{π}\text{)c}{}_{\mathrm{0k}}\text{]}$ decreases as ν increases. In the cases of j_νk and y_νk, these results hold also for k = 1.     

Highly edge‐connected detachments of graphs and digraphs

Let G = (V,E) be a graph or digraph and r : V → Z₊. An r‐detachment of G is a graph H obtained by ‘splitting’ each vertex ν ∈ V into r(ν) vertices. The vertices ν₁,…,ν_r(ν) obtained by splitting ν are called the pieces of ν in H. Every edge uν ∈ E corresponds to an edge of H connecting some piece of u to some piece of ν. Crispin Nash‐Williams 9 gave necessary and sufficient conditions for a graph to have a k‐edge‐connected r‐detachment. He also solved the version where the degrees of all the pieces are specified. In this paper, we solve the same problems for directed graphs. We also give a simple and self‐contained new proof for the undirected result. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 67–77, 2003