The Bergman kernel: Explicit formulas,deflation, Lu Qi-Keng problem and Jacobi polynomials

We investigate the Bergman kernel function for the intersection of two complex ellipsoids {(z,w ₁,w ₂) ∈ C ⁿ⁺²: |z ₁|²+...+|z _n |²+|w ₁| ^q < 1, |z ₁|²+...+|z _n |²+|w ₂| ^r < 1}. We also compute the kernel function for {(z ₁,w ₁,w ₂) ∈ C³: |z ₁|^2/n + |w ₁| ^q < 1, |z ₁|^2/n + |w ₂| ^r < 1} and show deflation type identity between these two domains. Moreover in the case that q = r = 2 we express the Bergman kernel in terms of the Jacobi polynomials. The explicit formulas of the Bergman kernel function for these domains enables us to investigate whether the Bergman kernel has zeros or not. This kind of problem is called a Lu Qi-Keng problem.     

On the rank of odd hyper-quasi-polynomials

Given any nonzero entire function g: ? → ?, the complex linear space F(g) consists of all entire functions f decomposable as f(z + w)g(z - w)=φ₁(z)ψ₁(w)+???+ φ_n(z)ψ_n(w) for some φ₁, ψ₁, …, φ_n, ψ_n: ? → ?. The rank of f with respect to g is defined as the minimum integer n for which such a decomposition is possible. It is proved that if g is an odd function, then the rank any function in F(g) is even.     

Hyperbolic Chevalley groups on ℂ<Superscript>2</Superscript>

Let Γ ? U (1, 1) be the subgroup generated by the complex reflections. Suppose that Γ acts discretely on the domain K = {(z ₁, z ₂) ∈ ?² ||z ₁|² ? |z ₂|² < 0} and that the projective group PΓ acts on the unit disk B = {|z ₁/z ₂| < 1} as a Fuchsian group of signature (n ₁, ..., n _s ), s ? 3, n _i ? 2. For such groups, we prove a Chevalley type theorem, i.e., find a necessary and sufficient condition for the quotient space K/Γ to be isomorphic to ?² ? {0}.     

Meromorphic Jost functions and asymptotic expansions for Jacobi parameters

We show that the parameters a _n , b _n of a Jacobi matrix have a complete asymptotic expansion

$a_n^2 - 1 = \sum\limits_{k = 1}^{K(R)} {p_k (n)\mu _k^{ - 2n} + O(R^{ - 2n} ),} b_n = \sum\limits_{k = 1}^{K(R)} {p_k (n)\mu _k^{ - 2n + 1} + O(R^{ - 2n} )} $

, where 1 < |µ_j| < R for j ? K(R) and all R, if and only if the Jost function, u, written in terms of z (where E = z + z ^?1) is an entire meromorphic function. We relate the poles of u to the µ_j’s.

     

A numerical algorithm for solving a nonstationary problem of optimal control

Let {φ _n ^(α,β) (z)} _n=0 ^∞ be a system of Jacobi polynomials orthonormal on the circle |z| = 1 with respect to the weight (1 ? cos τ)^α+1/2(1 + cos τ)^β+1/2 (α, β > ?1), and let \(\psi _n^{\left( {\alpha ,\beta } \right)*} \left( z \right): = z^n \overline {\psi _n^{\left( {\alpha ,\beta } \right)} \left( {{1 \mathord{\left/ {\vphantom {1 {\bar z}}} \right. \kern-\nulldelimiterspace} {\bar z}}} \right)}\)). We establish relations between the polynomial φ _n ^(α,?1/2) (z) and the nth (C, α ? 1/2)-mean of the Maclaurin series for the function (1 ? z)^?α?3/2 and also between the polynomial φ _n ^(α,?1/2)* (z) and the nth (C, α + 1/2)-mean of the Maclaurin series for the function (1 ? z)^?α?1/2. We use these relations to derive an asymptotic formula for φ _n ^(α,?1/2) (z); the formula is uniform inside the disk |z| < 1. It follows that φ _n ^(α,?1/2) (z) ≠ 0 in the disk |z| ≤ ρ for fixed φ ∈ (0, 1) and α > ?1 if n is sufficiently large.     

Linear combinations of prime powers in sums of terms of binary recurrence sequences

Let (U _n ) _n≥0 be a nondegenerate binary recurrence sequence with positive discriminant. Let p ₁ , . . . , p _s be fixed prime numbers, b ₁ , . . . , b _s be fixed nonnegative integers, and a ₁ , . . . , a _t be positive integers. In this paper, under certain assumptions, we obtain a finiteness result for the solution of the Diophantine equation \( {\alpha}_1{U}_{n1}+\cdots +{\alpha}_t{U}_{n1}={b}_1{p}_1^{z_1}+\cdots {b}_s{p}_s^{z_s}. \) Moreover, we explicitly solve the equation F _n1 + F _n2 = 2 ^z1 + 3 ^z2 in nonnegative integers n ₁, n ₂, z ₁, z ₂ with z ₂ ≥ z ₁. The main tools used in this work are the lower bound for linear forms in logarithms and the Baker–Davenport reduction method. This work generalizes the recent papers [E. Mazumdar and S.S. Rout, Prime powers in sums of terms of binary recurrence sequences, arXiv:1610.02774] and [C. Bertók, L. Hajdu, I. Pink, and Z. Rábai, Linear combinations of prime powers in binary recurrence sequences, Int. J. Number Theory, 13(2):261–271, 2017].     

Sets with even partition functions and 2-adic integers

For P ? \(\mathbb{F}_2 \)[z] with _P(0) = 1 and deg(_P) ≥ 1, let \(\mathcal{A}\) = \(\mathcal{A}\)(P) (cf. [4], [5], [13]) be the unique subset of ? such that Σ _n≥0 p(\(\mathcal{A}\), n)z ⁿ ≡ P(z) (mod 2), where p(\(\mathcal{A}\), n) is the number of partitions of n with parts in \(\mathcal{A}\). Let p be an odd prime and P ? \(\mathbb{F}_2 \)[z] be some irreducible polynomial of order p, i.e., p is the smallest positive integer such that P(z) divides 1 + z ^p in \(\mathbb{F}_2 \)[z]. In this paper, we prove that if m is an odd positive integer, the elements of \(\mathcal{A}\) = \(\mathcal{A}\)(P) of the form 2 ^k m are determined by the 2-adic expansion of some root of a polynomial with integer coefficients. This extends a result of F. Ben Saïd and J.-L. Nicolas [6] to all primes p.     

Coefficient estimates of negative powers and inverse coefficients for certain starlike functions

For ?1≤B<A≤1, let \(\mathcal {S}^{*}(A,B)\) denote the class of normalized analytic functions \(f(z)= z+{\sum }_{n=2}^{\infty }a_{n} z^{n}\) in |z|<1 which satisfy the subordination relation z f ^′(z)/f(z)?(1 + A z)/(1 + B z) and Σ^?(A,B) be the corresponding class of meromorphic functions in |z|>1. For \(f\in \mathcal {S}^{*}(A,B)\) and λ>0, we shall estimate the absolute value of the Taylor coefficients a _n (?λ,f) of the analytic function (f(z)/z)^?λ . Using this we shall determine the coefficient estimate for inverses of functions in the classes \(\mathcal {S}^{*}(A,B)\) and Σ^?(A,B).     

On almost good triples of vertices in edge regular graphs

Consider a connected edge regular graph Γ with parameters (v, k, λ) and put b ₁ = k?λ?1. A triple (u, w, z) of vertices is called (almost) good whenever d(u, w) = d(u, z) = 2 and µ(u, w)+µ(u, z) ≤ 2k ? 4b ₁ + 3 (and µ(u, w) + µ(u, z) = 2k ? 4b ₁ + 4). If k = 3b ₁ + γ with γ ≥ ?2, a triple (u, w, z) is almost good, and Δ = [u] ∩ [w] ∩ [z] then: either |Δ| ≤ 2; or Δ is a 3-clique and Γ is a Clebsch graph; or Δ is a 3-clique, k = 16, b ₁ = 6, and v = 31; or Δ is a 4-clique and Γ is a Schläfli graph.     

Solution of Functional Equations Related to Elliptic Functions

Functional equations of the form f(x + y)g(x ? y) = Σ _j=1 ⁿ α _j (x)β _j (y) as well as of the form f₁(x + z)f₂(y + z)f₃(x + y ? z) = Σ _j=1 ^m φ _j (x, y)ψ _j (z) are solved for unknown entire functions f, g,α _j , β _j : ? → ? and f₁, f₂, f₃, ψ _j : ? → ?, φ _j : ?² → ? in the cases of n = 3 and m = 4.     

Convergent expansions of the Bessel functions in terms of elementary functions

We consider the Bessel functions J _ν (z) and Y _ν (z) for R ν > ?1/2 and R z ≥ 0. We derive a convergent expansion of J _ν (z) in terms of the derivatives of \((\sin z)/z\), and a convergent expansion of Y _ν (z) in terms of derivatives of \((1-\cos z)/z\), derivatives of (1 ? e ^?z )/z and Γ(2ν, z). Both expansions hold uniformly in z in any fixed horizontal strip and are accompanied by error bounds. The accuracy of the approximations is illustrated with some numerical experiments.     

A pure smoothness condition for radó’s theorem for α-analytic functions

Let \(\Omega \subset {{\Bbb C}^n}\) be a bounded, simply connected ?-convex domain. Let α ∈ ?₊ⁿ and let f be a function on Ω which is separately \({C^{2{\alpha _j} - 1}}\)-smooth with respect to z_j (by which we mean jointly \({C^{2{\alpha _j} - 1}}\)-smooth with respect to Rez_j, Imz_j). If f is α-analytic on Ω\f^?1(0), then f is α-analytic on Ω. The result is well-known for the case α_i = 1, 1 ? i ? n, even when f a priori is only known to be continuous.     

The weighted Hardy spaces associated to self-adjoint operators and their duality on product spaces

Let L be a non-negative self-adjoint operator acting on L²(R ⁿ ) satisfying a pointwise Gaussian estimate for its heat kernel. Let w be an A _r weight on R ⁿ × R ⁿ , 1 < r < ∞. In this article we obtain a weighted atomic decomposition for the weighted Hardy space H _L,w ^p (R ⁿ ×R ⁿ ), 0 < p ≤ 1 associated to L. Based on the atomic decomposition, we show the dual relationship between H _L,w ¹ (R ⁿ × R ⁿ ) and BMO_L,w(R ⁿ × R ⁿ ).     

Curvature inequalities for operators of the Cowen–Douglas class

Let T be an operator tuple in the Cowen–Douglas class B _n (Ω) for Ω ? C ^m . The kernels Ker(T ? w) ^l , for w ∈ Ω, l = 1, 2, ···, define Hermitian vector bundles E _T ^l over Ω. We prove certain negativity of the curvature of E _T ^l . We also study the relation between certain curvature inequality and the contractive property of T when Ω is a planar domain.     

Generalizations of Schöbi’s Tetrahedral Dissection

Let v ₁,…,v _n be unit vectors in ? ⁿ such that v _i ?v _j =?w for i≠j, where \(-1. The points ∑ _i=1 ⁿ λ _i v _i (1≥λ ₁≥???≥λ _n ≥0) form a “Hill-simplex of the first type,” denoted by \(\mathcal {Q}_{n}(w)\). It was shown by Hadwiger in 1951 that \(\mathcal {Q}_{n}(w)\) is equidissectable with a cube. In 1985, Schöbi gave a three-piece dissection of \(\mathcal {Q}_{3}(w)\) into a triangular prism \(c\mathcal {Q}_{2}(\frac{1}{2})\times I\), where I denotes an interval and \(c=\sqrt{2(w+1)/3}\). In this paper, we generalize Schöbi’s dissection to an n-piece dissection of \(\mathcal {Q}_{n}(w)\) into a prism \(c\mathcal {Q}_{n-1}(\frac{1}{n-1})\times I\), where \(c=\sqrt{(n-1)(w+1)/n}\). Iterating this process leads to a dissection of \(\mathcal {Q}_{n}(w)\) into an n-dimensional rectangular parallelepiped (or “brick”) using at most n! pieces. The complexity of computing the map from \(\mathcal {Q}_{n}(w)\) to the brick is O(n ²). A second generalization of Schöbi’s dissection is given which applies specifically in ?⁴. The results have applications to source coding and to constant-weight binary codes.     

Circular Law for Noncentral Random Matrices

Let (X _jk )_j,k≥1 be an infinite array of i.i.d. complex random variables with mean 0 and variance 1. Let λ _n,1,…,λ _n,n be the eigenvalues of \((\frac{1}{\sqrt{n}}X_{jk})_{1\leqslant j,k\leqslant n}\). The strong circular law theorem states that, with probability one, the empirical spectral distribution \(\frac{1}{n}(\delta _{\lambda _{n,1}}+\cdots+\delta _{\lambda _{n,n}})\) converges weakly as n→∞ to the uniform law over the unit disc {z∈?,|z|≤1}. In this short paper, we provide an elementary argument that allows us to add a deterministic matrix M to (X _jk )_{1≤ j,k ≤ n} provided that Tr(MM ^*)=O(n ²) and rank(M)=O(n ^α ) with α<1. Conveniently, the argument is similar to the one used for the noncentral version of the Wigner and Marchenko–Pastur theorems.     

Meromorphic Solutions for a Class of Differential Equations and Their Applications

In this note, we study the admissible meromorphic solutions for algebraic differential equation fⁿf' + P_n?1(f) = R(z)e^α(z), where P_n?1(f) is a differential polynomial in f of degree ≤ n ? 1 with small function coefficients, R is a non-vanishing small function of f, and α is an entire function. We show that this equation does not possess any meromorphic solution f(z) satisfying N(r, f) = S(r, f) unless P_n?1(f) ≡ 0. Using this result, we generalize a well-known result by Hayman.     

Asymptotic behavior of the maximal zero of a polynomial orthogonal on a segment with a nonclassical weight

Let {p _n (t)} _n=0 ^t8 be a system of algebraic polynomials orthonormal on the segment [?1, 1] with a weight p(t); let {x _n,ν ^(p) } _ν=1 ⁿ be zeros of a polynomial p _n (t) (x _x,ν ^(p) = cosθ _n,ν ^(p) ; 0 < θ _n,1 ^(p) < θ _n,2 ^(p) < ... < θ _n,n ^(p) < π). It is known that, for a wide class of weights p(t) containing the Jacobi weight, the quantities θ _n,1 ^(p) and 1 ? x _n,1 ^(p) coincide in order with n ^?1 and n ^?2, respectively. In the present paper, we prove that, if the weight p(t) has the form p(t) = 4(1 ? t ²)^?1{ln²[(1 + t)/(1 ? t)] + π ²}^?1, then the following asymptotic formulas are valid as n → ∞:

$$\theta _{n,1}^{(p)} = \frac{{\sqrt 2 }}{{n\sqrt {\ln (n + 1)} }}\left[ {1 + {\rm O}\left( {\frac{1}{{\ln (n + 1)}}} \right)} \right],x_{n,1}^{(p)} = 1 - \left( {\frac{1}{{n^2 \ln (n + 1)}}} \right) + O\left( {\frac{1}{{n^2 \ln ^2 (n + 1)}}} \right).$$

     

Economical separability in free groups

Consider the rank n free group F _n with basis X. Bogopol’ski? conjectured in [1, Problem 15.35] that each element w ∈ F _n of length |w| ≥ 2 with respect to X can be separated by a subgroup H ≤ F _n of index at most C log |w| with some constant C. We prove this conjecture for all w outside the commutant of F _n , as well as the separability by a subgroup of index at most |w|/2 + 2 in general.