Elementary proof of the transformation formula for Lambert series involving generalized Dedekind sums

An elementary proof is given of the author's transformation formula for the Lambert series $\text{G}{}_{\mathrm{p}}\text{(x) =}\text{Σ}{}_{\mathrm{n?1}}{}^{\mathrm{<mtext>\infty </mtext>}}\text{n}{}^{\mathrm{?p}}\text{x}{}^{\mathrm{n}}\text{(1?x}{}^{\mathrm{n}}\text{)}$ relating G_p(e^2πiτ) to G_p(e^2πiAτ), where p > 1 is an odd integer and $\text{Aτ =}\text{(aτ + b)}\text{(cτ + d)}$ is a general modular substitution. The method extends Sczech's argument for treating Dedekind's function $\text{log}\text{η(τ) =}\text{πiτ}\text{12}\text{? G}{}_1\text{(e}{}^{\mathrm{2\pi i\tau }}\text{)}$, and uses Carlitz's formula expressing generalized Dedekind sums in terms of Eulerian functions.     

Randomness implies order

Let τ: [0, 1] → [0, 1] possess a unique invariant density $\text{f}{}^1$. Then given any ? > 0, we can find a density function p such that $\text{∥ p ? f}{}^1\text{∥ < ?,}\text{and}\text{p}$ is the invariant density of the stochastic difference equation x_{n + 1} = τ(x_n) + W, where W is a random variable. It follows that for all starting points $\text{x}{}_0\text{? [0, 1],}\text{lim}{}_{\mathrm{n\to \infty }}\text{(1}\text{n)}\text{∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{n\; ?\; 1}}\text{χ}{}_{\mathrm{B}}\text{(x}{}_{\mathrm{i}}\text{) = ∝}{}_{\mathrm{B}}\text{p(ξ) dξ}$.     

Limit laws for record values

{X_n,n?1} are i.i.d. random variables with continuous d.f. F(x). X_j is a record value of this sequence if X_j>max{X₁,…,X_j?1}. Consider the sequence of such record values {X_Ln,n?1}. Set R(x)=-log(1?F(x)). There exist B_n > 0 such that . in probability (i.p.) iff i.p. iff $\text{{R(kx)?R(x)}}\text{R}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(kx)}$ → ∞ as x→∞ for all k>1. Similar criteria hold for the existence of constants A_n such that X_Ln?A_n → 0 i.p. Limiting record value distributions are of the form N(-log(-logG(x))) where G(·) is an extreme value distribution and N(·) is the standard normal distribution. Domain of attraction criteria for each of the three types of limit laws can be derived by appealing to a duality theorem relating the limiting record value distributions to the extreme value distributions. Repeated use is made of the following lemma: If $\text{P}\text{{X}{}_{\mathrm{n}}\text{?x}=1?e}{}^{\mathrm{-x}}\text{,x?0}$, then X_Ln=Y₀+…+Y_n where the Y_j's are i.i.d. and $\text{P}\text{{Y}{}_{\mathrm{j}}\text{?x}=1?e}{}^{\mathrm{-x}}$.     

Decomposability of symmetric multilinear functions

Let V denote a finite dimensional vector space over a field K of characteristic 0, let T_n(V) denote the vector space whose elements are the K-valued n-linear functions on V, and let S_n(V) denote the subspace of T_n(V) whose members are the fully symmetric members of T_n(V). If $\text{L}$_n denotes the symmetric group on {1,2,…,n} then we define the projection $\text{P}{}_{\mathrm{<mtext>L</mtext>}}\text{: T}{}_{\mathrm{n}}\text{(V) → S}{}_{\mathrm{n}}\text{(V)}$ by the formula , where P_σ : T_n(V) → T_n(V) is defined so that P_σ(A)(y₁,y₂,…,y_n = A(y_σ(1),y_σ(2),…,y_σ(n)) for each A?T_n(V) and y_i?V, 1 ? i ? n. If $\text{x}{}_{\mathrm{i}}\text{? V}{}^1\text{, 1 ? i ? n}$, then x₁?x₂? … ?x_n denotes the member of T_n(V) such that $\text{(x}{}_1\text{?x}{}_2\text{· ? ? ?x}{}_{\mathrm{n}}\text{)(y}{}_1\text{,y}{}_2\text{,…,y}{}_{\mathrm{n}}\text{) = П}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{x}{}_{\mathrm{i}}\text{(y}{}_{\mathrm{i}}\text{)}$ for each y₁ ,₂,…,y_n in V, and x₁·x₂… x_n denotes $\text{P}{}_{\mathrm{<mtext>L</mtext>}}\text{(x}{}_1\text{?x}{}_2\text{? … ?x}{}_{\mathrm{n}}\text{)}$. If B? S_n(V) and there exists $\text{x}{}_{\mathrm{i}}\text{? V}{}^1\text{, 1 ? i ? n}$, such that B = x₁·x₂…x_n, then B is said to be decomposable. We present two sets of necessary and sufficient conditions for a member B of S_n(V) to be decomposable. One of these sets is valid for an arbitrary field of characteristic zero, while the other requires that K = R or C.     

Concavity of certain maps on positive definite matrices and applications to Hadamard products

If f is a positive function on (0, ∞) which is monotone of order n for every n in the sense of Löwner and if Φ₁ and Φ₂ are concave maps among positive definite matrices, then the following map involving tensor products:

$\text{(A,B)?f[Φ}{}_1\text{(A)}{}^{\mathrm{?1}}\text{?Φ}{}_2\text{(B)]·(Φ}{}_1\text{(A)?I)}$

is proved to be concave. If Φ₁ is affine, it is proved without use of positivity that the map

$\text{(A,B)?f[Φ}{}_1\text{(A)?Φ}{}_2\text{(B)}{}^{\mathrm{?1}}\text{]·(Φ}{}_1\text{(A)?I)}$

is convex. These yield the concavity of the map

$\text{(A,B)?A}{}^{\mathrm{1?p}}\text{?B}{}^{\mathrm{p}}$

(0<p?1) (Lieb's theorem) and the convexity of the map

$\text{(A,B)?A}{}^{\mathrm{1+p}}\text{?B}{}^{\mathrm{?p}}$

(0<p?1), as well as the convexity of the map

$\text{(A,B)?(A·log[A])?I?A?log[B]}$

.These concavity and convexity theorems are then applied to obtain unusual estimates, from above and below, for Hadamard products of positive definite matrices.     

A note on polynomial matrix functions over a finite field

Let F_n denote the ring of n×n matrices over the finite field F=GF(q) and let A(x)=A_Nx^N+ ?+ A₁x+A₀?F_n[x]. A function $\text{?:F}{}_{\mathrm{n}}\text{→F}{}_{\mathrm{n}}$ is called a right polynomial function iff there exists an A(x)?F_n[x] such that $\text{?(B)=A}{}_{\mathrm{N}}\text{B}{}^{\mathrm{N}}\text{+?+A}{}_1\text{B+ A}{}_0$ for every B?F_n. This paper obtains unique representations for and determines the number of right polynomial functions.     

Asymptotic results for partitions (II) and a conjecture of Bateman and Erdös

Asymptotic results are obtained for p_A^(k)(n), the kth difference of the function p_A(n) which is the number of partitions of n into integers from A. Under certain restrictions on A it is shown that

$\text{P}{}_{\mathrm{A}}{}^{\mathrm{(k+1)}}\text{(n)}\text{P}{}_{\mathrm{A}}{}^{\mathrm{(k)}}\text{(n)}\text{= O(n}{}^{\mathrm{?1/2}}\text{) (n→ ∫)}$

thereby verifying for these A a conjecture of Bateman and Erdös.     

Interpolation and inner maps that preserve measure

It is shown, for n ? m ? 1, that there exist inner maps Φ: Bⁿ → B^m with boundary values $\text{Φ}{}_1\text{: B}{}^{\mathrm{n}}\text{→ B}{}^{\mathrm{m}}$ such that $\text{σ}{}_{\mathrm{m}}\text{(A) = σ}{}_{\mathrm{n}}\text{(Φ}{}_1{}^{\mathrm{?1}}\text{(A))}$. where σ_n and σ_m are the Haar measures on ?Bⁿ and ?B^m, respectively, and A ? Bⁿ is an arbitrary Borel set.     

The algebra of hyperboloids of revolution

If we change the sign of p ? m columns (or rows) of an m × m positive definite symmetric matrix A, the resultant matrix B has p negative eigenvalues. We give systems of inequalities for the eigenvalues of B and of the matrix obtained from B by deleting one row and column. To obtain these, we first develop characterizations of the eigenvalues of B which are analogous to the minimum-maximum properties of the eigenvalues of a symmetric A, i.e. the Courant-Fischer theorem. These results arose from studying probability distributions on the hyperboloid of revolution

$\text{x}{}^2{}_1\text{+ ? + x}{}^2{}_{\mathrm{m?p}}\text{? x}{}^2{}_{\mathrm{m\; ?\; p\; +\; 1}}\text{? ? ? x}{}^2{}_{\mathrm{m}}\text{= 1}$

. By contrast, the familiar results are associated with the sphere x²₁ + ? + x²_m = 1.     

On the van der Waerden conjecture and zeros of polynomials

Let

$\text{F(x) =}\text{∑}\text{k=o}\text{n}\text{n}\text{k}\text{A}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}$

$\text{A}{}_{\mathrm{n}}\text{≠ 0,}$

and

$\text{G(x) =}\text{∑}\text{k=o}\text{n}\text{n}\text{k}\text{B}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}$

$\text{B}{}_{\mathrm{n}}\text{≠ 0,}$

be polynomials with real zeros satisfying A_n?1 = B_n?1 = 0, and let

$\text{H(x) =}\text{∑}\text{k=o}\text{n-2}\text{n}\text{k}\text{A}{}_{\mathrm{k}}\text{B}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}\text{.}$

Using the recently proved validity of the van der Waerden conjecture on permanents, some results on the real zeros of H(x) are obtained. These results are related to classical results on composite polynomials.     

Barriers for uniformly elliptic equations and the exterior cone condition

We consider the mixed boundary value problem $\text{Au = f}\text{in}\text{Ω, B}{}_0\text{u = g}{}_0\text{in}\text{Γ}{}^{\mathrm{?}}\text{, B}{}_1\text{u = g}{}_1\text{in}\text{Γ}{}^{\mathrm{+}}$, where Ω is a bounded open subset of $\text{R}$ⁿ whose boundary Γ is divided into disjoint open subsets Γ⁺ and Γ^? by an (n ? 2)-dimensional manifold ω in Γ. We assume A is a properly elliptic second order partial differential operator on $\text{Ω}$ and B_j, for j = 0, 1, is a normal jth order boundary operator satisfying the complementing condition with respect to A on $\text{Γ}{}^{\mathrm{+}}$. The coefficients of the operators and Γ⁺, Γ^? and ω are all assumed arbitrarily smooth. As announced in [Bull. Amer. Math. Soc.83 (1977), 391–393] we obtain necessary and sufficient conditions in terms of the coefficients of the operators for the mixed boundary value problem to be well posed in Sobolev spaces. In fact, we construct an open subset $\text{T}$ of the reals such that, if $\text{D}{}^{\mathrm{s}}\text{= {u ? H}{}^{\mathrm{s}}\text{(Ω): Au = 0}}$ then for $\text{s ? =}\text{1}\text{2}\text{(}\text{mod}\text{1), (B}{}_0\text{,B}{}_1\text{): D}{}^{\mathrm{s}}\text{→ H}{}^{\mathrm{<mtext>s\; ?\; </mtext><mtext>1</mtext><mtext>2</mtext>}}\text{(Γ}{}^{\mathrm{?}}\text{) × H}{}^{\mathrm{<mtext>s\; ?\; </mtext><mtext>3</mtext><mtext>2</mtext>}}\text{(Γ}{}^{\mathrm{+}}\text{)}$ is a Fredholm operator if and only if s ∈$\text{T}$ . Moreover, $\text{T}$ = ?_xew$\text{T}$_x, where the sets $\text{T}$_x are determined algebraically by the coefficients of the operators at x. If n = 2, $\text{T}$_x is the set of all reals not congruent (modulo 1) to some exceptional value; if n = 3, $\text{T}$_x is either an open interval of length 1 or is empty; and finally, if n ? 4, $\text{T}$_x is an open interval of length 1.     

Two-parameter Turan inequalities for ultraspherical and Laguerre polynomials

It is known that the classical orthogonal polynomials satisfy inequalities of the form U_n²(x) ? U_{n + 1}(x) U_{n ? 1}(x) > 0 when x lies in the spectral interval. These are called Turan inequalities. In this paper we will prove a generalized Turan inequality for ultraspherical and Laguerre polynomials. Specifically if P_n^λ(x) and L_n^α(x) are the ultraspherical and Laguerre polynomials and $\text{F}{}_{\mathrm{n}}{}^{\mathrm{\lambda }}\text{(x) =}\text{P}{}_{\mathrm{n}}{}^{\mathrm{\lambda }}\text{(x)}\text{P}{}_{\mathrm{n}}{}^{\mathrm{\lambda }}\text{(1)}\text{, G}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x) =}\text{L}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x)}\text{L}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(0)}\text{,}\text{then}\text{F}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x) F}{}_{\mathrm{n}}{}^{\mathrm{\beta }}\text{(x) ? F}{}_{\mathrm{n\; +\; 1}}{}^{\mathrm{\alpha }}\text{(x) F}{}_{\mathrm{n\; ?\; 1}}{}^{\mathrm{\beta }}\text{(x) > 0, ? 1 < x < 1, ?}\text{1}\text{2}\text{< α ? β ? α + 1}\text{and}\text{G}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x) G}{}_{\mathrm{n}}{}^{\mathrm{\beta }}\text{(x) ? G}{}_{\mathrm{n\; +\; 1}}{}^{\mathrm{\alpha }}\text{(x) G}{}_{\mathrm{n\; ?\; 1}}{}^{\mathrm{\beta }}\text{(x) > 0, x > 0, 0 < α ? β ? α + 1}$. We also prove the inequality $\text{(n + 1) F}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x) F}{}_{\mathrm{n}}{}^{\mathrm{\beta }}\text{(x) ? nF}{}_{\mathrm{n\; +\; 1}}{}^{\mathrm{\alpha }}\text{(x) F}{}_{\mathrm{n\; ?\; 1}}{}^{\mathrm{\beta }}\text{(x) > A}{}_{\mathrm{n}}\text{[F}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x)]}{}^2\text{, ?1 < x < 1, ?}\text{1}\text{2}\text{< α ? β < α + 1,}\text{where}\text{A}{}_{\mathrm{n}}$ is a positive constant depending on α and β.     

Transcendental continued fractions

In a previous paper it was proven that given the continued fractions

$\text{A = a}{}_1\text{+}\text{1}\text{a}{}_2\text{+}\text{1}\text{a}{}_3\text{+… and B = b}{}_1\text{+}\text{1}\text{b}{}_2\text{+}\text{1}\text{b}{}_3\text{+…}$

where the a's and b's are positive integers, then A, B, A ± B, $\text{A}\text{B}$ and AB are irrational numbers if $\text{a}{}_{\mathrm{n}}\text{2}\text{> b}{}_{\mathrm{n}}\text{> a}{}_{\mathrm{n?1}}{}^{\mathrm{5n}}$ for all n sufficiently large, and transcendental numbers if for all n sufficiently large. Using a more direct approach it is proven in this paper that A, B, A ± B, $\text{A}\text{B}$ and AB are transcendental numbers if a_n > b_n > a_n?1^(n?1)2 for all n sufficiently large.     

The sum of aΩ(n) over integers n ≤ x with all prime factors between a and y

For a > 0 let $\text{ψ}{}_{\mathrm{a}}\text{(x, y) =}\text{Σ}\text{a}{}^{\mathrm{\Omega (n)}}$, the sum taken over all n, 1 ≤ n ≤ x such that if p is prime and p|n then a < p ≤ y. It is shown for u < about ($\text{log log}\text{x}\text{log log log}\text{x}$) that $\text{ψ}{}_{\mathrm{a}}\text{(x, x}{}^{\mathrm{<mtext>1</mtext><mtext>u</mtext>}}\text{) ? x(}\text{log}\text{x)}{}^{\mathrm{a?1}}\text{p}{}_{\mathrm{a}}\text{(u)}$, where p_a(u) solves a delay differential equation much like that for the Dickman function p(u), and the asymptotic behavior of p_a(u) is worked out.     

Some divided difference inequalities for n-convex functions

Using results from the theory of B-splines, various inequalities involving the nth order divided differences of a function f with convex nth derivative are proved; notably, $\text{f}{}^{\mathrm{(n)}}\text{(z)}\text{n! ? [x}{}_0\text{,…, x}{}_{\mathrm{n}}\text{]f}\text{?}\text{∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{n}}\text{(f}{}^{\mathrm{(n)}}\text{(x}{}_{\mathrm{i}}\text{)}\text{(n + 1)!)}$, where z is the center of mass $\text{(}\text{1}\text{(n + 1))}\text{∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{n}}\text{x}{}_{\mathrm{i}}$.     

The asymptotic eigenvalue-distribution for a certain class of random matrices

For an n × n Hermitean matrix A with eigenvalues λ₁, …, λ_n the eigenvalue-distribution is defined by $\text{G(x, A) :=}\text{1}\text{n}$ · number {λ_i: λ_i ? x} for all real x. Let A_n for n = 1, 2, … be an n × n matrix, whose entries a_ik are for i, k = 1, …, n independent complex random variables on a probability space (Ω, $\text{R}$, p) with the same distribution F_a. Suppose that all moments $\text{E}$ | a | ^k, k = 1, 2, … are finite, $\text{E}$a=0 and $\text{E}$ | a | ². Let

$\text{M(A)=}\text{∑}\text{σ=1}\text{s}\text{θ}{}_{\mathrm{\sigma }}\text{P}{}_{\mathrm{\sigma }}\text{(A,A}{}^1\text{)}$

with complex numbers θ_σ and finite products P_σ of factors A and $\text{A}{}^1$ (= Hermitean conjugate) be a function which assigns to each matrix A an Hermitean matrix M(A). The following limit theorem is proved: There exists a distribution function G₀(x) = G₁x) + G₂(x), where G₁ is a step function and G₂ is absolutely continuous, such that with probability $\text{1 G(x, M(}\text{A}{}_{\mathrm{n}}\text{n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{))}$ converges to G₀(x) as n → ∞ for all continuity points x of G₀. The density g of G₂ vanishes outside a finite interval. There are only finitely many jumps of G₁. Both, G₁ and G₂, can explicitly be expressed by means of a certain algebraic function f, which is determined by equations, which can easily be derived from the special form of M(A). This result is analogous to Wigner's semicircle theorem for symmetric random matrices (E. P. Wigner, Random matrices in physics, SIAM Review9 (1967), 1–23). The examples $\text{A}{}_{\mathrm{r}}\text{A}{}^{\mathrm{1r}}$, $\text{A}{}^{\mathrm{r}}\text{+ A}{}^{\mathrm{1r}}$, $\text{A}{}^{\mathrm{r}}\text{A}{}^{\mathrm{1r}}\text{± A}{}^{\mathrm{1r}}\text{A}{}^{\mathrm{r}}$, r = 1, 2, …, are discussed in more detail. Some inequalities for random matrices are derived. It turns out that with probability 1 the sharpened form

$\text{lim sup}\text{n→∞}\text{∑}\text{i=1}{}^{\mathrm{n}}\text{|γ}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{|}{}^2\text{?6A}{}_{\mathrm{n}}\text{6}{}^2\text{? 0.8228…}$

of Schur's inequality for the eigenvalues λ_i⁽ⁿ⁾ of A_n holds. Consequently random matrices do not tend to be normal matrices for large n.     

Deterministic version of Wigner's semicircle law for the distribution of matrix eigenvalues

It is proved that Wigner's semicircle law for the distribution of eigenvalues of random matrices, which is important in the statistical theory of energy levels of heavy nuclei, possesses the following completely deterministic version. Let A_n=(a_ij), 1?i, ?n, be the nth section of an infinite Hermitian matrix, {λ⁽ⁿ⁾}_1?k?n its eigenvalues, and {u_k⁽ⁿ⁾}_1?k?n the corresponding (orthonormalized column) eigenvectors. Let $\text{v}{}^1{}_{\mathrm{n}}\text{=(a}{}_{\mathrm{n1}}\text{,a}{}_{\mathrm{n2}}\text{,?,a}{}_{\mathrm{n,n?1}}\text{)}$, put

$\text{X}{}_{\mathrm{n}}\text{(t)=[n(n-1)]}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{∑}\text{k=1}\text{[(n-1)t]}\text{|v}{}_{\mathrm{n}}{}^1\text{u}{}_{\mathrm{f}}{}^{\mathrm{(n-1)}}\text{|}{}^2\text{,}\text{0?t?1}$

(bookeeping function for the length of the projections of the new row v¹_n of A_n onto the eigenvectors of the preceding matrix A_n?1), and let finally

$\text{F}{}_{\mathrm{n}}\text{(x)=n}{}^{-1}\text{(}\text{number of}\text{λ}{}_{\mathrm{k}}{}^{\mathrm{(n)}}\text{?x}\text{n}\text{,1?k?n)}$

(empirical distribution function of the eigenvalues of $\text{A}{}_{\mathrm{n}}\text{n}$. Suppose (i) $\text{lim}{}_{\mathrm{n}}\text{a}{}_{\mathrm{nn}}\text{n}\text{=0}$, (ii) lim_nX_n(t)=Ct(0<C<∞,0?t?1). Then

$\text{F}{}_{\mathrm{n}}\text{?W(·,C)}\text{(n→∞)}$

,where W is absolutely continuous with (semicircle) density

$\text{w(x,C)=}\text{(2Cπ)}{}^{-1}\text{(4C-x}{}^2{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{for}\text{|x|?2}\text{C}\text{0}\text{for}\text{|x|?2}\text{C}$