A recursive formula for even order harmonic series

A useful recursive formula for obtaining the infinite sums of even order harmonic series Σ^∞_n=1 (1/n^2k), k = 1, 2, …, is derived by an application of Fourier series expansion of some periodic functions. Since the formula does not contain the Bernoulli numbers, infinite sums of even order harmonic series may be calculated by the formula without the Bernoulli numbers. Infinite sums of a few even order harmonic series, which are calculated using the recursive formula, are tabulated for easy reference.     

Certain summation formulas involving harmonic numbers and generalized harmonic numbers

Harmonic numbers and generalized harmonic numbers have been studied since the distant past and involved in a wide range of diverse fields such as analysis of algorithms in computer science, various branches of number theory, elementary particle physics and theoretical physics. Here we aim at presenting further interesting identities about certain finite or infinite series involving harmonic numbers and generalized harmonic numbers by applying an algorithmic method to a known summation formula for the hypergeometric function ₅F₄(1).     

The Carnot-Carathéodory Distance and the Infinite Laplacian

In ℝ ⁿ equipped with the Euclidean metric, the distance from the origin is smooth and infinite harmonic everywhere except the origin. Using geodesics, we find a geometric characterization for when the distance from the origin in an arbitrary Carnot-Carathéodory space is a viscosity infinite harmonic function at a point outside the origin. We show that at points in the Heisenberg group and Grushin plane where this condition fails, the distance from the origin is not a viscosity infinite harmonic subsolution. In addition, the distance function is not a viscosity infinite harmonic supersolution at the origin.     

On the representation of biharmonic functions with singularities in ℝ n

Biharmonic functions are defined on Euclidean spaces, Riemannian manifolds, infinite trees, and more generally on abstract harmonic spaces. In this note, we consider biharmonic functions b defined on annular sets Ω \ K and obtain Laurent-type decompositions for b in the Euclidean spaces and in infinite trees. Particular importance is given to the investigation when b extends as a distribution on Ω.     

Tools for Malliavin Calculus in UMD Banach Spaces

In this paper we study the Malliavin derivatives and Skorohod integrals for processes taking values in an infinite dimensional space. Such results are motivated by their applications to SPDEs and in particular financial mathematics. Vector-valued Malliavin theory in Banach space E is naturally restricted to spaces E which have the so-called umd property, which arises in harmonic analysis and stochastic integration theory. We provide several new results and tools for the Malliavin derivatives and Skorohod integrals in an infinite dimensional setting. In particular, we prove weak characterizations, a chain rule for Lipschitz functions, a sufficient condition for pathwise continuity and an Itô formula for non-adapted processes.     

Spectral bounds for the independence ratio and the chromatic number of an operator

We define the independence ratio and the chromatic number for bounded, self-adjoint operators on an L ²-space by extending the definitions for the adjacency matrix of finite graphs. In analogy to the Hoffman bounds for finite graphs, we give bounds for these parameters in terms of the numerical range of the operator. This provides a theoretical framework in which many packing and coloring problems for finite and infinite graphs can be conveniently studied with the help of harmonic analysis and convex optimization. The theory is applied to infinite geometric graphs on Euclidean space and on the unit sphere.     

Further summation formulae related to generalized harmonic numbers

By employing the univariate series expansion of classical hypergeometric series formulae, Shen [L.-C. Shen, Remarks on some integrals and series involving the Stirling numbers and ζ(n), Trans. Amer. Math. Soc. 347 (1995) 1391-1399] and Choi and Srivastava [J. Choi, H.M. Srivastava, Certain classes of infinite series, Monatsh. Math. 127 (1999) 15-25; J. Choi, H.M. Srivastava, Explicit evaluation of Euler and related sums, Ramanujan J. 10 (2005) 51-70] investigated the evaluation of infinite series related to generalized harmonic numbers. More summation formulae have systematically been derived by Chu [W. Chu, Hypergeometric series and the Riemann Zeta function, Acta Arith. 82 (1997) 103-118], who developed fully this approach to the multivariate case. The present paper will explore the hypergeometric series method further and establish numerous summation formulae expressing infinite series related to generalized harmonic numbers in terms of the Riemann Zeta function ζ(m) with m=5,6,7, including several known ones as examples.     

Structural properties of Dirichlet series with harmonic coefficients

An infinite family of functional equations in the complex plane is obtained for Dirichlet series involving harmonic numbers. Trigonometric series whose coefficients are linear forms with rational coefficients in hyperharmonic numbers up to any order are evaluated via Bernoulli polynomials, Gauss sums, and special values of L-functions subject to the parity obstruction. This in turn leads to new representations of Catalan’s constant, odd values of the Riemann zeta function, and polylogarithmic quantities. Consequently, a dichotomy result is deduced on the transcendentality of Catalan’s constant and a series with hyperharmonic terms. Moreover, making use of integrals of smooth functions, we establish Diophantine-type approximations of real numbers by values of an infinite family of Dirichlet series built from representations of harmonic numbers.     

Z-measures on partitions related to the infinite Gelfand pair (S(2∞),H(∞))

The paper deals with the z-measures on partitions with the deformation (Jack) parameters 2 or 1/2. We provide a detailed explanation of the representation-theoretic origin of these measures, and of their role in the harmonic analysis on the infinite symmetric group.     

Harmonic functions on topological groups and symmetric spaces

Let G be a metric group, not necessarily locally compact, acting on a metric space X, for instance, a right coset space of G. We introduce and develop a basic structure theory for harmonic functions on X which is applicable to infinite dimensional Riemannian symmetric spaces.     

Infinite-dimensional diffusions as limits of random walks on partitions

Starting with finite Markov chains on partitions of a natural number n we construct, via a scaling limit transition as n → ∞, a family of infinite-dimensional diffusion processes. The limit processes are ergodic; their stationary distributions, the so-called z-measures, appeared earlier in the problem of harmonic analysis for the infinite symmetric group. The generators of the processes are explicitly described.     

Identities and congruences for Ramanujan’s ω(q)

Recently, the authors constructed generalized Borcherds products where modular forms are given as infinite products arising from weight 1/2 harmonic Maass forms. Here we illustrate the utility of these results in the special case of Ramanujan’s mock theta function ω(q). We obtain identities and congruences modulo 512 involving the coefficients of ω(q).     

Nonstandard integration theory in topological vector lattices

This paper develops a Daniell-Stone integration theory in topological vector lattices. Starting with an internal, vector valued, positive linear functionalI on an internal lattice of vector valued functions, we produce a nonstandard hull valued integralJ satisfying the monotone convergence theorem. Nonstandard hulls form a natural extension of infinite dimensional spaces and are equivalent to Banach space ultrapower constructions. The first application of our integral is a construction of Banach limits for bounded, vector valued sequences. The second example yields an integral representation for bounded and quasibounded harmonic functions similar to that of the Martin boundary. The third application uses our general integral to extend the Bochner integral.     

Universal Polynomial Expansions of Harmonic Functions

Let Ω be a domain in ? ^N such that $\left(\mathbb{R}^{N}\cup\lbrace\infty\rbrace\right)\setminus\Omega$ is connected. It is known that, for each w?∈?Ω, there exist harmonic functions on Ω that are universal at w, in the sense that partial sums of their homogeneous polynomial expansion about w approximate all plausibly approximable functions in the complement of Ω. Under the assumption that Ω omits an infinite cone, it is shown that the connectedness hypothesis on $\left(\mathbb{R}^{N}\cup\lbrace\infty\rbrace\right)\setminus\Omega$ is essential, and that a harmonic function which is universal at one point is actually universal at all points of Ω.     

On Stirling numbers and Euler sums

In this paper, we propose another yet generalization of Stirling numbers of the first kind for noninteger values of their arguments. We discuss the analytic representations of Stirling numbers through harmonic numbers, the generalized hypergeometric function and the logarithmic beta integral. We present then infinite series involving Stirling numbers and demonstrate how they are related to Euler sums. Finally, we derive the closed form for the multiple zeta function ζ(p, 1,…, 1) for Re(p)>1.     

Some Remarks on the Eigenvalue Multiplicities of the Laplacian on Infinite Locally Finite Trees

We consider the continuous Laplacian on an infinite uniformly locally finite network under natural transition conditions as continuity at the ramification nodes and the classical Kirchhoff flow condition at all vertices in a L ^∞-setting. The characterization of eigenvalues of infinite multiplicity for trees with finitely many boundary vertices (von Below and Lubary, Results Math 47:199–225, 2005, 8.6) is generalized to the case of infinitely many boundary vertices. Moreover, it is shown that on a tree, any eigenspace of infinite dimension contains a subspace isomorphic to ${\ell^\infty({\mathbb N})}$ . As for the zero eigenvalue, it is shown that a locally finite tree either is a Liouville space or has infinitely many linearly independent bounded harmonic functions if the edge lengths do not shrink to zero anywhere. This alternative is shown to be false on graphs containing circuits.     

Chaotic vibrations of flexible infinite length cylindrical panels using the Kirchhoff–Love model

Treated as continuous deformable systems with an infinite number of degrees of freedom, flexible infinite length cylindrical panels subject to harmonic load are studied. Using the finite difference method with respect to spatial coordinates, the continuous system is reduced to lumped one governed by ordinary differential equations. These equations are transformed to a normal form and then solved numerically using the fourth order Runge–Kutta method. In order to trace and explain vibrational behaviour, dependencies w_max(q₀) and Lyapunov exponents are calculated for panels with parameter value k_x = 48. The corresponding charts of the control parameters {q₀, ω_q} are also reported. Novel scenarios yielding chaotic dynamics exhibited by cylindrical panels are illustrated and discussed.     

Bubbling analysis for approximate Lorentzian harmonic maps from Riemann surfaces

For a sequence of approximate harmonic maps \((u_n,v_n)\) (meaning that they satisfy the harmonic system up to controlled error terms) from a compact Riemann surface with smooth boundary to a standard static Lorentzian manifold with bounded energy, we prove that identities for the Lorentzian energy hold during the blow-up process. In particular, in the special case where the Lorentzian target metric is of the form \(g_N -\beta dt^2\) for some Riemannian metric \(g_N\) and some positive function \(\beta \) on N, we prove that such identities also hold for the positive energy (obtained by changing the sign of the negative part of the Lorentzian energy) and there is no neck between the limit map and the bubbles. As an application, we complete the blow-up picture of singularities for a harmonic map flow into a standard static Lorentzian manifold. We prove that the energy identities of the flow hold at both finite and infinite singular times. Moreover, the no neck property of the flow at infinite singular time is true.     

Two-dimensional steady-state general solution for isotropic thermoelastic materials with applications. II. Green’s function for two-phase infinite plane

Green’s function for isotropic thermoelastic two-phase infinite plane under a line heat source is established in this paper. By virtue of the fourth compact general solutions in Part I which is expressed in three harmonic functions, six new suitable harmonic functions with undetermined constants are constructed for the two semi-infinite planes of the two-phase infinite plane, respectively. The corresponding thermoelastic field can be obtained by substituting these harmonic functions into the general solution, and the undetermined constants can be determined by compatibility conditions and the equilibrium conditions. Numerical results are given graphically by contours.     

Hausdorff Dimension of Harmonic Measure for Self-Conformal Sets

Under some technical assumptions it is shown that the Hausdorff dimension of the harmonic measure on the limit set of a conformal infinite iterated function system is strictly less than the Hausdorff dimension of the limit set itself if the limit set is contained in a real-analytic curve, if the iterated function system consists of similarities only, or if this system is irregular. As a consequence of this general result the same statement is proven for hyperbolic and parabolic Julia sets, finite parabolic iterated function systems and generalized polynomial-like mappings. Also sufficient conditions are provided for a limit set to be uniformly perfect and for the harmonic measure to have the Hausdorff dimension less than 1. Some results in the spirit of Przytycki et al. (Ann. of Math.130 (1989), 1-40; Stud. Math.97 (1991), 189-225) are obtained.