Infinite Sum Research Articles

Casimir energy of hyperbolic orbifolds with conical singularities

In this article, we obtain the explicit expression of the Casimir energy for compact hyperbolic orbifold surfaces in terms of the geometrical data of the surfaces with the help of zeta-regularization techniques. The orbifolds may have finitely many conical singularities. In computing the contribution to the energy from a conical singularity, we derive an expression of an elliptic orbital integral as an infinite sum of special functions. We prove that this sum converges exponentially fast. Additionally, we show that under a natural assumption known to hold asymptotically on the growth of the lengths of primitive closed geodesics of the (2, 3, 7)-triangle group orbifold, its Casimir energy is positive (repulsive).

Tunnelling amplitudes through localised external potentials from Feynman diagram summation

AbstractCurrently there is no general theory of quantum tunnelling of a particle through a potential barrier which is compatible with QFT. We present a complete calculation of tunnelling amplitudes for a scalar field for some simple potentials using quantum field-theoretic methods. Using the perturbative S-matrix formalism, starting with the Klein–Gordon Lagrangian, we show that an infinite summation of Feynman diagrams can recover tunnelling amplitudes consistent with relativistic quantum mechanics. While this work does not include many-particle effects arising from a fully quantised QFT, it is necessary to investigate QFT corrections to tunnelling amplitudes.

Some identities on degenerate harmonic and degenerate higher-order harmonic numbers

The harmonic numbers and higher-order harmonic numbers appear frequently in several areas which are related to combinatorial identities, many expressions involving special functions in analytic number theory, and analysis of algorithms. The aim of this paper is to study the degenerate harmonic and degenerate higher-order harmonic numbers, which are respectively degenerate versions of the harmonic and higher-order harmonic numbers, in connection with the degenerate zeta and degenerate Hurwitz zeta function. Here the degenerate zeta and degenerate Hurwitz zeta function are respectively degenerate versions of the Riemann zeta and Hurwitz zeta function. We show that several infinite sums involving the degenerate higher-order harmonic numbers can be expressed in terms of the degenerate zeta function. Furthermore, we demonstrate that an infinite sum involving finite sums of products of the degenerate harmonic numbers can be represented by using the degenerate Hurwitz zeta function.

PowerPoint as a Tool to Support the Creation of Dynamic Models for Teaching: The Case of the Sum of an Infinite Geometric Sequence

PowerPoint is a learning aid, developed for presentations. PowerPoint has “dynamic” features for creating presentations with dynamic actions. The knowledge of the infinite backward multiplier is difficult to understand. One of the difficulties in understanding this knowledge is infinity, “dynamic”. In this study, we use the dynamic features of PowerPoint to design the sum of an infinite geometric sequence teaching. The study was conducted with 32 students in 11th grade. Students experimented with the dynamic model in PowerPoint. Research results show that the flexible use of PowerPoint’s dynamic features allows the creation of dynamic models of the sum of an infinite geometric sequence. Dynamic models have made it easier for students to build knowledge, creating geometric images of infinite sums. Moreover, with the dynamic features of PowerPoint, creating artistic images, attracts students to actively participate in learning.

Explicit evaluation of a family of Berndt-type integrals

In this paper, we compute a family of infinite integrals involving hyperbolic and trigonometric functions, which we will call Berndt-type integrals. By contour integration, these integrals are first converted to some hyperbolic (infinite) sums, all of which can be calculated in closed forms by two different methods. The first method is indirect and we need to apply some previous results which are computational in nature, and therefore it does not provide general formulas and is inexplicit. For the second method, by comparing the Fourier series expansion and Maclaurin series expansion of one of the Jacobi elliptic functions we are able to directly express the same type of hyperbolic sums more clearly and concisely. When a parameter in these series is set to π, these sums can be expressed as rational polynomials of Γ4(1/4) and 1/π which give rise to the closed formulas of the Berndt-type integrals we are interested in. Moreover, we obtain a structural theorem on the original Berndt integral.

Analysis Of The Asymmetry Factor In Photophoresis Problems With Uniform Polychromatic Fields And Spherical Absorbers

Photophoresis with plane waves and spherical absorbers is traditionally investigated by considering monochromatic radiation. The asymmetry factor, which is proportional to the photophoretic force, can be expressed in using the Mie theory in terms of an infinite sum of terms depending on the wavelength of the impinging light. Here, we analyze an extension of the photophoresis problem for a polychromatic uniform optical field. Such an analysis demands the introduction of a spectrum which describes the weight of each plane wave for the total, polychromatic field. We expect that this work can find applications or pave the way for more advanced investigations in diverse areas of physics and engineering such as optical levitation, optical trap displays and spectroscopy.

Black hole bulk-cone singularities

Lorentzian correlators of local operators exhibit surprising singularities in theories with gravity duals. These are associated with null geodesics in an emergent bulk geometry. We analyze singularities of the thermal response function dual to propagation of waves on the AdS Schwarzschild black hole background. We derive the analytic form of the leading singularity dual to a bulk geodesic that winds around the black hole. Remarkably, it exhibits a boundary group velocity larger than the speed of light, whose dual is the angular velocity of null geodesics at the photon sphere. The strength of the singularity is controlled by the classical Lyapunov exponent associated with the instability of nearly bound photon orbits. In this sense, the bulk-cone singularity can be identified as the universal feature that encodes the ubiquitous black hole photon sphere in a dual holographic CFT. To perform the computation analytically, we express the two-point correlator as an infinite sum over Regge poles, and then evaluate this sum using WKB methods. We also compute the smeared correlator numerically, which in particular allows us to check and support our analytic predictions. We comment on the resolution of black hole bulk-cone singularities by stringy and gravitational effects into black hole bulk-cone “bumps”. We conclude that these bumps are robust, and could serve as a target for simulations of black hole-like geometries in table-top experiments.

Exact results for giant graviton four-point correlators

We study the four-point correlator O2O2DD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left\\langle {\\mathcal{O}}_2{\\mathcal{O}}_2\\mathcal{DD}\\right\\rangle $$\\end{document} in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills theory (SYM) with SU(N) gauge group, where O2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{O}}_2 $$\\end{document} represents the superconformal primary operator with dimension two, while D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{D} $$\\end{document} denotes a determinant operator of dimension N, which is holographically dual to a giant graviton D3-brane extending along S5. We analyse the integrated correlator associated with this observable, obtained after integrating out the spacetime dependence over a supersymmetric invariant measure. Similarly to other classes of integrated correlators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM, this integrated correlator can be computed through supersymmetric localisation on the four-sphere. Employing matrix-model recursive techniques, we demonstrate that the integrated correlator can be reformulated as an infinite sum of protected three-point functions with known coefficients. This insight allows us to circumvent the complexity associated with the dimension-N determinant operator, significantly streamlining the large-N expansion of the integrated correlator. In the planar limit and beyond, we derive exact results for the integrated correlator valid for all values of the ’t Hooft coupling, and investigate the resurgent properties of their strong coupling expansion. Additionally, in the large-N expansion with fixed (complexified) Yang-Mills coupling, we deduce the SL(2, ℤ) completion of these results in terms of the non-holomorphic Eisenstein series. The proposed modular functions are confirmed by explicit instanton calculations from the matrix model, and agree with expectations from the holographic dual picture of known results in type IIB string theory.

Infinite sums of reciprocal quadratics

Infinite sums of reciprocal quadratics

Non-Parametric Estimation of the Renewal Function for Multidimensional Random Fields

This paper addresses the almost sure convergence and the asymptotic normality of an estimator of the multidimensional renewal function based on random fields. The estimator is based on a sequence of non-negative independent and identically distributed (i.i.d.) multidimensional random fields and is expressed as infinite sums of k-folds convolutions of the empirical distribution function. It is an extension of the work from the case of the two-dimensional random fields to the case of the d-dimensional random fields where d>2. This is established by the definition of a “strict order relation”. Concrete applications are given.

New prime number theory

This paper introduces a novel approach to estimating the sum of prime numbers by leveraging insights from partition theory, prime number gaps, and the angles of triangles. The methodology is applied to infinite sums and the nth sum, and several ways of defining the nth sum of a prime number are proposed. By using the Ramanujan infinite series of natural numbers, it is possible to derive an infinite series of prime numbers.

Exploring the Extended Beta-Logarithmic Function: Matrix Arguments and Properties

The beta-logarithmic function substantially generalizes the standard beta function, which is widely recognized for its significance in many applications. This article is devoted to the study of a generalization of the classical beta-logarithmic function in a matrix setting called the extended beta-logarithmic matrix function. The proofs of some essential properties of this extension, such as convergence, partial derivative formulas, functional relations, integral representations, inequalities, and finite and infinite sums, are established. Moreover, an application of the extended beta-logarithmic function in matrix arguments is proposed in probability theory. Further, numerical examples and graphical presentations of the new generalization are obtained.

Some infinite series summations involving linear recurrence relations of order 2 and 3

This paper extends known results of second and third order recursive sequences through extensive formulations of properties of the roots of their characteristic equations, some are old but most are new. They are applied to novel studies of $\sum_{n=0}^{\infty}{\frac{a_{mn}}{10^{n+1}}, \ m=1,2,3}$, including their convergence criteria, and applied to many standard sequences, as particular cases of a generic $\left\{a_n\right\}$. The detailed development of the algebra of the pertinent theorems, and their associated lemmas and corollaries, should open up new vistas for interested number theorists with the concluding results on series values.

Spinning particle geometries in AdS3/CFT2

We study spinning particle/defect geometries in the context of AdS3/CFT2. These solutions lie below the BTZ threshold, and can be obtained from identifications of AdS3. We construct the Feynman propagator by solving the bulk equation of motion in the spinning particle geometry, summing over the modes of the fields and passing to the boundary. The quantization of the scalar fields becomes challenging when confined to the regions that are causally well-behaved. If the region containing closed timelike curves (CTCs) is included, the normalization of the scalar fields enjoys an analytical simplification and the propagator can be expressed as an infinite sum over image geodesics. In the dual CFT2, the propagator can be recast as the HHLL four-point function, where by taking into account the PSL(2, ℤ) modular images, we recover the bulk computation. We comment on the casual behavior of bulk geometries associated with single-trace operators of spin scaling with the central charge below the BTZ threshold.

A study of Horn matrix functions and Horn confluent matrix functions

Abstract In this paper, we give the matrix version of Horn’s hypergeometric function and their confluent cases. We also discuss the regions of convergence, system of matrix differential equations of bilateral type, differential formulae and infinite summation formulae satisfied by these hypergeometric matrix functions. With the study of these 23 matrix functions, matrix generalization of Horn’s list of 34 hypergeometric series will be completed.

On a finite sum of cosecants appearing in various problems

Finite trigonometric sums is a challenging and often quite difficult object of study. In this paper we investigate the finite sum of cosecants ∑csc⁡(φ+aπl/n), where the summation index l runs through 1 to n−1 and φ and a are arbitrary parameters, as well as several closely related sums, such as similar sums of a series of secants, of tangents and of cotangents. These trigonometric sums appear in various problems in mathematics, physics, and a variety of related disciplines. Their particular cases were fragmentarily considered in previous works, and it was noted that even a simple particular case ∑csc⁡(πl/n) does not have a closed–form, i.e. a compact summation formula (similar sums of sines, of cosines, of tangents, of cotangents and even of secants, if they exist, possess such expressions). In this paper, we derive several alternative representations for the above–mentioned sums, study their properties, relate them to many other finite and infinite sums, obtain their complete asymptotic expansions for large n and provide accurate upper and lower bounds (e.g. the typical relative error for the upper bound is lesser than 2×10−9 for n⩾10 and lesser than 7×10−14 for n⩾50, which is much better than the bounds we could find in previous works). Our researches reveal that these sums are deeply related to several special numbers and functions, especially to the digamma function (furthermore, as a by-product, we obtain several interesting summation formulæ for the digamma function). Asymptotical studies show that these sums may have qualitatively different behaviour depending on the choice of φ and a. In particular, as n increases some of them may become sporadically large and we identify the terms of the asymptotics responsible for such a behaviour. Finally, we also provide several historical remarks related to various sums considered in the paper. We show that some results in the field either were rediscovered several times or can easily be deduced from various known formulæ, including some formulæ dating back to the XVIIIth century.

Infinite Sums Involving Powers of Gibonacci Polynomials

Infinite Sums Involving Powers of Gibonacci Polynomials

A new method for solving the linearized 1D Vlasov–Poisson system yielding a new class of solutions

We describe a new method for solving the linearized 1D Vlasov–Poisson system by using properties of Cauchy-type integrals. Our method remedies critical flaws of the two standard methods, reveals a previously unrecognized Gaussian-in-time-like decay, and can also account for an externally applied electric field. The Landau approximation involves deforming the Bromwich contour around the poles closest to the real axis due to the analytically continued dielectric function, finding the long-time behavior for a stable system: Landau damping. Jackson's generalization encircles all poles while sending the contour to infinity, assuming its contribution vanishes, which is not true in general. This gives incorrect solutions for physically reasonable configurations and can exhibit pathological behavior, of which we show examples. The van Kampen method expresses the solution for a stable equilibrium as a continuous superposition of waves, resulting in an opaque integral. Case's generalization includes unstable systems and predicts a decaying discrete mode for each growing discrete mode, an apparent contradiction to both the Jackson solution and ours. We show, without imposing additional constraints, that the decaying modes are never present in the time evolution due to an exact cancellation with part of the continuum. Our solution is free of integral expressions, is obtained using algebra and Laurent series expansions, does not rely on analytic continuations, and results in a correct asymptotically convergent form in the case of infinite sums. The analysis used can be readily applied in higher-dimensional, electromagnetic systems and also provides a new technique for evaluating certain inverse Laplace transforms.

Some properties of classes of minimal numberings of arithmetical set families

We prove that for each u ⩾ 2 the class of all single-valued Σ0 ucomputable numberings of any infinite family of total functions is effectively infinite and the class of all its Σ0 u-1-computable numberings is generated by the downward closure with respect to the reducibility of the set of all infinite direct sums of uniformly Σ0 u-1-computable sequences of its single-valued numberings. It is established that if u > 2, then the class of all Σ0 u-computable numberings of any infinite family is generated by infinite direct sums of uniformly Σ0 u-computable and uniformly Σ0 u-minimal sequences of its numberings.

A compendium of logarithmic corrections in AdS/CFT

We study the logarithmic corrections to various CFT partition functions in the context of the AdS4/CFT3 correspondence for theories arising on the worldvolume of M2-branes. We utilize four-dimensional gauged supergravity and heat kernel methods and present general expressions for the logarithmic corrections to the gravitational on-shell action and black hole entropy for a number of different supergravity backgrounds. We outline several subtle features of these calculations and contrast them with a similar analysis of logarithmic corrections performed directly in the eleven-dimensional uplift of a given four-dimensional supergravity background. We find results consistent with AdS/CFT provided that the infinite sum over KK modes on the internal space is regularized in a specific manner. This analysis leads to an explicit expression for the logarithmic correction to the Bekenstein-Hawking entropy of large Kerr-Newmann and Reissner-Nordström black holes in AdS4. Our results also have important implications for effective field theory coupled to gravity in AdS4 and for the existence of scale-separated AdS4 vacua in string theory, which come in the form of new constraints on the field content and mass spectrum of matter fields.