Higher Transcendental Functions Research Articles

Hypergeometric structures in Feynman integrals

For the precision calculations in perturbative Quantum Chromodynamics (QCD) gigantic expressions (several GB in size) in terms of highly complicated divergent multi-loop Feynman integrals have to be calculated analytically to compact expressions in terms of special functions and constants. In this article we derive new symbolic tools to gain large-scale computer understanding in QCD. Here we exploit the fact that hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package Sigma in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code HypSeries transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code solvePartialLDE is designed. Generalized hypergeometric functions, Appell-, Kampé de Fériet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton–type functions are considered. We illustrate the algorithms by examples.

Higher Transcendental Functions and Their Multi-Disciplinary Applications

This volume consists of a collection of 17 peer-reviewed and accepted submissions from authors around the world (including several invited feature articles) to the Special Issue of the journal Mathematics, on the general subject-area of “Higher Transcendental Functions and Their Multi-Disciplinary Applications” [...]

Some Families of Generating Functions Associated with Orthogonal Polynomials and Other Higher Transcendental Functions

In this invited survey-cum-expository review article, we present a brief and comprehensive account of some general families of linear and bilinear generating functions which are associated with orthogonal polynomials and such other higher transcendental functions as (for example) hypergeometric functions and hypergeometric polynomials in one, two and more variables. Many of the results as well as the methods and techniques used for their derivations, which are presented here, are intended to provide incentive and motivation for further research on the subject investigated in this article.

A Guide to Special Functions in Fractional Calculus

Dedicated to the memory of Professor Richard Askey (1933–2019) and to pay tribute to the Bateman Project. Harry Bateman planned his “shoe-boxes” project (accomplished after his death as Higher Transcendental Functions, Vols. 1–3, 1953–1955, under the editorship by A. Erdélyi) as a “Guide to the Functions”. This inspired the author to use the modified title of the present survey. Most of the standard (classical) Special Functions are representable in terms of the Meijer G-function and, specially, of the generalized hypergeometric functions pFq. These appeared as solutions of differential equations in mathematical physics and other applied sciences that are of integer order, usually of second order. However, recently, mathematical models of fractional order are preferred because they reflect more adequately the nature and various social events, and these needs attracted attention to “new” classes of special functions as their solutions, the so-called Special Functions of Fractional Calculus (SF of FC). Generally, under this notion, we have in mind the Fox H-functions, their most widely used cases of the Wright generalized hypergeometric functions pΨq and, in particular, the Mittag–Leffler type functions, among them the “Queen function of fractional calculus”, the Mittag–Leffler function. These fractional indices/parameters extensions of the classical special functions became an unavoidable tool when fractalized models of phenomena and events are treated. Here, we try to review some of the basic results on the theory of the SF of FC, obtained in the author’s works for more than 30 years, and support the wide spreading and important role of these functions by several examples.

Weakest link concept for springs fatigue

ABSTRACTIn the present manuscript, a new approach is developed to account the stress gradient effect on fatigue life of springs. The new formula of crack growths per cycle is introduced. The expressions for spring length over the number of cycles are derived in terms of higher transcendental function. The closed-form solutions are used for the estimation of the fatigue life of heterogeneously stressed structural members. The probability distribution of the fatigue limit for heterogeneously stressed structural elements is evaluated. The proposed approach for the stress gradient sensitivity of fatigue life is based on the weakest link concept. The weakest link approach is applied to calculate the number of cycles to crack initiation of structural elements under different probability levels. The effect of stress ratio on elongation of crack is discussed. The developed theory is applied to helical springs under cyclic load. The fatigue sensitivity to stress concentration is addressed in application to springs. Effect of fluctuating stresses on fatigue life of springs is combined with the influence of heterogeneous stress distribution (stress gradient) over the cross-section of wire and time-varying stresses. These two factors lead to complicated evaluation for fatigue life of helical springs. The stress field is inhomogeneous over the cross-section of the wire of spring. The stress distribution is uniquely defined by ratio of the diameter of wire to the diameter of spring body. The calculated lifetimes are compared with the lifetimes obtained from experiments performed on helical springs subjected to cyclic load of constant amplitude.

Chaotic processes using the two-parameter derivative with non-singular and non-local kernel: Basic theory and applications.

After having the issues of singularity and locality addressed recently in mathematical modelling, another question regarding the description of natural phenomena was raised: How influent is the second parameter β of the two-parameter Mittag-Leffler function Eα,β(z), z∈ℂ? To answer this question, we generalize the newly introduced one-parameter derivative with non-singular and non-local kernel [A. Atangana and I. Koca, Chaos, Solitons Fractals 89, 447 (2016); A. Atangana and D. Bealeanu (e-print)] by developing a similar two-parameter derivative with non-singular and non-local kernel based on Eα , β(z). We exploit the Agarwal/Erdelyi higher transcendental functions together with their Laplace transforms to explicitly establish the Laplace transform's expressions of the two-parameter derivatives, necessary for solving related fractional differential equations. Explicit expression of the associated two-parameter fractional integral is also established. Concrete applications are done on atmospheric convection process by using Lorenz non-linear simple system. Existence result for the model is provided and a numerical scheme established. As expected, solutions exhibit chaotic behaviors for α less than 0.55, and this chaos is not interrupted by the impact of β. Rather, this second parameter seems to indirectly squeeze and rotate the solutions, giving an impression of twisting. The whole graphics seem to have completely changed its orientation to a particular direction. This is a great observation that clearly shows the substantial impact of the second parameter of Eα , β(z), certainly opening new doors to modeling with two-parameter derivatives.

Mathematical Solvability of a Caputo Fractional Polymer Degradation Model Using Further Generalized Functions

The continuous fission equation with derivative of fractional orderα, describing the polymer chain degradation, is solved explicitly. We prove that, whether the breakup rate depends on the size of the chain breaking up or not, the evolution of the polymer sizes distribution is governed by a combination of higher transcendental functions, namely, Mittag-Leffler function, the further generalizedG-function, and the Pochhammer polynomial. In particular, this shows the existence of an eigenproperty; that is, the system describing fractional polymer chain degradation contains replicated and partially replicated fractional poles, whose effects are given by these functions.

The uniformization of certain algebraic hypergeometric functions

The hypergeometric functions Fn−1n are higher transcendental functions, but for certain parameter values they become algebraic, because the monodromy of the defining hypergeometric differential equation becomes finite. It is shown that many algebraic Fn−1nʼs, for which the finite monodromy is irreducible but imprimitive, can be represented as combinations of certain explicitly algebraic functions of a single variable; namely, the roots of trinomials. This generalizes a result of Birkeland, and is derived as a corollary of a family of binomial coefficient identities that is of independent interest. Any tuple of roots of a trinomial traces out a projective algebraic curve, and it is also determined when this so-called Schwarz curve is of genus zero and can be rationally parametrized. Any such parametrization yields a hypergeometric identity that explicitly uniformizes a family of algebraic Fn−1nʼs. Many examples of such uniformizations are worked out explicitly. Even when the governing Schwarz curve is of positive genus, it is shown how it is sometimes possible to construct explicit single-valued or multivalued parametrizations of individual algebraic Fn−1nʼs, by parametrizing a quotiented Schwarz curve. The parametrization requires computations in rings of symmetric polynomials.

Integral representations and summations of the modified Struve function

It is known that Struve function $\mathbf H_\nu$ and modified Struve function $\mathbf L_\nu$ are closely connected to Bessel function of the first kind $J_\nu$ and to modified Bessel function of the first kind $I_\nu$ and possess representations through higher transcendental functions like generalized hypergeometric ${}_1F_2$ and Meijer $G$ function. Also, the NIST project and Wolfram formula collection contain a set of Kapteyn type series expansions for $\mathbf L_\nu(x)$. In this paper firstly, we obtain various another type integral representation formulae for $\mathbf L_\nu(x)$ using the technique developed by D. Jankov and the authors. Secondly, we present some summation results for different kind of Neumann, Kapteyn and Schl\"omilch series built by $I_\nu(x)$ and $\mathbf L_\nu(x)$ which are connected by a Sonin--Gubler formula, and by the associated modified Struve differential equation. Finally, solving a Fredholm type convolutional integral equation of the first kind, Bromwich--Wagner line integral expressions are derived for the Bessel function of the first kind $J_\nu$ and for an associated generalized Schl\"omilch series.

Error-rate analysis of the OFDM for correlated Nakagami-m fading channel by using maximal-ratio combining diversity

In this paper, we have analyzed the performance of correlated Nakagami-m fading channel by using the maximal-ratio-combing diversity at the receiver. A closed-form mathematical expression is derived for the average bit error rate (BER) for binary phase-shift keying (BPSK) and average symbol-error-rate (SER) for M-Quardrature amplitude modulation (M-QAM) scheme in terms of the higher transcendental function such as Appell hypergeometric function by using the well-known moment generating function (MGF)-based approach with arbitrary fading index for the orthogonal frequency division multiplexing (OFDM) communication systems. Moreover, we also derived an expression for the outage probability and the proposed numerical results are compared with the reported literature.

Impedance of an antenna-array element

A number of expressions for calculating the active impedances of the radiating elements of large linear and planar antenna arrays with allowance for the interaction of the entire set of the elements are presented. The expressions are applicable not only to the radiating elements distant from the edge of the array but also to the edge and corner elements. The expressions are based on the formulas for certain sums of elementary and higher transcendental functions that give simple closed-form expressions for the sums of mutual impedances. As an illustration, the resistances of weakly directional radiating elements in dense arrays, where the mutual coupling of the elements is extremely strong and results in important effects, are calculated.

Subordination to periodic processes and synchronization

We study the subordination to a process that is periodic in the natural time scale, and equivalent to a clock with N states. The rationale for this investigation is given by a set of many interacting clocks with N states. The natural time scale representation corresponds to the dynamics of an individual clock with no interaction with the other clocks of this set. We argue that the cooperation among the clocks of this set has the effect of generating a global clock, whose times of sojourn in each of its N states are described by a distribution density with an inverse power law form and power index μ < 2 . This is equivalent to extending the widely used subordination method from fluctuation–dissipation processes to periodic processes, thereby raising the question of whether special conditions exist of perfect synchronization, signaled by regular oscillations, and especially by oscillations with no damping. We study first the case of a Poisson subordination function. We show that in spite of the random nature of the subordination method the procedure has the effect of creating damped oscillations, whose damping vanishes in the limiting case of N ≫ 1 , thereby suggesting a condition of perfect synchronization in this limit. The Bateman’s mathematical arguments [H. Bateman, Higher Transcendental Functions, vol. III, Robert K Krieger, Publishing Company, Inc. Krim.Fr. Drive Malabar, FL; Copyright 1953 by McGraw-Hill Book Company Inc.] indicate that the condition of perfect synchronization is possible also in the non-Poisson case, with μ < 2 , although it may lie beyond the range of computer simulation. To make the theoretical predictions accessible to numerical simulation, we use a subordination function whose survival probability is a Mittag–Leffler exponential function. This method prevents us from directly establishing the macroscopic coherence emerging from μ = 2 , which generates a perfect form of 1 / f noise. However, it affords indirect evidence that perfect synchronization signaled by undamped regular oscillations may be produced in this case. Furthermore, we explore a condition characterized by an excellent agreement between theory and numerical simulation, where the long-time region relaxation, with a perfect inverse power law decay, emerging from the subordination to ordinary fluctuation–dissipation processes, is replaced by exponentially damped regular oscillations.

Closed-Form Symbol Error Probabilities of STBC and CDD MC-CDMA With Frequency-Correlated Subcarriers Over Nakagami-$m$ Fading Channels

The performances of cyclic delay diversity and space-time block-coded multicarrier code-division multiple-access (MC-CDMA) systems operating in a Nakagami-m fading channel with correlated subcarriers are investigated. By using the moment-generating function of the correlated Nakagami-m random variable, explicit closed-form formulas for the exact average symbol error rate (SER) of M-ary signals for an MC-CDMA system employing the aforementioned diversity techniques are derived with a maximum ratio combining detection technique. The average SERs are expressed in terms of higher transcendental functions, such as the Gauss and Appell hypergeometric functions. Both numerical and simulation results are presented, showing excellent agreement.

Closed-form expressions for symbol error probability of orthogonal space–time block codes over Rician–Nakagami channels

The performances of orthogonal space–time block codes (OSTBCs) over Rician–Nakagami channels are investigated. In particular, we derive closed-form symbol error probability (SEP) expressions for OSTBC systems in which M-ary phase-shift-keying modulation and M-ary quadrature-amplitude modulation are used. These SEP results are expressed in terms of Lauricella's multivariate hypergeometric functions, which can be easily evaluated numerically. When the Rician–Nakagami channel degenerates to the Rician–Rayleigh channel, or equivalently the Rayleigh fading channel, the closed-form SEP expressions are rewritten in terms of higher transcendental functions, that is, Gauss hypergeometric function and Appell hypergeometric function.

Comparisons between predicted and measured acoustic scattering from a wedge

Our earlier work has mainly dealt with developing a mathematical approach for predicting acoustc scattering from a wedge for various boundary conditions and wedge material properties. We initially reported scattering for wedge surfaces that satisfy a Dirichlet or a Neumann boundary condition. Recently, our approach was extended to include wedge boundary conditions that can be characterized by an impedance, including absorption. Numerical predictions are found using the Kontorovich-Lebedev transform [A. Erdelyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953)] based upon MacDonald functions, requiring very high precision numerical techniques. We now apply this numerical approach to calculate the monostatic scattering amplitude for a fabricated wedge structure that has been measured in a nonlaboratory environment. Comparisons will be presented and discussed of the results obtained from the predicted and measured monostatic scattering pressures for a range of insonification frequencies at selected offset distances and angles relative to the edge of the wedge.

Structural Aspects of Numerical Loop Calculus

From higher transcendental functions to smooth integral representations in the evaluation of multi-loop Feynman diagrams: sorrily abandoning the analytical way.

Analysis of a beam-column system under varying axial forces of elliptic type: the exact solution of Lamé’s equation

We investigate the dynamical response of a beam-column system with hinged ends subjected to an axial pulsating force of elliptic type. It is shown that the resulting equation is of the form d 2y 1 dτ 2 +y 1[d 1+d 2cn 2(τ,k 2)]=0, which is the well-known Lamé equation [Higher Transcendental Functions, Bateman Manuscript Project, edited by McGraw-Hill, New York, vol. 3]. In this paper, we obtain the general exact solution of this equation that reveals stable behavior of the beam-column system if the assigned initial conditions are of the form y 1(0)= y 10 and y ̇ 1(0)=0 . It is also found that at a certain value of the modulus of the elliptic force, the lateral vibrational frequency is independent of the material properties of the beam-column system.

Further results on acoustic scattering from the edge of a wedge

Acoustic monostatic and bistatic scattering from the edge of a solid wedge immersed in water is analyzed further, always assuming a line source parallel to the edge of a wedge. Our previous work on the subject is extended by emphasizing the dependence of the scattering amplitude on the wedge material, which may be uniform or layered, and which can be characterized by surface impedance. Impedance values are obtained for multiple layers by the method of Brekhovskihk. Our numerical analysis is based upon Macdonald functions [A. Erdelyi, Higher Transcendental Functions (McGraw–Hill, New York, 1953)] via an integral representation that we were able to render convergent. The scattering amplitude is integral bilinear in the Macdonald functions and dependent upon the wedge impedance, and has been evaluated numerically. [Work supported by NAVSEA.]

On the Convergence of Borel Approximants

For a wide class of functions having a representation as Laplace integrals, we obtain expansions in terms of certain higher transcendental functions. Compared with the classical method of expansions as factorial series, these series converge considerably faster and, at the same time, in larger regions.

Is Normal Distribution due to Karl Gauss? Euler, his Family of Gamma Functions, and Place in History of Statistics

While Gauss (1777–1855) has primacy in explicating properties of the normal distribution, it is Euler (1707–1783) who is predominant with respect to analytical formulation of this function. This case is discussed within the context of uniform rendering of computer algorithms for higher transcendental functions.