Four-term Recurrence Relation Research Articles

On polynomials generated by a four-term recurrence relation and linear coefficients

On polynomials generated by a four-term recurrence relation and linear coefficients

Four-Term Recurrence for Fast Krawtchouk Moments Using Clenshaw Algorithm

Krawtchouk polynomials (KPs) are discrete orthogonal polynomials associated with the Gauss hypergeometric functions. These polynomials and their generated moments in 1D or 2D formats play an important role in information and coding theories, signal and image processing tools, image watermarking, and pattern recognition. In this paper, we introduce a new four-term recurrence relation to compute KPs compared to their ordinary recursions (three-term) and analyse the proposed algorithm speed. Moreover, we use Clenshaw’s technique to accelerate the computation procedure of the Krawtchouk moments (KMs) using a fast digital filter structure to generate a lattice network for KPs calculation. The proposed method confirms the stability of KPs computation for higher orders and their signal reconstruction capabilities as well. The results show that the KMs calculation using the proposed combined method based on a four-term recursion and Clenshaw’s technique is reliable and fast compared to the existing recursions and fast KMs algorithms.

Principal Solutions of Recurrence Relations and Irrationality Questions in Number Theory

We apply the theory of disconjugate linear recurrence relations to the study of irrational quantities in number theory. In particular, for an irrational number associated with solutions of linear three-term recurrence relations, we show that there exists a four-term linear recurrence relation whose solutions show that the number has an irrational square if and only if the four-term recurrence relation has a principal solution of a certain type. The result is extended to higher-order recurrence relations, and a transcendence criterion can also be formulated in terms of these principal solutions. The method generates new series expansions of positive integer powers of ζ(3) and ζ(2) in terms of Apéry’s now classic sequences.

Three- and four-term recurrence relations for Horn's hypergeometric function $H_4$

Three- and four-term recurrence relations for hypergeometric functions of the second order (such as hypergeometric functions of Appell, Horn, etc.) are the starting point for constructing branched continued fraction expansions of the ratios of these functions. These relations are essential for obtaining the simplest structure of branched continued fractions (elements of which are simple polynomials) for approximating the solutions of the systems of partial differential equations, as well as some analytical functions of two variables. In this study, three- and four-term recurrence relations for Horn's hypergeometric function $H_4$ are derived. These relations can be used to construct branched continued fraction expansions for the ratios of this function and they are a generalization of the classical three-term recurrent relations for Gaussian hypergeometric function underlying Gauss' continued fraction.

Context-Free Grammars for Several Triangular Arrays

In this paper, we present a unified grammatical interpretation of the numbers that satisfy a kind of four-term recurrence relation, including the Bell triangle, the coefficients of modified Hermite polynomials, and the Bessel polynomials. Additionally, as an application, a criterion for real zeros of row-generating polynomials is also presented.

Quantum Rajeev–Ranken model as an anharmonic oscillator

The Rajeev–Ranken (RR) model is a Hamiltonian system describing screw-type nonlinear waves of wavenumber k in a scalar field theory pseudodual to the 1 + 1D SU(2) principal chiral model. Classically, the RR model is Liouville integrable. Here, we interpret the model as a novel 3D cylindrically symmetric quartic oscillator with an additional rotational energy. The quantum theory has two dimensionless parameters. Upon separating variables in the Schrödinger equation, we find that the radial equation has a four-term recurrence relation. It is of type [0, 1, 16] and lies beyond the ellipsoidal Lamé and Heun equations in Ince’s classification. At strong coupling λ, the energies of highly excited states are shown to depend on the scaling variable λk. The energy spectrum at weak coupling and its dependence on k in a double-scaling strong coupling limit are obtained. The semi-classical Wentzel-Kramers-Brillouin (WKB) quantization condition is expressed in terms of elliptic integrals. Numerical inversion enables us to establish a (λk)2/3 dispersion relation for highly energetic quantized “screwons” at moderate and strong coupling. We also suggest a mapping between our radial equation and the one of Zinn-Justin and Jentschura that could facilitate a resurgent WKB expansion for energy levels. In another direction, we show that the equations of motion of the RR model can also be viewed as Euler equations for a step-3 nilpotent Lie algebra. We use our canonical quantization to uncover an infinite dimensional reducible unitary representation of this nilpotent algebra, which is then decomposed using its Casimir operators.

Plancherel–Rotach type asymptotic formulae for multiple orthogonal Hermite polynomials and recurrence relations

We study the asymptotic properties of multiple orthogonal Hermite polynomials which are determined by the orthogonality relations with respect to two Hermite weights (Gaussian distributions) with shifted maxima. The starting point of our asymptotic analysis is a four-term recurrence relation connecting the polynomials with adjacent numbers. We obtain asymptotic expansions as the number of the polynomial and its variable grow consistently (the so-called Plancherel–Rotach type asymptotic formulae). Two techniques are used. The first is based on constructing expansions of bases of homogeneous difference equations, and the second on reducing difference equations to pseudodifferential ones and using the theory of the Maslov canonical operator. The results of these approaches agree.

Generalized Hypergeometric Function 3F2 Ratios and Branched Continued Fraction Expansions

The paper is related to the classical problem of the rational approximation of analytic functions of one or several variables, particulary the issues that arise in the construction and studying of continued fraction expansions and their multidimensional generalizations—branched continued fraction expansions. We used combinations of three- and four-term recurrence relations of the generalized hypergeometric function 3F2 to construct the branched continued fraction expansions of the ratios of this function. We also used the concept of correspondence and the research method to extend convergence, already known for a small region, to a larger region. As a result, we have established some convergence criteria for the expansions mentioned above. It is proved that the branched continued fraction expansions converges to the functions that are an analytic continuation of the ratios mentioned above in some region. The constructed expansions can approximate the solutions of certain differential equations and analytic functions, which are represented by generalized hypergeometric function 3F2. To illustrate this, we have given a few numerical experiments at the end.

Some properties of 2-orthogonal polynomials associated with inverse Gaussian distribution

In this work, we discuss several desirable properties of the inverse Gaussian (IG) family involving orthogonal polynomials. In particular, we discuss the cumulant-generating function and associated polynomials. A particularly important property is that the associated polynomials satisfy a four-term recurrence relation. Then we state and calculate the famous creation and annihilation operators of the Heisenberg-Weyl Lie algebra that acting on IG polynomials Pn as the multiplicative operator X and the derivative one D on the monomials xn . Finally, I used the Maple computer algebra system to program and Calculate the six first IG polynomials and their roots numerically.

GLOBAL ATTRACTIVITY IN A FOUR-TERM RECURRENCE RELATION


 
 
 Positive solutions of the four-term recurrence relation $x_{n+1}=f(x_n)g(x_{n-1}, x_{n-2})$ are shown to converge to its positive equalibrium points under relative mild conditions. 
 
 

Exact solutions of Mathieu’s equation

Abstract Mathieu’s equation originally emerged while studying vibrations on an elliptical drumhead, so naturally, being a linear second-order ordinary differential equation with a Cosine periodic potential, it has many useful applications in theoretical and experimental physics. Unfortunately, there exists no closed-form analytic solution of Mathieu’s equation, so that future studies and applications of this equation, as evidenced in the literature, are inevitably fraught by numerical approximation schemes and nonlinear analysis of so-called stability charts. The present research work, therefore, avoids such analyses by making exceptional use of Laurent series expansions and four-term recurrence relations. Unexpectedly, this approach has uncovered two linearly independent solutions to Mathie’s equation, each of which is in closed form. An exact and general analytic solution to Mathieu’s equation, then, follows in the usual way of an appropriate linear combination of the two linearly independent solutions.

Representation of a quotient of solutions of a four-term linear recurrence relation in the form of a branched continued fraction

The quotient of two linearly independent solutions of a four-term linear recurrence relation is represented in the form of a branched continued fraction with two branches of branching by analogous with continued fractions. Formulas of partial numerators and partial denominators of this branched continued fraction are obtained. The solutions of the recurrence relation are canonic numerators and canonic denominators of $\mathcal{B}$-figured approximants. Two types of figured approximants $\mathcal{A}$-figured and $\mathcal{B}$-figured are often used. A $n$th $\mathcal{A}$-figured approximant of the branched continued fraction is obtained by adding a next partial quotient to the $(n-1)$th $\mathcal{A}$-figured approximant. A $n$th $\mathcal{B}$-figured approximant of the branched continued fraction is a branched continued fraction that is a part of it and contains all those elements that have a sum of indexes less than or equal to $n$. $\mathcal{A}$-figured approximants are widely used in proving of formulas of canonical numerators and canonical denominators in a form of a determinant, $\mathcal{B}$-figured approximants are used in solving the problem of corresponding between multiple power series and branched continued fractions. A branched continued fraction of the general form cannot be transformed into a constructed branched continued fraction. For calculating canonical numerators and canonical denominators of a branched continued fraction with $N$ branches of branching, $N>1$, the linear recurrent relations do not hold. $\mathcal{B}$-figured convergence of the constructed fraction in a case when coefficients of the recurrence relation are real positive numbers is investigated.

Exact results for the anisotropic honeycomb lattice Green function with applications to three-step Pearson random walks

The mathematical properties of the lattice Green function for the anisotropic honeycomb lattice are studied, where is a complex variable which lies in a plane, and is a real anisotropy parameter with . This double integral defines a single-valued analytic function provided that a cut is made along the real axis from to . In order to analyse the behaviour of along the edges of the cut it is convenient to define the limit function where . It is proved that the limit functions and can be sectionally evaluated exactly for all , in terms of various elliptic integrals of the first kind , where is a rational function of and u.Next, it is demonstrated that is a solution of a second order linear differential equation with eight ordinary regular singular points and two apparent singular points. It is shown that the apparent singularities can be removed by constructing a particular differential equation of third order. The series solution where and is investigated. In particular, we show that, in general, satisfies a four-term linear recurrence relation. This result is used to determine the asymptotic behaviour of as .Integral representations are established for and . It is found that where J0(z) and Y0(z) denote Bessel functions of the first and second kind, respectively, and .Finally, the results are applied to the lattice Green function for the anisotropic simple cubic lattice, and to the theory of Pearson random walks in a plane.

The massive Dirac equation in the Kerr-Newman-de Sitter and Kerr-Newman black hole spacetimes

Exact solutions of the Dirac general relativistic equation that describe the dynamics of a massive, electrically charged particle with half-integer spin in the curved spacetime geometry of an electrically charged, rotating Kerr-Newman-(anti) de Sitter black hole are investigated. We first, derive the Dirac equation in the Kerr-Newman-de Sitter (KNdS) black hole background using a generalised Kinnersley null tetrad in the Newman-Penrose formalism. Subsequently in this frame and in the KNdS black hole spacetime, we prove the separation of the Dirac equation into ordinary differential equations for the radial and angular parts. Under specific transformations of the independent and dependent variables we prove that the transformed radial equation for a massive charged spin fermion in the background KNdS black hole constitutes a highly non-trivial generalisation of Heun’s equation since it possess five regular finite singular points. Using a Regge-Wheeler-like independent variable we transform the radial equation in the KNdS background into a Schrödinger like differential equation and investigate its asymptotic behaviour near the event and cosmological horizons. For the case of a massive fermion in the background of a Kerr-Newman (KN) black hole we first prove that the radial and angular equations that result from the separation of Dirac’s equation reduce to the generalised Heun differential equation (GHE). The local solutions of such GHE are derived and can be described by holomorphic functions whose power series coefficients are determined by a four-term recurrence relation. In addition using asymptotic analysis we derive the solutions for the massive fermion far away from the KN black hole and the solutions near the event horizon . The determination of the separation constant as an eigenvalue problem in the KN background is investigated. Using the aforementioned four-term recursion formula we prove that in the non-extreme KN geometry there are no bound states with , where ω and μ are the energy and mass of the fermion respectively.

SERIES SOLUTIONS OF CONFLUENT HEUN EQUATIONS IN TERMS OF INCOMPLETE GAMMA-FUNCTIONS

We present a simple systematic algorithm for construction of expansions of the solutions of ordinary differential equations with rational coefficients in terms of mathematical functions having indefinite integral representation. The approach employs an auxiliary equation involving only the derivatives of a solution of the equation under consideration. Using power-series expansions of the solutions of this auxiliary equation, we construct several expansions of the four confluent Heun equations' solutions in terms of the incomplete Gamma-functions. In the cases of single- and double-confluent Heun equations the coefficients of the expansions obey four-term recurrence relations, while for the bi- and tri-confluent Heun equations the recurrence relations in general involve five terms. Other expansions for which the expansion coefficients obey recurrence relations involving more terms are also possible. The particular cases when these relations reduce to ones involving less number of terms are identified. The conditions for deriving closed-form finite-sum solutions via right-hand side termination of the constructed series are discussed.

Appell Hypergeometric Expansions of the Solutions of the General Heun Equation

Starting from a second-order Fuchsian differential equation having five regular singular points, an equation obeyed by a function proportional to the first derivative of the solution of the Heun equation, we construct several expansions of the solutions of the general Heun equation in terms of Appell generalized hypergeometric functions of two variables of the first kind. Several cases when the expansions reduce to those written in terms of simpler mathematical functions such as the incomplete Beta function or the Gauss hypergeometric function are identified. The conditions for deriving finite-sum solutions via termination of the series are discussed. In general, the coefficients of the expansions obey four-term recurrence relations; however, there exist certain choices of parameters for which the recurrence relations involve only two terms, though not necessarily successive. For such cases, the coefficients of the expansions are explicitly calculated and the general solution of the Heun equation is constructed in terms of the Gauss hypergeometric functions.

Hierarchies of sum rules for squares of spherical Bessel functions

ABSTRACTA four-term recurrence relation for squared spherical Bessel functions is shown to yield closed-form expressions for several types of finite weighted sums of these functions. The resulting sum rules, which may contain an arbitrarily large number of terms, are found to constitute three independent hierarchies. Their use leads to an efficient numerical evaluation of these sums.

Exact and approximate solutions to Schrödinger’s equation with decatic potentials

Abstract The one-dimensional Schrödinger’s equation is analysed with regard to the existence of exact solutions for decatic polynomial potentials. Under certain conditions on the potential’s parameters, we show that the decatic polynomial potential V (x) = ax 10 + bx 8 + cx 6 + dx 4 + ex 2, a > 0 is exactly solvable. By examining the polynomial solutions of certain linear differential equations with polynomial coefficients, the necessary and sufficient conditions for corresponding energy-dependent polynomial solutions are given in detail. It is also shown that these polynomials satisfy a four-term recurrence relation, whose real roots are the exact energy eigenvalues. Further, it is shown that these polynomials generate the eigenfunction solutions of the corresponding Schrödinger equation. Further analysis for arbitrary values of the potential parameters using the asymptotic iteration method is also presented.

Four-term recurrence relations for γ-forms

Four-term recurrence relations for γ-forms

Four-term leaping recurrence relations

Consider linear three-term recurrence relation Z n = a 1 Z n−1 + a 2 Z n−2 (n ≥ 2). Then, from our former theorem, the three-term leaping recurrence relation holds for any positive integer k. In this article we shall consider the linear four-term recurrence relation Z n = a 1 Z n−1 + a 2 Z n−2 + a 3 Z n−3 (n ≥ 3). Then we can have the four-term leaping recurrence relation Z n = b 1 Z n−k + b 2 Z n−2k + b 3 Z n−3k for a given positive integer k with 1 < k < n/3.