General Recurrence Relation Research Articles

Variational Iteration Technique and Some Methods for the Approximate Solution of Nonlinear Equations

In this paper, we suggest and analyze a new recurrence relation which generates the iterative methods of higher order for solving nonlinear equation f(x) = 0. This general recurrence relation is obtained by using variational iteration technique. We purpose some new iterative methods for solving nonlinear equations. We also test different examples to illustrate the efficienc y of these new methods. Comparison with other similar methods is also given. These new methods can also be considered as alternative to the existing methods. This technique can be applied to suggest a wide class of new iterative methods for solving nonlinear equations.

Closed-form solution of a general three-term recurrence relation

We present a closed-form solution for n th term of a general three-term recurrence relation with arbitrary given n-dependent coefficients. The derivation and corresponding proof are based on two approaches, which we develop and describe in detail. First, the recursive-sum theory, which gives the exact solution in a compact finite form using a recursive indexing. Second, the discrete dimensional-convolution procedure, which transforms the solution to the non-recursive expression of n, including a finite number of elementary operations and functions.

Moments of Progressive Type-II Right Censored Order Statistics from a General Class of Doubly Truncated Continuous Distributions

In this paper we derive some general recurrence relations between moments of progressively Type-II right censored order statistics from a general class of doubly truncated distributions, thus unifying the earlier results in this direction due to several authors.

Current records and record range with some applications

In a sequence of independent and identically distributed (iid) random variables, the upper (lower) current records and record range are studied. We derive general recurrence relations between the single and product moments for the upper and lower current records based on Weibull and positive Weibull distributions, as well as Pareto and negative Pareto distributions, respectively. Moreover, some asymptotic results for general current records are established. In addition, a recurrence relation and an explicit formula for the moments of record range based on the exponential distribution are given. Finally, numerical examples are presented to illustrate and corroborate theoretical results.

Forms representing forms and linear spaces on hypersurfaces

Monotone variational recurrence relations such as the Frenkel-Kontorova lattice, arise in solid state physics, conservative lattice dynamics and as Hamiltonian twist maps. For such recurrence relations, Aubry-Mather theory guarantees the existence of solutions of every rotation number. They are the action minimizers that constitute the Aubry-Mather set. When the rotation number is irrational, the Aubry-Mather set is either connected or a Cantor set. A connected Aubry-Mather set is called a minimal foliation. In the case of twist maps, it describes an invariant circle, while in solid state physics it corresponds to a continuum of ground states. A Cantor Aubry-Mather set is called a minimal lamination. In this paper we prove that when the rotation number of a minimal foliation is either rational or easy to approximate by rational numbers, then the foliation can be destroyed into a lamination by an arbitrarily small smooth perturbation of the recurrence relation. This generalizes a theorem of Mather for twist maps to general recurrence relations.

A Recurrence Relation of Hypergeometric Series Through Record Statistics and a Characterization

In this paper, a new general recurrence relation of hypergeometric series is derived using distribution function of upper record statistics. The result can be extended easily to k-records statistics . A characterization is given on the basis of this recurrence relation.

A general recurrence relation for the weight-functions in Mühlbach–Neville–Aitken representations with application to WENO interpolation and differentiation

In several applications, such as WENO interpolation and reconstruction [C.W. Shu, SIAM Rev. 51 (2009) 82–126], we are interested in the analytical expression of the weight-functions which allow the representation of the approximating function on a given stencil (Chebyshev-system) as the weighted combination of the corresponding approximating functions on substencils (Chebyshev-subsystems). We show that the weight-functions in such representations [G. Mühlbach, Numer. Math. 31 (1978) 97–110] can be generated by a general recurrence relation based on the existence of a 1-level subdivision rule. As an example of application we apply this recurrence to the computation of the weight-functions for Lagrange interpolation [E. Carlini, R. Ferretti, G. Russo, SIAM J. Sci. Comput. 27 (2005) 1071–1091] for a general subdivision of the stencil {xi-M-,…,xi+M+} of M+1≔M-+M++1 distinct ordered points into Ks+1⩽M≔M-+M+>1 (Neville) substencils {xi-M-+ks,…,xi+M+-Ks+ks} (ks∈{0,…,Ks}) all containing the same number of M-Ks+1 points but each shifted by 1 cell with respect to its neighbour, and give a general proof for the conditions of positivity of the weight-functions (implying convexity of the combination), extending previous results obtained for particular stencils and subdvisions Liu (2009) [Y.Y. Liu, C.W. Shu, M.P. Zhang, Acta Math. Appl. Sin. 25 (2009) 503–538]. Finally, we apply the recurrence relation to the representation by combination of substencils of derivatives of arbitrary order of the Lagrange interpolating polynomial.

On the nested sums and orthogonal polynomials

The nested sums applied to a general three-term recurrence relation permits us to give compact representations of orthogonal polynomial sequences, which satisfy such kind of linear recurrence. We illustrate this model on particular examples of classical as well as nonclassical orthogonal polynomials. Some related nontrivial combinatorial identities are also obtained.

Recurrence relations for one-dimensional harmonic oscillator matrix elements of Gaussian and exponential operators

This paper reports the development of several general recurrence relations that can be used to evaluate one-dimensional, three centre harmonic oscillator matrix elements of the operators and f = exp(−cx C ). The matrix elements have the general form ⟨φ m (a 1/2 x A )|g(or f)|φ n (b 1/2 x B )⟩; φ m is the harmonic oscillator basis function for an eigenstate m. The coordinates are x A = x − A x , and so on, where A x , B x , and C x are points of reference for the displacement of a common atom whose instantaneous coordinate is x. A typical case might be that of a hydrogen atom referred to two wells located at A x and B x , and a second atom located at C x on the x axis. The recurrence relations apply to all cases including the two centre A x = B x and overlap integrals, A x ≠ B x , c = 0, and C x = 0. Moreover, the recurrence relations can generate matrix elements to any order. The applications of some of these recursions are illustrated with several examples: (1) the variational treatment of the Morse oscillator using one-dimensional harmonic oscillator basis functions; (2) the development of a model of the Morse oscillator in Gaussian coordinates together with (3) the variational analysis of that model. In addition, (4) the simplest version of a symmetric double potential well system is examined using both the Morse oscillator and the model potential.

The solution of the time-fractional diffusion equation by the generalized differential transform method

In the present paper, a general recurrence relation for determining the solutions of the time-fractional diffusion equation is obtained with the generalized differential transform method. The obtained relation will help us to solve time-fractional diffusion equations with various external forces and initial conditions. Four illustrative examples are given.

Flow and pressure distribution in linear discrete “ladder-type” fluidic circuits: An analytical approach

This article proposes fully analytical solutions for a certain class of networks or circuits for fluid flow, called “ladders” by analogy with the designation used in electrical engineering. Fluidic ladders comprise a discrete number of parallel channels, the ends of which are connected to a straight distributor manifold and to a straight collector manifold. The hydrodynamics are assumed to be purely linear, i.e. viscous laminar flow is assumed everywhere, inertial effects and non-linear contributions of branching singularities are neglected. The known and relatively simple case of the classical electric ladders is taken as a starting point to formulate and solve Kirchhoff's equations together with Ohm's law. The solutions for the steady-state flow-rates in each branch of the ladder are in the form of polynomials of dimensionless resistance ratios. The polynomials and their coefficients are shown to obey simple and general recurrence relations, which allow any size of ladder to be solved. A number of special cases are investigated, from a unique resistance (homogeneous ladders) to two or three different resistances, or a resistance distribution allowing a homogeneous distribution of flow among the parallel channels.

Asymptotic behaviours and general recurrence relations

In this paper, we present some new results which give greater efficiency to our method – expanded in preceding papers, for studying some asymptotic behaviours of polynomials generated by recurrence relations. More precisely, we give some new results which show that, for the asymptotic of the polynomials satisfying recurrence relations, some interesting well-known studies in the literature can be extended to recurrence relations of other kinds. For example, in contrast with what is known, we explicitly show that there exist many families of polynomials, say { M k } and { N k } with M k a perturbation of N k such that lim k → ∞ M k ( z ) / N k ( z ) = 1 uniformly in a subset – which may be very large – of the complex plane and with conditions on perturbations which, to our knowledge, are new. We illustrate that by some surprising examples. This work also allows us to extend our analysis to other classes of orthogonal polynomials.

Universal structure and universal equations (PDE) for unitary ensembles

Random matrix ensembles with unitary invariance of measure (UE) are described in a unified way using a combination of Tracy–Widom (TW) and Adler–Shiota–van Moerbeke approaches to the derivation of partial differential equations (PDEs) for spectral gap probabilities. First, general three-term recurrence relations for UE restricted to subsets of real line, or, in other words, for functions in the resolvent kernel, are obtained. Using them, simple universal relations between all TW dependent variables and one-dimensional Toda lattice τ-functions are found. A universal system of PDE for UE is derived from previous relations, which leads also to a single independent PDE for spectral gap probability of various UE. Thus, orthogonal function bases and Toda lattice are seen at the core of correspondence of different approaches. Moreover, Toda-AKNS system provides a common structure of PDE for unitary ensembles. Interestingly, this structure can be seen in two very different forms: one arises from orthogonal function-Toda lattice considerations, while the other comes from Schlesinger equations for isomonodromic deformations and their relation to TW equations. The simple example of Gaussian matrices most neatly exposes this structure.

Recurrence relations for polynomial sequences via Riordan matrices

We give recurrence relations for any family of generalized Appell polynomials unifying so some known recurrences for many classical sequences of polynomials. Our main tool to get our goal is the Riordan group. We use the product of Riordan matrices to interpret some relationships between different polynomial families. Moreover using the Hadamard product of series we get a general recurrence relation for the polynomial sequences associated to the so called generalized umbral calculus.

Recurrence for values of the zeta function

Using the Padé approximation of the exponential function, we obtain a general recurrence relation for values of the zeta function which contains, as particular cases, many of relations already proved. Applications to Bernoulli polynomials are given. At last, we derive some new recurrence relations with gap of length 4 for zeta numbers.

On the use of shifted Jacobi polynomials in accurate evaluation of roots and weights of Rys polynomials

In this paper it is shown that shifted Jacobi polynomials G(n)(p,q,x) can be used in connection with the Gaussian quadrature modified moment technique to greatly enhance the accuracy of evaluation of Rys roots and weights used in Gaussian integral evaluation in quantum chemistry. A general four-term inhomogeneous recurrence relation is derived for the shifted Jacobi polynomial modified moments over the Rys weight function e(-Tx)/square root x. It is shown that for q=1/2 this general four-term inhomogeneous recurrence relation reduces to a three-term p-dependent inhomogeneous recurrence relation. Adjusting p to proper values depending on the Rys exponential parameter T, the method is capable of delivering highly accurate results for large number of roots and weights in the most difficult to treat intermediate T range. Examples are shown, and detailed formulas together with practical suggestions for their efficient implementation are also provided.

Long Days on the Fibonacci Clock

The rule generating the sequence is extraordinarily simple, but its repeated application produces rich mathematics. If we leave the rule alone and change the start ing values, we again find sequences with interesting properties?such as the Lucas If we introduce modular arithmetic, there are new questions to answer. In what follows, I study sequences in ?p = Z/pZ, the integers mod p, where p is a prime. I like to call these the Fibonacci numbers. Regardless of the start ing pair, the sequence will repeat [10]. The question I want to answer is What is the maximum period for any sequence in Z/pZT In what follows, I present a particular point of view about the sequence in a way that gives some insight into both the standard sequence and its variations. Specifically, the sequence is interpreted in terms of a matrix acting on a finite set, an idea that is related to group actions and to (discrete) dynamical systems. The underlying set is the two-dimensional vector space ?p2; the matrix M from (2) provides the rule for the process. Iterations of the system correspond to powers of the matrix. Periods in the system are related, then, to powers of M that are equivalent modulo p to the identity matrix. The point of view works for all cases, even for the generalized numbers, where weights are allowed in the recursion formula. For a thorough look at dynamical systems and number theory, Silverman's book [8] is an excellent source. Most of what is contained here is not new. Searching Mathematical Reviews turns up dozens of articles about periods of numbers in Z/mZ, including many where m doesn't even have to be a prime or a power of a prime. The most-referenced article is Wall's article [10] in the monthly in 1960. Wall established many funda mental results, and posed some tantalizing questions. In particular, he showed that the period divides (p ? 1) when 5 is a quadratic residue mod p and divides (2/7 + 2) when it is not, but he did not find the maximal periods. Wall's investigation was motivated by a search for methods of generating pseudorandom Later, Brent [1], also motivated by pseudorandom numbers, considered the special properties of sequences modulo a power of 2. The story, however, begins even before the days of Mathematical Reviews. In the 1930s, Ward [11] considered periods, both minimal and maximal, and other characteristics of sequences arising from rather general recurrence relations, not just the relation. Kaiman and Mena's article in an earlier issue of this Magazine [5] examines many of the famous properties of the num bers as specific instances of properties of general second-order recurrences. Ward's results built on even earlier work by Carmichael [2] and others. For the early history of the subject, the curious reader should consult Dickson's history [3], particularly Volume I, Chapter XVII, where elements of the problem are traced back to Gauss and Lagrange. Earlier in the Magazine, Vella and Vella [9] looked at possible periods in

The Dying Rabbit Problem Revisited

AbstractIn this paper we study a generalization of the Fibonacci sequence in which rabbits are mortal and take more that two months to become mature. In particular we give a general recurrence relation for these sequences (improving the work in [Hoggat and Lind, Fibonacci Quart. 7: 482–487, 1969]) and we calculate explicitly their general term (extending the work in [Miles, Amer. Math. Monthly 67: 745–752, 1960]). In passing, and as a technical requirement, we also study the behavior of the positive real roots of the characteristic polynomial of the considered sequences.

Perturbed recurrence relations III the general case—some new applications

In this paper, we give some new extensions and some new applications of our results on the perturbation of coefficients and the order of a general recurrence relation—for example we will give some new results for the asymptotic properties, for the zeros and for the differential equations of the polynomials which satisfy the perturbed recurrence relation.

Shortened recurrence relations for Bernoulli numbers

Starting with two little-known results of Saalschütz, we derive a number of general recurrence relations for Bernoulli numbers. These relations involve an arbitrarily small number of terms and have Stirling numbers of both kinds as coefficients. As special cases we obtain explicit formulas for Bernoulli numbers, as well as several known identities.