Explicit Recursive Formulas Research Articles

On the generalization of the Euler polynomials with the real parameters

In this study, we give multiplication formula for generalized Euler polynomials of order α and obtain some explicit recursive formulas. The multiple alternating sums with positive real parameters a and b are evaluated in terms of both generalized Euler and generalized Bernoulli polynomials of order α . Finally we obtained some interesting special cases.

A simple Bayesian approach to multiple change-points

After a brief review of previous frequentist and Bayesian approaches to multiple change-points, we describe a Bayesian model for multiple parameter changes in a multiparameter exponential family. This model has attractive sta- tistical and computational properties and yields explicit recursive formulas for the Bayes estimates of the piecewise constant parameters. Efficient estimators of the hyperparameters of the Bayesian model for the parameter jumps can be used in conjunction, yielding empirical Bayes estimates. The empirical Bayes approach is also applied to solve long-standing frequentist problems such as significance testing of the null hypothesis of no change-points versus multiple change-point alterna- tives, and inference on the number and locations of change-points that partition the unknown parameter sequence into segments of equal values. Simulation studies of performance and an illustrative application to the British coal mine data are also given. Extensions from the exponential family to general parametric families and from independent observations to genearlized linear time series models are then provided.

Pricing in an equilibrium based model for a large investor

We study a financial model with a non-trivial price impact effect. In this model we consider the interaction of a large investor trading in an illiquid security, and a market maker who is quoting prices for this security. We assume that the market maker quotes the prices such that by taking the other side of the investor’s demand, the market maker will arrive at maturity with the maximal expected utility of the terminal wealth. Within this model we provide an explicit recursive pricing formula for an exponential utility function, as well as an asymptotic expansion for the price for a “small” simple demand.

Darboux covariant Lax pairs and infinite conservation laws of the (2+1)-dimensional breaking soliton equation

In this paper, the binary Bell polynomials are applied to succinctly construct bilinear formulism, bilinear Bäcklund transformations, Lax pairs, and Darboux covariant Lax pairs for the (2+1)-dimensional breaking soliton equation. An extra auxiliary variable is introduced to get the bilinear formulism. The infinitely local conservation laws of the equation are found by virtue of its Lax equation and a generalized Miura transformation. All conserved densities and fluxes are given with explicit recursion formulas.

On Explicit Recursive Formulas in the Spectral Perturbation Analysis of a Jordan Block

Let $A(\varepsilon)$ be an analytic square matrix and $\lambda_{0}$ an eigenvalue of $A(0)$ of algebraic multiplicity $m\geq1$. Then under the condition $\frac{\partial}{\partial\varepsilon}\det(\lambda I-A(\varepsilon))|_{(\varepsilon,\lambda)=(0,\lambda_{0})}\neq0$, we prove that the Jordan normal form of $A(0)$ corresponding to the eigenvalue $\lambda_{0}$ consists of a single $m\times m$ Jordan block, the perturbed eigenvalues near $\lambda_{0}$ and their corresponding eigenvectors can be represented by a single convergent Puiseux series containing only powers of $\varepsilon^{1/m}$, and there are explicit recursive formulas to compute all the Puiseux series coefficients from just the derivatives of $A(\varepsilon)$ at the origin. Using these recursive formulas we calculate the series coefficients up to the second order and list them for quick reference. This paper gives, under a generic condition, explicit recursive formulas to compute the perturbed eigenvalues and eigenvectors for non-self-adjoint analytic perturbations of matrices with nonderogatory eigenvalues.

The all-loop integrand for scattering amplitudes in planar $ \mathcal{N} = 4 $ SYM

We give an explicit recursive formula for the all ℓ-loop integrand for scattering amplitudes in \( \mathcal{N} = 4 \) SYM in the planar limit, manifesting the full Yangian symmetry of the theory. This generalizes the BCFW recursion relation for tree amplitudes to all loop orders, and extends the Grassmannian duality for leading singularities to the full amplitude. It also provides a new physical picture for the meaning of loops, associated with canonical operations for removing particles in a Yangian-invariant way. Loop amplitudes arise from the “entangled” removal of pairs of particles, and are naturally presented as an integral over lines in momentum-twistor space. As expected from manifest Yangian invariance, the integrand is given as a sum over non-local terms, rather than the familiar decomposition in terms of local scalar integrals with rational coefficients. Knowing the integrands explicitly, it is straightforward to express them in local forms if desired; this turns out to be done most naturally using a novel basis of chiral, tensor integrals written in momentum-twistor space, each of which has unit leading singularities. As simple illustrative examples, we present a number of new multi-loop results written in local form, including the 6- and 7-point 2-loop NMHV amplitudes. Very concise expressions are presented for all 2-loop MHV amplitudes, as well as the 5-point 3-loop MHV amplitude. The structure of the loop integrand strongly suggests that the integrals yielding the physical amplitudes are “simple”, and determined by IR-anomalies. We briefly comment on extending these ideas to more general planar theories.

Labeled floor diagrams for plane curves

Floor diagrams are a class of weighted oriented graphs introduced by E. Brugallé and the second author. Tropical geometry arguments lead to combinatorial descriptions of (ordinary and relative) Gromov–Witten invariants of projective spaces in terms of floor diagrams and their generalizations. In a number of cases, these descriptions can be used to obtain explicit (direct or recursive) formulas for the corresponding enumerative invariants. In particular, we use this approach to enumerate rational curves of given degree passing through a collection of points on the complex plane and having maximal tangency to a given line. Another application of the combinatorial approach is a proof of a conjecture by P. Di Francesco–C. Itzykson and L. Göttsche that in the case of a fixed cogenus, the number of plane curves of degree d passing through suitably many generic points is given by a polynomial in d , assuming that d is sufficiently large. Furthermore, the proof provides a method for computing these “node polynomials”. A labeled floor diagram is obtained by labeling the vertices of a floor diagram by the integers 1; . . . ; d in a manner compatible with the orientation.We show that labeled floor diagrams of genus 0 are equinumerous to labeled trees, and therefore counted by the celebrated Cayley formula. The corresponding bijections lead to interpretations of the Kontsevich numbers (the genus-0 Gromov– Witten invariants of the projective plane) in terms of certain statistics on trees.

The multiplication formulas for the Apostol–Bernoulli and Apostol–Euler polynomials of higher order

This article obtains the multiplication formulas for the Apostol–Bernoulli and Apostol–Euler polynomials of higher order and deduces some explicit recursive formulas. The λ-multiple power sum and λ-multiple alternating sum that are clearly evaluated associated with the Apostol–Bernoulli and Apostol–Euler polynomials of higher order, respectively, are also introduced. Some earlier results of Carlitz and Howard can be deduced.

An explicit method for numerical simulation of wave equations

In this paper, a method to develop a hierarchy of explicit recursion formulas for numerical simulation in an irregular grid for scalar wave equations is presented and its accuracy is illustrated via 2-D and 1-D models. Approaches to develop the stable formulas which are of 2M-order accuracy in both time and space with M being a positive integer for regular grids are discussed and illustrated by constructing the second order (M = 1) and the fourth order (M = 2) recursion formulas.

Some statistical applications of Faa di Bruno

The formula of Faa di Bruno is used to calculate higher order derivatives of a composition of functions. In this paper, we first review the multivariate version due to Constantine and Savits [A multivariate Faa di Bruno formula with applications, Trans. AMS 348 (1996) 503–520]. We next derive some useful recursion formulas. These results are then applied to obtain both explicit expressions and recursive formulas for the multivariate Hermite polynomials and moments associated with a multivariate normal distribution. Finally, an explicit expression is derived for the formal Edgeworth series expansion of the distribution of a normalized sum of iid random variables.

High-order solutions of three-dimensional rough-surface scattering problems at high-frequencies. Part II: The vector electromagnetic case

We introduce a new numerical scheme for three-dimensional electromagnetic rough-surface scattering simulations with the capability of delivering very accurate results from low to high frequencies at a cost that is independent of the wavelength of radiation. The method is an extension of the ideas and techniques introduced in the first paper of this series (Waves in Random and Complex Media, 15 (2005), pp. 1–16) to the vector electromagnetic case, and it is based on the solution of an integral equation formulation of the scattering problem. As in the scalar case, the solution of the integral equation (i.e. the current) is expressed as a slow modulation of an oscillatory exponential of known phase, and explicit recursive formulae are derived for the successive terms in a series expansion of the slow envelope in inverse powers of the wavenumber. As we show, and in spite of the considerably more involved nature of the derivations and resulting formulae, the performance of the method retains the favourable characteristics that were demonstrated in the treatment of acoustic scattering problems. In particular, results with full double-precision accuracy are presented for various geometries, incidences and polarizations.

Quantum integrability of the deformed elliptic Calogero–Moser problem

The integrability of the deformed quantum elliptic Calogero–Moser problem introduced by Chalykh, Feigin, and Veselov is proven. Explicit recursive formulas for the integrals are found. For integer values of the parameter this implies the algebraic integrability of the systems.

Interacting Fock Expansion of Lévy White Noise Functionals

Consider the Levy white noise space (S*,ℬ(S*),μ), where S* is the Schwartz distributions over Rd and μ is a Levy white noise measure lifted from a 1-dimensional infinitely divisible distribution with finite moments. We give explicit forms and recursion formulas of moment and renormalization kernels for the Levy white noise measure. By defining inner products (⋅,⋅)[n] in n-particle spaces, we establish an interacting Fock space ⊕n=0∞ℋ(n) and the interacting Fock expansions for Levy white noise functionals. The usual Fock space Γ(H)=⊕n=0∞\(H^{\hat \otimes n} \) can be viewed as a quotient space of the interacting Fock space. As a particular case, we give the interacting Fock expansion for gamma white noise functionals.

Wavelength assignment method for WDM network of star topology

A novel wavelength assignment method for a bidirectional two-fibre WDM network of star topology is proposed. It is shown that the minimal number of wavelengths required for an n-node star network is n (when n is odd) or n−1 (when n is even). A simple and explicit recursive formula, which completes wavelength assignment quickly with a time complexity of O(n2 log2(n)), is given. Simulation results show that the present method can give an optimal wavelength assignment much faster than conventional methods for a star network.

A matching pursuit technique for computing the simplest normal forms of vector fields

This paper presents a matching pursuit technique for computing the simplest normal forms of vector fields. First a simple, explicit recursive formula is derived for general differential equations, which reduces computation to the minimum. Then a matching pursuit technique is introduced and applied to the Takens–Bogdanov dynamical singularity. It is shown that unlike other methods for computing normal forms, the technique using matching pursuit does not need any algebraic constraints which are required for the existence of the simplest normal form. The efficient method and matching pursuit technique, which have been implemented using Maple, can be “automatically” executed on various computer systems. A number of examples are presented to demonstrate the advantages of the technique.

AN EFFICIENT METHOD FOR COMPUTING THE SIMPLEST NORMAL FORMS OF VECTOR FIELDS

A computationally efficient method is proposed for computing the simplest normal forms of vector fields. A simple, explicit recursive formula is obtained for general differential equations. The most important feature of the approach is to obtain the "simplest" formula which reduces the computation demand to minimum. At each order of the normal form computation, the formula generates a set of algebraic equations for computing the normal form and nonlinear transformation. Moreover, the new recursive method is not required for solving large matrix equations, instead it solves linear algebraic equations one by one. Thus the new method is computationally efficient. In addition, unlike the conventional normal form theory which uses separate nonlinear transformations at each order, this approach uses a consistent nonlinear transformation through all order computations. This enables one to obtain a convenient, one step transformation between the original system and the simplest normal form. The new method can treat general differential equations which are not necessarily assumed in a conventional normal form. The method is applied to consider Hopf and Bogdanov–Takens singularities, with examples to show the computation efficiency. Maple programs have been developed to provide an "automatic" procedure for applications.

Lacunary Polynomial Sums

Lacunary sums of multinomial coefficients from homogeneous polynomials are examined by explicit equations and recursive formulas. We also determine the special relationship between certain lacunary sums and the Fibonacci and Lucas numbers.

Variants of the Greville Formula with Applications to Exact Recursive Least Squares

In this paper, we present an order-recursive formula for the pseudoinverse of a matrix. It is a variant of the well-known Greville [SIAM Rev., 2 (1960), pp. 578--619] formula. Three forms of the proposed formula are presented for three different matrix structures. Compared with the original Greville formula, the proposed formulas have certain merits. For example, they reduce the storage requirements at each recursion by almost half; they are more convenient for deriving recursive solutions for optimization problems involving pseudoinverses. Regarding applications, using the new formulas, we derive recursive least squares (RLS) procedures which coincide exactly with the batch LS solutions to the problems of the unconstrained LS, weighted LS, and LS with linear equality constraints, respectively, including their simple and exact initializations. Compared with previous results, e.g., Albert and Sittler [J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1965), pp. 384--417], our derivation of the explicit recursive formulas is much easier, and the recursions take a much simpler form. New findings include that the linear equality constrained LS and the unconstrained LS can have an identical recursion---their only difference is the initial conditions. In addition, some robustness issues, in particular, during the exact initialization of the RLS are studied.

State‐space Models with Finite Dimensional Dependence

We consider nonlinear state‐space models, where the state variable (ζt) is Markov, stationary and features finite dimensional dependence (FDD), i.e. admits a transition function of the type: π(ζt|ζt−1) =π(ζt)a′(ζt)b(ζt−1), where π(ζt) denotes the marginal distribution of ζt, with a finite number of cross‐effects between the present and past values. We discuss various characterizations of the FDD condition in terms of the predictor space and nonlinear canonical decomposition. The FDD models are shown to admit explicit recursive formulas for filtering and smoothing of the observable process, that arise as an extension of the Kitagawa approach. The filtering and smoothing algorithms are given in the paper.JEL. C4.

Recursive analytical formula for the Green’s function of a Hamiltonian having a sum of one-dimensional arbitrary delta-function potentials

The Green's functions of one-dimensional Hamiltonians containing, respectively, one and two delta-function potentials are derived by analytically summing over the corresponding Lippmann-Schwinger series. A generalization of this procedure leads to an explicit recursive formula for the Green's function ${G}^{(n+1)}$ corresponding to a Hamiltonian containing a sum of $n+1$ delta-function potentials of arbitrary positions and strengths in terms of ${G}^{(n)}$ and the additional $n+1$ delta-function potential parameters.