Asymptotic behaviours and general recurrence relations

In this letter, based on Hamiltonian identity we propose a generalized expression of the second hypervirial for an arbitrary central potential wave functions in higher dimensions D and then present a useful general recurrence relation among the off-diagonal multipolar matrix elements. We show that this new proposed recurrence formula is very powerful in deriving the recurrence relations among the hydrogenic matrix elements. Some important recurrence relations can be simply obtained. It is interesting to find that the selection rule is independent of the central potential V(r) for f = rk with k = 0, 2 and that the diagonal matrix elements of V′(r) are independent of the system eigenvalues for k = 1. The application of the general recurrence relation to the isotropic harmonic oscillator case is discussed briefly.

CHAPTER 4 - Recurrence Relations and Algebraic Equations

Recursiveness in matrix rational interpolation problems

Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: Continuous case

Some properties of generalized secant integrals: extended definition and recurrence relations

Connection and linearization coefficients of the Askey–Wilson polynomials

In recent years, one of the most interesting developments in quantum mechanics has been the construction of new exactly solvable potentials connected with the appearance of families of exceptional orthogonal polynomials (EOP) in mathematical physics. In contrast with families of (Jacobi, Laguerre and Hermite) classical orthogonal polynomials, which start with a constant, the EOP families begin with some polynomial of degree greater than or equal to one, but still form complete, orthogonal sets with respect to some positive-definite measure. We show how they may appear in the bound-state wavefunctions of some rational extensions of well-known exactly solvable quantum potentials. Such rational extensions are most easily constructed in the framework of supersymmetric quantum mechanics (SUSYQM), where they give rise to a new class of translationally shape invariant potentials. We review the most recent results in this field, which use higher-order SUSYQM. We also comment on some recent re-examinations of the shape invariance condition, which are independent of the EOP construction problem.

‎In this article‎, ‎an efficient numerical method based on a new class of orthogonal polynomials‎, ‎namely Chelyshkov polynomials‎, ‎has been presented to approximate solution of time-fractional telegraph (TFT) equations‎. ‎The fractional operational matrix of the Chelyshkov polynomials along with the typical collocation method is used to reduces TFT equations to a system of algebraic equations‎. ‎The error analysis of the proposed collocation method is also investigated‎. ‎A comparison with other published results confirms that the presented Chelyshkov collocation approach is efficient and accurate for solving TFT equations‎. ‎Illustrative examples are included to demonstrate the efficiency of the Chelyshkov method‎.

The underlying stochastic nature of the requirements for the Solvency II regulations has introduced significant challenges if the required calculations are to be performed correctly, without resorting to excessive approximations, within practical timescales. It is generally acknowledged by practising actuaries within UK life offices that it is currently impossible to correctly fulfil the requirements imposed by Solvency II using existing computational techniques based on commercially available valuation packages. Our work has already shown that it is possible to perform profitability calculations at a far higher rate than is achievable using commercial packages. One of the key factors in achieving these gains is to calculate reserves using recurrence relations that scale linearly with the number of time steps. Here, we present a general vector recurrence relation which can be used for a wide range of non-unit linked policies that are covered by Solvency II; such contracts include annuities, term assurances, and endowments. Our results suggest that by using an optimised parallel implementation of this algorithm, on an affordable hardware platform, it is possible to perform the ‘brute force’ approach to demonstrating solvency in a realistic timescale (of the order of a few hours).

An algebraic approach to obtain ladder operators for hydrogenlike wave functions from the algebraic representation of the ${\mathit{L}}_{\mathit{n}}^{\mathit{k}}$ Laguerre associated polynomials directly involved is presented. The method leads to diverse ladder operators depending on the manner as the operational form of ${\mathit{L}}_{\mathit{n}}^{\mathit{k}}$ is used. As an example, this work deals with linear creation and annihilation operators of two kinds: some that influence the n energy quantum number and others acting on the l orbital number according to usual ladder properties. In addition, as a useful application of the raising and lowering operators thus obtained, generalized recurrence relations and closed formulas for the calculation of multipole matrix elements are algebraically deduced by means of a procedure that takes advantage of the relationship between the ladder operators associated with the bra and those related to the ket.

Generating function, recurrence relations, differential relations

Wronskian Appell polynomials and symmetric functions

This paper continues the analysis of a class of orthogonal polynomials in two variables on a region bounded by two straight lines and a parabola touching these lines, which was introduced by the first author. An explicit series expansion for these polynomials is obtained, which generalizes Constantine’s expansion of hypergeometric functions of $(2 \times 2)$ matrix argument in terms of James’ zonal polynomials. In two special cases the orthogonal polynomials turn out to be Appell’s hypergeometric $F_4 $-functions and certain hypergeometric functions in two variables of order three, respectively.

By using the calculus of finite differences methods and the umbral calculus, we construct recurrence relations for a new class of special numbers. Using this recurrence relation, we define generating functions for this class of special numbers and also new classes of special polynomials. We investigate some properties of these generating functions. By using these generating functions with their functional equations, we obtain many new and interesting identities and relations related to these classes of special numbers and polynomials, the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers. Finally, some derivative formulas and integral formulas for these classes of special numbers and polynomials are given. In general, this article includes results that have the potential to be used in areas such as discrete mathematics, combinatorics analysis and their applications.

For a class of polynomial quantum Hamiltonians used in models of combination scattering in quantum optics, we obtain the asymptotic behavior of the spectrum for large occupation numbers in the secondary quantization representation. Hamiltonians of this class can be diagonalized using a special system of polynomials determined by recurrence relations with coefficients depending on a parameter (occupation number). For this system of polynomials, we determine the asymptotic behavior a discrete measure with respect to which they are orthogonal. The obtained limit measures are interpreted as equilibrium measures in extremum problems for a logarithmic potential in an external field and with constraints on the measure. We illustrate the general case with an exactly solvable example where the Hamiltonian can be diagonalized by the canonical Bogoliubov transformation and the special orthogonal polynomials degenerate into the Krawtchouk classical discrete polynomials.