Related Polynomials Research Articles

On quasi-orthogonal polynomials: Their differential equations, discriminants and electrostatics

In this paper, we develop a general theory of quasi-orthogonal polynomials. We first derive three-term recurrence relation and second-order differential equations for quasi-orthogonal polynomials. We also give an expression for their discriminants in terms of the recursion coefficients of the corresponding orthogonal polynomials. In addition, we investigate an electrostatic equilibrium problem where the equilibrium position of movable charges is attained at the zeros of the quasi-orthogonal polynomials. The examples of the Freud weight w ( x ) = e − x 4 + 2 t x 2 and the Jacobi weight w ( x ) = ( 1 − x ) α ( 1 + x ) β are discussed in some detail. Finally, we consider the nonlinear orthogonality preserving transformation and related matrix problem.

Recurrence relations of the multi-indexed orthogonal polynomials V: Racah and q -Racah types

In previous papers, we discussed the recurrence relations of the multi-indexed orthogonal polynomials of the Laguerre, Jacobi, Wilson, and Askey-Wilson types. In this paper, we explore those of the Racah and q-Racah types. For the M-indexed (q-)Racah polynomials, we derive 3 + 2M term recurrence relations with variable dependent coefficients and 1 + 2L term (L ≥ M + 1) recurrence relations with constant coefficients. Based on the latter, the generalized closure relations and the creation and annihilation operators of the quantum mechanical systems described by the multi-indexed (q-)Racah polynomials are obtained. In Appendix B and Appendix C, we present a proof and some data of the recurrence relations with constant coefficients for the multi-indexed Wilson and Askey-Wilson polynomials.

Characterisation of titanium oxide layers using Raman spectroscopy and optical profilometry: Influence of oxide properties

Abstract This study evaluates the use of a combination of Raman spectroscopy and optical profilometry as a surface characterisation technique for the examination of oxide layers grown on titanium metal substrates. The titanium oxide layers with thickness of up to 8 µm, were obtained using a low-pressure oxygen microwave plasma treatment of the titanium metal substrate. The effect of the microwave plasma processing conditions (input power, pressure and treatment time) on the Raman bandwidth, intensity and peak position was evaluated. Also, the effect of these processing conditions on the surface roughness parameters (Sa, Sdq, Ssk and Sku) of the oxide layers was investigated. Analysis of the peak positions of Eg and A1g modes indicated that the effects of input power and chamber pressure was to induce a shift towards the lower frequency with increasing input power and pressure (1–2 kPa). The intensity of the Raman bands was found to be dependent on the morphology and surface chemistry of the oxide layer. The intensity of Raman band (A1g), was found to be particularly influenced by the average surface roughness (Sa) and the crystallite size. Exponential and polynomial relations were found to correlate with these properties. A two-latent variable Partial Least Squares Regression model developed on Raman spectral data could predict surface roughness with a coefficient of determination (R2) of approx. 0.87 when applied to the testing of an independent set of titanium oxide test coatings.

Identities and relations for Fubini type numbers and polynomials via generating functions and p-adic integral approach

The Fubini type polynomials have many application not only especially in combinatorial analysis, but also other branches of mathematics, in engineering and related areas. Therefore, by using the p-adic integrals method and functional equation of the generating functions for Fubini type polynomials and numbers, we derive various different new identities, relations and formulas including well-known numbers and polynomials such as the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers of the second kind, the ?-array polynomials and the Lah numbers.

Some relations for universal Bernoulli polynomials

Some relations for universal Bernoulli polynomials

Associated Hermite Polynomials Related to Parabolic Cylinder Functions

In analogy to the role of Lommel polynomials in relation to Bessel functions Jv(z) the theory of Associated Hermite polynomials in the scaled form with parmeter v to Parabolic Cylinder functions Dv(z) is developed. The group-theoretical background with the 3-parameter group of motions M(2) in the plane for Bessel functions and of the Heisenberg-Weyl group W(2) for Parabolic Cylinder functions is discussed and compared with formulae, in particular, for the lowering and raising operators and the eigenvalue equations. Recurrence relations for the Associated Hermite polynomials and for their derivative and the differential equation for them are derived in detail. Explicit expressions for the Associated Hermite polynomials with involved Jacobi polynomials at argument zero are given and by means of them the Parabolic Cylinder functions are represented by two such basic functions.

Chebyshev Polynomials with Applications to Two-Dimensional Operators

A new application of Chebyshev polynomials of second kind Un(x) to functions of two-dimensional operators is derived and discussed. It is related to the Hamilton-Cayley identity for operators or matrices which allows to reduce powers and smooth functions of them to superpositions of the first N-1 powers of the considered operator in N-dimensional case. The method leads in two-dimensional case first to the recurrence relations for Chebyshev polynomials and due to initial conditions to the application of Chebyshev polynomials of second kind Un(x). Furthermore, a new general class of Generating functions for Chebyshev polynomials of first and second kind Un(x) comprising the known Generating function as special cases is constructed by means of a derived identity for operator functions f(A) of a general two-dimensional operator A. The basic results are Formulas (9.5) and (9.6) which are then specialized for different examples of functions f(x). The generalization of the theory for three-dimensional operators is started to attack and a partial problem connected with the eigenvalue problem and the Hamilton-Cayley identity is solved in an Appendix. A physical application of Chebyshev polynomials to a problem of relativistic kinematics of a uniformly accelerated system is solved. All operator calculations are made in coordinate-invariant form.

A class of big (p,q)-Appell polynomials and their associated difference equations

In the present paper, we introduce and investigate the big (p,q)-Appell polynomials. We prove an equivalance theorem satisfied by the big (p, q)-Appell polynomials. As a special case of the big (p,q)- Appell polynomials, we present the corresponding equivalence theorem, recurrence relation and difference equation for the big q-Appell polynomials. We also present the equivalence theorem, recurrence relation and differential equation for the usual Appell polynomials. Moreover, for the big (p; q)-Bernoulli polynomials and the big (p; q)-Euler polynomials, we obtain recurrence relations and difference equations. In the special case when p = 1, we obtain recurrence relations and difference equations which are satisfied by the big q-Bernoulli polynomials and the big q-Euler polynomials. In the case when p = 1 and q ? 1-, the big (p,q)-Appell polynomials reduce to the usual Appell polynomials. Therefore, the recurrence relation and the difference equation obtained for the big (p; q)-Appell polynomials coincide with the recurrence relation and differential equation satisfied by the usual Appell polynomials. In the last section, we have chosen to also point out some obvious connections between the (p; q)-analysis and the classical q-analysis, which would show rather clearly that, in most cases, the transition from a known q-result to the corresponding (p,q)-result is fairly straightforward.

Structure Relations of Classical Orthogonal Polynomials in the Quadratic and q-Quadratic Variable

We prove an equivalence between the existence of the first structure relation satisfied by a sequence of monic orthogonal polynomials $\{P_n\}_{n=0}^{\infty}$, the orthogonality of the second derivatives $\{\mathbb{D}_{x}^2P_n\}_{n= 2}^{\infty}$ and a generalized Sturm-Liouville type equation. Our treatment of the generalized Bochner theorem leads to explicit solutions of the difference equation [Vinet L., Zhedanov A., J. Comput. Appl. Math. 211 (2008), 45-56], which proves that the only monic orthogonal polynomials that satisfy the first structure relation are Wilson polynomials, continuous dual Hahn polynomials, Askey-Wilson polynomials and their special or limiting cases as one or more parameters tend to $\infty$. This work extends our previous result [arXiv:1711.03349] concerning a conjecture due to Ismail. We also derive a second structure relation for polynomials satisfying the first structure relation.

Algebra of Symmetries of Three-Frequency Resonance: Reduction of a Reducible Case to an Irreducible Case

For the three-frequency quantum resonance oscillator, the reducible case, where the frequencies are integer and at least one pair of frequencies has a nontrivial common divisor, is studied. It is shown how the description of the algebra of symmetries of such an oscillator can be reduced to the irreducible case of pairwise coprime integer frequencies. Polynomial algebraic relations are written, and irreducible representations and coherent states are constructed.

Construction of all polynomial relations among Dedekind eta functions of level N

We describe an algorithm that, given a positive integer N, computes a Gröbner basis of the ideal of polynomial relations among Dedekind η-functions of level N, i.e., among the elements of {η(δ1τ),…,η(δnτ)} where 1=δ1<δ2<…<δn=N are the positive divisors of N.More precisely, we find a finite generating set (which is also a Gröbner basis) of the ideal ker⁡ϕ whereϕ:Q[E1,…,En]→Q[η(δ1τ),…,η(δnτ)],Ek↦η(δkτ),k=1,…,n.

Polynomial relations between matrices of graphs

AbstractWe derive a correspondence between the eigenvalues of the adjacency matrix and the signless Laplacian matrix of a graph when is ‐biregular by using the relation . This motivates asking when it is possible to have for a polynomial, , and matrices associated to a graph . It turns out that, essentially, this can only happen if is either regular or biregular.

How the spin-orbit interaction appears with the same order of a non-commutative change of framework for relativistic fermions

In this research, in a difficult but absolutely precise way of calculation, we show how a very tiny amount of a non-commutative change of quantum space would appear almost as big as a normal physical interaction, namely the Rashba spin-orbit interaction, for relativistic fermions. Hence, in order to show that, we firstly solve a relativistic equation of motion of a Dirac particle, influenced by a typical harmonic energy-dependent interaction for commutative and non-commutative frameworks via the Nikiforov–Uvarov exact approach. Then to study perturbation effects of a spin-orbit interaction, we apply it for both mentioned frameworks, obtaining their energy polynomial relations and discriminant formula to precisely extract all physical-admissible roots of their quartic equations. In this step, we analyze the behaviors of their quartic eigenvalue polynomials in four sections and accurately compare them one by one. Finally, we distinctly show that the magnitude of the physical spin-orbit perturbation appears, almost of the same order of imposing a non-commutative geometry change of framework, as an outstanding result.

Existence of a pair of new recurrence relations for the Meixner-Pollaczek polynomials

We report on existence of pair of new recurrence relations (or difference equations) for the Meixner-Pollaczek polynomials. Proof of the correctness of these difference equations is also presented. Next, we found that subtraction of the forward shift operator for the Meixner-Pollaczek polynomials from one of these recurrence relations leads to the difference equation for the Meixner-Pollaczek polynomials generated via $\cosh$ difference differentiation operator. Then, we show that, under the limit $\varphi \to 0$, new recurrence relations for the Meixner-Pollaczek polynomials recover pair of the known recurrence relations for the generalized Laguerre polynomials. At the end, we introduced differentiation formula, which expresses Meixner-Pollaczek polynomials with parameters $\lambda>0$ and $0 \lt \varphi \lt \pi$ via generalized Laguerre polynomials.

Understanding Fundamental Properties and Atmospheric Features of Subdwarfs via a Case Study of SDSS J125637.13–022452.4∗

Abstract We present the distance-calibrated spectral energy distribution (SED) of the sdL3.5 subdwarf SDSS J125637.13−022452.4 (J1256−0224) using its Gaia DR2 parallax. We report the bolometric luminosity and semi-empirical fundamental parameters, as well as updated UVW velocities. The SED of J1256−0224 is compared to field-age and low-gravity dwarfs of the same effective temperature (T eff) and bolometric luminosity. In the former comparison, we find that the SED of J1256−0224 is brighter than the field source in the optical, but dims in comparison beyond the J band, where it becomes fainter than the field from the H through W2 bands. Compared to the young source, it is fainter at all wavelengths. We conclude that J1256−0224 is depleted of condensates compared to both objects. A near-infrared band-by-band analysis of the spectral features of J1256−0224 is done and is compared to the equivalent T eff sample. From this analysis, we find a peculiar behavior of the J-band K i doublets whereby the 1.17 μm doublet is stronger than the field or young source, as expected, while the 1.25 μm doublet shows indications of low gravity. In examining a sample of four other subdwarfs with comparable data, we confirm this trend across different subtypes indicating that the 1.25 μm doublet is a poor indicator of gravity for low-metallicity objects. In the K-band analysis of J1256−0224, we detect the 2.29 μm CO line of J1256−0224, previously unseen in the low-resolution SpeX data. We also present fundamental parameters using Gaia parallaxes for nine additional subdwarfs with spectral types M7–L7 for comparison. The 10 subdwarfs are placed in a temperature sequence, and we find a poor linear correlation with spectral type. We present polynomial relations for absolute magnitude in JHKW1W2, effective temperature, and bolometric luminosity versus spectral type for subdwarfs.

Identities and recurrence relations of special numbers and polynomials of higher order by analysis of their generating functions

The aim of this is to give generating functions for new families of special numbers and polynomials of higher order. By using these generating functions and their functional equations, we derive identities and relations for these numbers and polynomials. Relations between these new families of special numbers and polynomials and Bernoulli numbers and polynomials are given. Finally, recurrence relations and derivative formulas, which are related to these numbers and polynomials, are given.

A determinant approach to q-Bessel polynomials and applications

The article aims to introduce the q-analogues of well known Bessel polynomials and to study their important properties. A new recurrence relation and determinant definition for the q-Bessel polynomials are derived. A hybrid form of q-Bessel polynomials namely, the extended q-Eulerian type-Bessel polynomials are introduced and are characterized by means of series expansion and determinant definition. As an special case, the characterizations for the extended q-Euler–Bessel polynomials are given. Further, the 2Dq-Bessel polynomials are introduced by means of generating function and determinant definition and some identities for these q-polynomials are proved.

Efficient and robust recurrence relations for the Zernike circle polynomials and their derivatives in Cartesian coordinates.

For some time it has been known and recommended that the calculation of Zernike polynomials to radial orders higher than 8 to 10 should be performed using recurrence relations rather than explicit expressions due increasingly large cancellation errors. This paper presents a set of simple recurrence relations that can be used for the unit-normalized Zernike polynomials in polar coordinates and easily adapted to Cartesian coordinates as well. The recurrence relations are also well suited for the calculation of the Cartesian derivatives of the Zernike polynomials. The recurrence relations are easily extended to arbitrarily high orders. Assessments of the precision achievable with standard 64-bit floating point arithmetic show that Zernike polynomials up to radial order 30 can be calculated over the unit disc with errors not exceeding 5E-14, and up to radial order 50 with errors not exceeding 1.2E-13. Comparison with the Zernike capability in OpticStudio (Zemax) shows that the recurrence relations are superior in performance (both speed and precision) over the existing algorithm implemented in the software. General pseudo-code for the calculation of Zernike polynomials and their derivatives is also presented.

Fourier transform of hydrogen-type atomic orbitals

Hydrogen-type atomic orbitals (HTOs) are an important type of exponential-type orbital. These orbitals have some mathematical properties and they are used usually in the theoretical atomic and molecular investigations as special functions to figure out analytical expressions. The Fourier transform method is a great way to convert basis functions into the momentum space, because their Fourier transforms are easier to use in mathematical calculations. In this paper, we obtain new and useful mathematical representations for the Fourier transform of HTOs related with Gegenbauer polynomials and hypergeometric functions, by using recurrence relations of Laguerre polynomials, Rayleigh expansion and some properties of normalized HTOs.

New empirical relation for chemical shift and effective charge in the X-ray absorption edge shifts

A new relation, ΔE = aebq, between the chemical shift ΔE and effective charge ‘q’ is proposed. It has been shown that the relation generates polynomial relations, between ΔE and ‘q’ used by earlier investigators and addresses their short-comings effectively. Further, four possible sign combinations of ‘q’ and ΔE are accounted for using the proposed equation. Units of arbitrary constants ‘a’ and ‘b’ are derived. The tendency of ‘q’ to attain saturation value is also explained. The success of the new relation has been tested using its linearized form ln ∣ ΔE ∣ = ln ∣ a ∣ + bq for a large number of compounds, by taking their experimental X-ray absorption edge shifts along with corresponding values of ‘q’ from the available literature. In the process, the claim of earlier workers regarding linear relationship between ΔE and ‘q’ and ΔE and ‘qX’, where ‘X’ is the electronegativity, has been found incorrect in the case of 6p6 isoelectronic series of compounds.