Lommel Polynomials Research Articles

Coefficients bounds for a subclass of q-bi-starlike functions associated with the generalized q-Lommel polynomials

ABSTRACT Orthogonal q-polynomials, both new and old, have witnessed a huge and revived attention in recent years, because of their applications in many diverse areas of mathematics and other sciences. In Geometric Function Theory, different subclasses of analytic and bi-univalent functions have been investigated and studied involving different orthogonal q-polynomials. In our present investigation, motivated by these recent research going on, first, we define some new subclasses of q-bi-starlike functions with the help of certain q-derivative operator which involving the generalized q-Lommel polynomials and q-Chebyshev polynomials. We then obtain the initial coefficients bounds for our defined function classes. Furthermore, the Fekete–Szegö inequalities are obtained for these defined function classes.

The Lommel polynomials and related formulas

Properties of the Lommel polynomials R n , ν ( z ) which appear in the theory of the Bessel functions, are studied. New functional relations for hypergeometric and Horn functions, including reduction formulas, are derived.

On the generalized interlacing property for the zeros of Bessel functions

This paper investigates a generalized interlacing property between Bessel functions, particularly Jν and Jμ, where the difference |ν−μ| exceeds 2. This interlacing phenomenon is marked by a compensatory interaction with the zeros of Lommel polynomials, extending our understanding beyond the traditional |ν−μ|≤2 regime. The paper extends the generalized interlacing property to cylinder functions and the derivative of Bessel functions, as an application. It is also discussed that Siegel's extension of Bourget hypothesis to rational numbers of ν cannot be further improved to arbitrary real numbers.

Infinite Series Based on Bessel Zeros

An interesting series based on Bessel function roots (zeros) is discussed and numerically analyzed. The novel-derived simplified general solutions are based on Lommel polynomials. This kind of series can have a large practical use in many scientific areas, such as solid mechanics, fluid mechanics, thermodynamics, electronics, physics, etc. Some practical examples connected with fluid mechanics are provided in this paper. The errors in Afanasiev solutions are corrected. In addition, the main solution for the series analyzed by Baricz and Angel is presented.

The asymptotic zero distribution of Lommel polynomials as functions of their order with a variable complex argument

We study the asymptotic distribution of roots of Lommel polynomials as polynomials of the order with a variable and purely imaginary argument. The roots are complex and accumulate on certain curves in the complex plane. We prove existence of the weak limit of corresponding root-counting measures and deduce formulas for the supporting curves and density. The obtained result represents a solvable example of a more general problem which is still open. Numerical illustrations of the main result are also involved.

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.

Doubly infinite Jacobi matrices revisited: Resolvent and spectral measure

We study the resolvent and spectral measure of certain doubly infinite Jacobi matrices via asymptotic solutions of two-sided difference equations. By finding the minimal (or subdominant) solutions or calculating the continued fractions for the difference equations, we derive explicit formulas for the matrix entries of resolvent of doubly infinite Jacobi matrices corresponding to Lommel polynomials, associated ultraspherical polynomials, and Al-Salam–Ismail polynomials. The spectral measures are then obtained by inverting Stieltjes transformations.

The radius of convexity of normalized Bessel functions

The radius of convexity of two normalized Bessel functions of the first kind are determined in the case when the order is between $-2$ and $-1.$ Our methods include the minimum principle for harmonic functions, the Hadamard factorization of some Dini functions, properties of the zeros of Dini functions via Lommel polynomials and some inequalities for complex and real numbers.

Asymptotic expansion of the modified Lommel polynomials $h_{n,\nu }(x)$ and their zeros

Asymptotic expansion of the modified Lommel polynomials $h_{n,\nu }(x)$ and their zeros

Orthogonal polynomials associated with Coulomb wave functions

A new class of orthogonal polynomials associated with Coulomb wave functions is introduced. These polynomials play a role analogous to that the Lommel polynomials have in the theory of Bessel functions. The orthogonality measure for this new class is described in detail. In addition, the orthogonality measure problem is discussed on a more general level. Apart from this, various identities derived for the new orthogonal polynomials may be viewed as generalizations of certain formulas known from the theory of Bessel functions. A key role in these derivations is played by a Jacobi (tridiagonal) matrix JL whose eigenvalues coincide with the reciprocal values of the zeros of the regular Coulomb wave function FL(η,ρ). The spectral zeta function corresponding to the regular Coulomb wave function or, more precisely, to the respective tridiagonal matrix is studied as well.

High-index asymptotics of spherical Bessel products averaged with modulated Gaussian power laws

Bessel integrals of type \({\int_0^\infty {g(k)j_l^{(m)}(k)j_l^{(n)} (k)k^{2}{\rm d}k}}\) are investigated, where the kernel g(k) is a modulated Gaussian power-law distribution \({k^\mu{e}^{-ak^{2}-(b+{{\rm i}} \omega)k}}\), and the jl(m) are multiple derivatives of spherical Bessel functions. These integrals define the multipole moments of Gaussian random fields on the unit sphere, arising in multipole fits of temperature and polarization power spectra of the cosmic microwave background. Two methods allowing efficient numerical calculation of these integrals are presented, covering Bessel indices l in the currently accessible multipole range 0 ≤ l ≤ 104 and beyond. The first method is based on a representation of spherical Bessel functions by Lommel polynomials. Gaussian power-law averages can then be calculated in closed form as finite Hankel series of parabolic cylinder functions, which allow high-precision evaluation. The second method is asymptotic, covering the high-l regime, and is applicable to general distribution functions g(k) in the integrand; it is based on the uniform Nicholson approximation of the Bessel derivatives in conjunction with an integral representation of squared Airy functions. A numerical comparison of these two methods is performed, employing Gaussian power laws and Kummer distributions to average the Bessel products.

A divergent generating function that can be summed and analysed analytically

We study a recurrence relation, originating in combinatorial problems, where the generating function, as a formal power series, satisfies a differential equation that can be solved in a suitable domain; this yields an analytic function in a domain, but the solution is singular at the origin and the generating function has radius of convergence 0. Nevertheless, the solution to the recurrence can be obtained from the analytic solution by finding an asymptotic series expansion. Conversely, the analytic solution can be obtained by summing the generating function by the Borel summation method. This is an explicit example, which we study detail, of a behaviour known to be typical for a large class of holonomic functions. We also express the solution using Bessel functions and Lommel polynomials.

Stieltjes transforms of a class of probability measures

Let the probability measures μN,N=2,3,… be defined by μN({λk,N})=1/N,μN(A)=0 for λk,N∉A, where λk,N,k=1,2,…,N are the zeros of the orthogonal polynomial PN+1(x), which is obtained recursively by P0(x)=0,P1(x)=1,anPn+1(x)+an−1Pn−1(x)+bnPn(x)=xPn(x). Conditions on an and bn are found such that the sequence μN,N=2,3,… converges weakly to the Dirac measure at the point zero. This is achieved through the convergence of the sequence of Stieltjes transforms ∫−∞∞dμN(t)/(λ−t) to the function 1/λ. Typical examples are the Al-Salam and Carlitz polynomials, the Wall polynomials, the Lommel polynomials and the Tricomi–Carlitz polynomials.

Orthogonality and Completeness of $q$-Fourier Type Systems

We establish orthogonality and completeness of the system of q -exponential functions { \mathcal E_q(\cdot; i\omega_n) } using orthogonality and dual orthogonality of a q -analogue of Lommel polynomials. We also set up a very general procedure by which one can produce similar orthogonal systems using bilinear generating functions formed by products of two complete orthogonal function systems.

A class of phase-like states

We investigate a class of phase-like states |ν;e with an integer parameter ν = 0,1,... which contains the coherent phase states |e as the special case ν = 0. Many characteristics, such as the number statistics and the Susskind-Glogower phase distribution, can be calculated in a closed way and are discussed. The parameter ν can be continuously interpolated and extended to a real parameter up to arbitrary ν>-1. Problems of the definition of phase variances are discussed. The class of states is compared with the so-called `phase-optimized' states and it is found that the case ν = 2 of them is near to these states. The properties of the considered class of states |ν;e provide additional arguments to consider the coherent phase states corresponding to ν = 0 as the genuine phase-optimized states. Since the Lommel polynomials play a role in the Fock-state representation of `phase-optimized' states as is shown and to prevent misinterpretation, it is proposed to rename these states `Lommel states'. In the limiting case of high mean number, they make the transition to Chebyshev states of second kind and are near to the case ν = 1 of the class of phase-like states |ν;e. The mathematical tools connected with a generalization of the geometric series are given in an appendix.

The Asymptotic Zero Distribution of Orthogonal Polynomials with Varying Recurrence Coefficients

We study the zeros of orthogonal polynomials pn, N, n=0, 1, …, that are generated by recurrence coefficients an, N and bn, N depending on a parameter N. Assuming that the recurrence coefficients converge whenever n, N tend to infinity in such a way that the ratio n/N converges, we show that the polynomials pn, N have an asymptotic zero distribution as n/N tends to t>0 and we present an explicit formula for the limiting measure. This formula contains the asymptotic zero distri- butions for various special classes of orthogonal polynomials that were found earlier by different methods, such as Jacobi polynomials with varying parameters, discrete Chebyshev polynomials, Krawtchouk polynomials, and Tricomi–Carlitz polynomials. We also give new results on zero distributions of Charlier polynomials, Stieltjes–Wigert polynomials, and Lommel polynomials.

Some Basic Lommel Polynomials

The basic Lommel polynomials associated to the1ϕ1q-Bessel function and the Jacksonq-Bessel functions are considered as orthogonal polynomials inqν, whereνis the order of the corresponding basic Bessel functions. The corresponding moment problems are both indeterminate and determinate depending on a parameter. Using techniques of Chihara and Maki we derive an explicit orthogonality measure, which is discrete and unbounded. For the indeterminate moment problem this measure is N-extremal. Some results on the zeros of the basic Bessel functions, both as functions of the order and of the argument are obtained. Precise asymptotic behaviour of the zeros of the1ϕ1q-Bessel function is obtained.

On the area swept under the occupation process of an M/M/1 queue in a busy period

We compute in this paper the distribution of the area \mathcal{A} swept under the occupation process of an M/M/1 queue during a busy period. For this purpose, we use the expression of the Laplace transform \mathcal{A}^\star of the random variable \mathcal{A} established in earlier studies as a fraction of Bessel functions. To get information on the poles and the residues of \mathcal{A}^\star, we take benefit of the fact that this function can be represented by a continued fraction. We then show that this continued fraction is the even part of an S fraction and we identify its successive denominators by means of Lommel polynomials. This allows us to numerically evaluate the poles and the residues. Numerical evidence shows that the poles are very close to the numbers \sigma_n=-(1+\rho)/n as n \to \infty. This motivated us to formulate some conjectures, which lead to the derivation of the asymptotic behaviour of the poles and the residues. This is finally used to derive the asymptotic behaviour of the probability survivor function \mathbf{P}\{\mathcal{A}>x\}. The outstanding property of the random variable \mathcal{A} is that the poles accumulate at 0 and its tail does not exhibit a nice exponential decay but a decay of the form cx^{-{1}/{4}}{\rm e}^{-\gamma \sqrt{x}} for some positive constants c and \gamma, which indicates that the random variable \mathcal{A} has a Weibull-like tail.

Turán inequalities for symmetric orthogonal polynomials

A method is outlined to express a Turán determinant of solutions of a three term recurrence relation as a weighted sum of squares. This method is shown to imply the positivity of Turán determinants of symmetric Pollaczek polynomials, Lommel polynomials and q‐Bessel functions.

Combinatorial Interpretation and Operator Calculus of Lommel Polynomials

In this paper we present interpretations of Lommel polynomials and their derivatives. A combinatorial interpretation uses matchings in graphs. This gives an interpretation for the derivatives as well. Then Lommel polynomials are considered from the point of view of operator calculus. A step-3 nilpotent Lie algebra and finite-difference operators arise in the analysis.