Asymptotic estimates for Laguerre polynomials

Number of terms required in the fourier expansion of the reflection function for optically thick atmospheres

A good portion of Gatteschi’s research publications—about 65%—is devoted to asymptotics of special functions and their zeros. Most prominently among the special functions studied figure classical orthogonal polynomials, notably Jacobi polynomials and their special cases, Laguerre polynomials, and Hermite polynomials by implication. Other important classes of special functions dealt with are Bessel functions of the first and second kind, Airy functions, and confluent hypergeometric functions, both in Tricomi’s and Whittaker’s form. This work is reviewed here, and organized along methodological lines.

Given an entire function f ( z ) f(z) that has only negative zeros, we shall prove that all the functions of type f ( m ) ( x ) / f ( n ) ( x ) , m > n f^{(m)}(x)/f^{(n)}(x),\ m>n are completely monotonic. Examples of this type are given for Laguerre polynomials, ultraspherical polynomials, Jacobi polynomials, Stieltjes-Wigert polynomials, q q -Laguerre polynomials, Askey-Wilson polynomials, Ramanujan function, q q -exponential functions, q q -Bessel functions, Euler’s gamma function, Airy function, modified Bessel functions of the first and the second kind, and the confluent basic hypergeometric series.

We have found new exact solutions of the coupled nonlinear Schroedinger equation of Benney-Roskes which is written as i u t -β u x x +γ u y y +δ u * u u -2 w u =0, β w x x +γ w y y -βδ( u * u ) x x =0. Three types of new solutions have been obtained. They are described by the Hermite polynomials, the Bessel function of order ±1/4 and ±1/3 (Airy function). Physically these solutions represent exploding and decaying solitons.

For three-dimensional Schrodinger equations, we study how to localize exact solutions represented as the product of an Airy function (Berry-Balazs solutions) and a Bessel function and known as Airy-Bessel beams in the paraxial approximation in optics. For this, we represent such solutions in the form of Maslov’s canonical operator acting on compactly supported functions on special Lagrangian manifolds. We then use a result due to Hormander, which permits using the formula for the commutation of a pseudodifferential operator with Maslov’s canonical operator to “move” the compactly supported amplitudes outside the canonical operator and thus obtain effective formulas preserving the structure based on the Airy and Bessel functions. We discuss the influence of dispersion effects on the obtained solutions.

Linear second-order differential equations having a large real parameter and turning point in the complex plane are considered. Classical asymptotic expansions for solutions involve the Airy function and its derivative, along with two infinite series, the coefficients of which are usually difficult to compute. By considering the series as asymptotic expansions for two explicitly defined analytic functions, Cauchy’s integral formula is employed to compute the coefficient functions to a high order of accuracy. The method employs a certain exponential form of Liouville–Green expansions for solutions of the differential equation, as well as for the Airy function. We illustrate the use of the method with the high accuracy computation of Airy-type expansions of Bessel functions of complex argument.

Wronskian type determinants of orthogonal polynomials, Selberg type formulas and constant term identities

The existing studies on the behavior of cracks in continuously graded materials assume the elastic properties to vary in the plane of the crack. In the case of a plate graded along the thickness and having a crack in its plane, the elastic properties will vary along the crack front. The present study aims at investigating the effect of elastic gradients along the crack front on the structure of the near-tip stress fields in such transversely graded materials. The first four terms in the expansion of the stress field are obtained by the eigenfunction expansion approach (Hartranft and Sih, 1969, “The Use of Eigen Function Expansion in the General Solution of Three Dimensional Crack Problems,” J. Math. Mech., 19(2), pp. 123–138) assuming an exponential variation of the elastic modulus. The results of this part of the study indicated that for an opening mode crack, the angular structure of the first three terms in the stress field expansion corresponding to r(−1∕2), r0, and r1∕2 are identical to that given by Williams’s solution for homogeneous material (Williams, 1957, “On the Stress Distribution at the Base of a Stationary Crack,” ASME J. Appl. Mech., 24, pp. 109–114). Transversely graded plates having exponential gradation of elastic modulus were prepared, and the stress intensity factor (SIF) on the compliant and stiffer face of the material was determined using strain gauges for an edge crack subjected to pure bending. The experimental results indicated that the SIF can vary as much as two times across the thickness for the gradation and loading considered in this study.

Note on some formulæ involving the Laguerre and Hermitian polynomials and Bessel functions

The generation of undulator X-ray radiation harmonics in free-electron lasers (FELs), which has been observed in different experiments, has been given an analytical explanation. Expressions for spectrum lines and radiation intensity are written in explicit form in terms of generalized Bessel and Airy functions with regard to the spreads of electron energy, electron beam size, emittance, spectrum line splitting, and magnetic field constant components. The theory proposed here adequately explains the spectrum and intensity of radiation harmonics in all experiments with a SACLA FEL, a Linac coherent light source (LCLS) based on an FEL, PAL-XFEL, SwissFEL, and other equipment conducted under different conditions in a wide wavelength range, including hard X-ray radiation.

Buckling and postbuckling of a long, hanging column lowered into a fluid

In the mid 1980s it was conjectured that every bispectral meromorphic function ψ ⁢ ( x , y ) {\psi(x,y)} gives rise to an integral operator K ψ ⁢ ( x , y ) {K_{\psi}(x,y)} which possesses a commuting differential operator. This has been verified by a direct computation for several families of functions ψ ⁢ ( x , y ) {\psi(x,y)} where the commuting differential operator is of order ≤ 6 {\leq 6} . We prove a general version of this conjecture for all self-adjoint bispectral functions of rank 1 and all self-adjoint bispectral Darboux transformations of the rank 2 Bessel and Airy functions. The method is based on a theorem giving an exact estimate of the second- and first-order terms of the growth of the Fourier algebra of each such bispectral function. From it we obtain a sharp upper bound on the order of the commuting differential operator for the integral kernel K ψ ⁢ ( x , y ) {K_{\psi}(x,y)} leading to a fast algorithmic procedure for constructing the differential operator; unlike the previous examples its order is arbitrarily high. We prove that the above classes of bispectral functions are parametrized by infinite-dimensional Grassmannians which are the Lagrangian loci of the Wilson adelic Grassmannian and its analogs in rank 2.

We derive an exact expression for the second term in the activity expansion of S (Q,ω), the Fourier transform of Van Hove’s correlation function, for a quantum system of bosons or fermions (spin neglected). The results for both the self and distinct parts of this function are given in terms of standard two‐particle T matrices. The detailed‐balance condition and zeroth and first moment relations are confirmed. An expression for the second term in the activity expansion of S (Q), the structure factor, is obtained. The derivation follows directly from the definition of S (Q,ω) using formal operator techniques; in addition, we show that the conserving T approximation for two‐particle Green’s functions yields the same result.

Theory of generalized hermite polynomials

Special functions such as hypergeometric, zeta, Bessel, Whittaker, Struve, Airy, Weber-Hermite and Mittag-Leffler functions are obtained as a solution to complex differential equations in engineering, science and technology. In this work, generalized Euler-type integrals involving four parameters Mittag-Leffler function are proposed. Some special cases of this type of generalized integrals that are corresponding to well-known results in the literature are also inferred.