[en] Characteristics and advantages of a basis set of Hermite Gaussian functions modulated by plane waves (THGF) are discussed and analysed in a specific case. The expressions of the prototype integrals necessary in variational calculations for bound and continuum states are given

[en] Various methods are discussed for numerical evaluation of the Voigt integral and alternatives are suggested to Gauss-Hermite quadrature which are valid for all values of Lorentzian and instrumental widths. (Auth.)

No abstract available

[en] A new expansion is presented for the product of two confluent hypergeometric functions with different argument, using the expansion for the generalized Laguerre polynomials. Applications are discussed

[en] Several systems of infinite nonlinear algebraic equations and sum rules satisfied by the zeros of confluent hypergeometric function PHI(a,c;x) are reported. The special cases for Bessel functions and Laguerre polynomials are obtained. (author)

[en] A method for finding relations of roots of polynomials is presented. Our approach allows us to get a number of relations between the zeros of the classical polynomials as well as the roots of special polynomials associated with rational solutions of the Painleve equations. We apply the method to obtain the relations for the zeros of several polynomials. These are: the Hermite polynomials, the Laguerre polynomials, the Yablonskii-Vorob'ev polynomials, the generalized Okamoto polynomials, and the generalized Hermite polynomials. All the relations found can be considered as analogues of generalized Stieltjes relations

[en] New nonlinear connection formulae of the q-orthogonal polynomials, such as continuous q-Laguerre, continuous big q-Hermite, q-Meixner-Pollaczek and q-Gegenbauer polynomials, in terms of their respective classical analogues are obtained using a special realization of the q-exponential function as infinite multiplicative series of ordinary exponential function

[en] Generalised Hermite–Gaussian modes (gHG modes), an extended notion of Hermite–Gaussian modes (HG modes), are formed by the summation of normal HG modes with a characteristic function α, which can be used to unite conventional HG modes and Laguerre–Gaussian modes (LG modes). An infinite number of normalised orthogonal modes can thus be obtained by modulation of the function α. The gHG mode notion provides a useful tool in analysis of the deformation and transformation phenomena occurring in propagation of HG and LG modes with astigmatic perturbation. (paper)

[en] Some theorems and conjectures concerning the zeros of the classical polynomials (of Hermite, Laguerre and Jacobi) are announced. The theorems, whose proof is not reported, have been obtained from the relationship between the motion of the zeros of special solutions of simple linear partial differential equations (in x and t), and one-dimensional many-body problems. (author)

[en] It is proved that the n zeros of simple combinations of Laguerre polynomials satisfy a simple system of n nonlinear equations