Abstract. The trajectories followed in the complex plane by all the zeros of the Hankel function and those of its derivative, when the order varies continuously along real values, are discussed.1. Introduction. Many physical problems require a good knowledge of the location of zeros of the Hankel function and/or those of its derivative. For instance, the trajectories of the zeros of 7/"(l)(z), for varying real order v, are the ^-trajectories of the S-matrix singularities for quantum scattering by a hard sphere. Also, the zeros of H¡;X)iz) and id/dz)HJ,x\z) give, respectively, the poles and zeros of the logarithmic derivative of the external Schrödinger wave function in a short-range potential, which should match, at the edge of the potential, with the logarithmic derivative of the internal wave function.Information provided by classical treatises [7], [4] on special functions about the zeros of H¡;x\z) and id/dz)H¡¡x\z) is rather insufficient. A more recent updated revision of the topic has been published by Luke [6]. In the case of integer order, v = n, two types of zeros of #"(1)(z) or of id/dz)H^x\z) are found [1, pp. 373-374], [3] (in the principal Riemann sheet, | arg z |< tt):(1) An infinite number of zeros for |Rez|>n just below the negative real semi axis.(2) A group of A? zeros for | Re z | < n which he along the lower half of the boundary of an eye-shaped domain around z = 0.

Recently, it has been proved that a nonlinear quantum oscillator, generalization of the isotonic one, is exactly solvable for certain values of its parameters. Here we show that the Schrödinger equation for such an oscillator can be transformed into a confluent Heun equation. We give a very simple and efficient algorithm to solve it numerically, no matter what the values of the parameters are. Algebraic quasi-polynomial solutions, for particular values of the parameters, are found.

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Useful expressions of the derivatives, to any order, of Pochhammer and reciprocal Pochhammer symbols with respect to their arguments are presented. They are building blocks of a procedure, recently suggested, for obtaining the ε-expansion of functions of the hypergeometric class related to Feynman integrals. The procedure is applied to some examples of such kind of functions taken from the literature.

Abstract. The eigenstates of a real or complex cubic anharmonic oscillator are investigated using an original and alternative method. The procedure consists of determining global solutions of the Schrödinger equation that comply with the pertinent boundary conditions and allows us to obtain, in a very simple way, the eigenenergies and eigenfunctions of the Hamiltonian. Scattering by a real cubic potential is investigated as a particular case.