Gegenbauer expansion to model the incident wave-field of a high-order Bessel vortex beam in spherical coordinates

The aim of this short communication is to report that Gegenbauer's (partial-wave) expansion, that may be used (under some specific conditions) to represent the incident field of an acoustical (or optical) high-order Bessel beam (HOBB) in spherical coordinates, anticipates earlier expressions for undistorted waves. The incident wave-field is written in terms of the spherical Bessel function of the first kind, the gamma function as well as the Gegenbauer or ultraspherical functions given in terms of the associated Legendre functions when the order m of the HOBB is an integer number. Expressions for high-order and zero-order Bessel beams as well as for plane progressive waves reported in prior works can be deduced from Gegenbauer's partial-wave expansion by appropriate choice of the beams' parameters. Hence the value of this note becomes historical. In addition, Gegenbauer's expansion in spherical coordinates may be used to advantage to model the wave-field of a fractional HOBB at the origin (i.e. z=0).