## 1 Introduction  
When light is used for steering micromachine components and generating multiple optical traps for microscopic particles, laser beam modes are a convenient tool, as they conserve their structure both upon free-space propagation and at the focus of a spherical lens. Another benefit is that mode symmetry can be adjusted by varying certain parameters and passing from symmetry in the Cartesian coordinates to circular symmetry via elliptical symmetry. Ince-Gaussian (IG) modes can be utilized for this purpose, as they are transformed into either Hermite-Gaussian (HG) beams or Laguerre-Gaussian (LG) beams when the ellipticity parameter is changed.  
An initial solution to the Whittaker equation in the form of IG functions was obtained by [PERSON] in [1, 2]. In [PERSON]'s book [3], this solution was obtained in separated variables in elliptical coordinates. The applications of IG beams in optics research were considered in [4]. In [5], expressions for the amplitudes of IG beams were obtained for \(p=0\), \(1\), and \(2\), independent of the ellipticity parameter. In [6], elegant IG beams were investigated. These works have created an impetus for the broad adoption of IG beams, along with other well-known laser modes (LG and HG), in optics. In [7], IG beams were generated with a digital hologram. In [8], vector (classically entangled) IG beams were generated as superpositions of a right-handed, circularly polarized, even IG beam and a left-handed, circularly polarized, odd IG beam. IG beams with quantum entanglement were generated in [9]. Nonlinear transformation of IG beams by spontaneous parametric down-conversion was investigated in [10; 11]. In [12], IG beams were generated also parametrically, but without conversion. In [13], elegant IG beams were studied in a parabolic medium. Propagation of IG beams in uniaxial crystals was studied in [14]. IG beams are used for underwater data transmission [15]. Propagation of IG beams in a turbulent atmosphere was considered in [16].  
As seen from the above brief review of works dealing with IG beams, they have been actively studied in optics. However, no analytical representation of these beams via LG or HG modes with an explicit dependence on the ellipticity parameter has yet been proposed. In this work, in an effort to fill in this gap, we derive a number of particular analytical formulae for even and odd IG modes with the indices \(p=3\), \(4\), \(5\), and \(6\), which are expressed via LG and HG modes and exhibit explicit dependence on the ellipticity parameter \(\varepsilon\). This explicit dependence of the expansion coefficients on the ellipticity parameter makes it possible to control the intensity pattern of IG beams by continuously varying this parameter. In this work, for the first time in the context of optics, we consider a situation wherein the ellipticity parameter \(\varepsilon\) can be not only positive, but negative as well. We analyze how the IG beam changes with changes in the sign of this parameter. The simultaneous representation of IG modes via several LG modes and several HG modes, as proposed herein, suggests that when the ellipticity parameter \(\varepsilon\) approaches zero, any IG mode is converted into a specific LG mode (more exactly, to its real or imaginary part), and when the ellipticity parameter \(\varepsilon\) approaches infinity, the IG mode aligns with a specific HG mode.