The paper is devoted to the study of the analytic and some geometrical aspects of the equation \[ \vec r'''+ \biggl( \alpha^2 s^2+ {{C^2} \over {\alpha^2}} \biggr) \vec r''+ 3\alpha^2 s\vec r' =0 \] in terms of the position vector \(\vec r(s)\) in the three-dimensional space. The equation is solved by means of power series in terms of the Pollaczek polynomials. The obtained solutions are then related to Appell’s hypergeometric functions of two variables and corresponding confluent forms. Finally, a concept of the approximate calculation of nonlinear splines via cubic splines is introduced and a geometrical interpretation involving an apple is included.
For the entire collection see [Zbl 0827.00050].