Unique condition for generalized Laguerre functions to solve pole position problem

Previous research indicates that solution to pole optimization problem for the generalized Laguerre functions can be found by vanishing at least one of two clearly stated Laguerre coefficients. The aim of this paper is to prove uniqueness of a certain coefficient leading to the optimal solution. To achieve this purpose, we employed connection coefficients method to work out specific recurrence relations suitable for the continuous generalized Laguerre functions in the case of the optimal pole position. The proposed results were extended to the discrete Laguerre functions using modified bilinear transform and introducing the rational z-transform of the Meixner-like functions. The findings of this research present a postulated and proved theorem and conducted computational experiments to support the theoretical results.