We considered a Hankel transform evaluation of Narayana and shifted Narayana polynomials. Those polynomials arises from Narayana numbers and have many combinatorial properties. A mainly used tool for the evaluation is the method based on orthogonal polynomials. Furthermore, we provided a Hankel transform evaluation of the linear combination of two consecutive shifted Narayana polynomials, using the same method (based on orthogonal polynomials) and previously obtained moment representation of Narayana and shifted Narayana polynomials.
The paper summarises basic properties of orthogonal polynomials and their use for approximation of functions representing a surface shape of optical components. The approximation of least-squares is demonstrated including its properties, and a strategy of a generation of orthogonal polynomials on a selected region is shown as well. The second part of the paper deals with mathematical description of aspherical optical surfaces. and Práce shrnuje základní vlastnosti ortogonálních polynomů a jejich využití pro aproximaci funkcí, které vyjadřují tvar ploch v rámci optické praxe. Aproximace funkce je představena ve smyslu nejmenších čtverců, jsou určeny její vlastnosti a možnost generace ortogonálních polynomů na libovolné oblasti. V druhé polovině práce jsou shrnuty možnosti matematického vyjádření asférických ploch v optice.