Various functions that the students learn up till the first year of universities, such as trigonometric functions and exponential functions, are called “elementary functions” and serve as the basis for formulating changes and shapes. But there are many phenomena such as large oscillations of a pendulum or the shape the moving rope takes when used in rope-skipping, which cannot be set to formulae without using “special functions.” Our laboratory aims to discover unique special functions and their applications by researching the mathematical structure and characteristics of certain relational expressions (functional equations, in particular differential equations).