Local approximations of mixed-type functions in non-newtonyan analysis

Non-Newtonian analysis is a branch of mathematics based on the generalization of the classical concepts of differentiation and integration through alternative generator functions. It provides effective tools, particularly for the study of rapidly growing functions such as geometric, bigeometric, and related function classes. In this thesis, classical sums and alpha sums of functions defined within classical analysis and functions generated by different generators in non-Newtonian analysis are investigated. In the existing literature, the local behavior of such mixed-type functions is generally examined using the classical Taylor approach. However, this approach does not adequately reflect the intrinsic structure of the non-Newtonian components. The main objective of this study is to develop a new first-order approximation method for the non-Newtonian sums of functions defined by different generators. In the proposed method, the classical and non-Newtonian components of a function are treated separately, and each component is linearized according to its own analytical framework. The resulting approximation is compared with the classical first-order Taylor approximation, and detailed error analyses are carried out. The effectiveness of the proposed method is demonstrated through various numerical examples. In particular, it is shown that lower approximation errors can be achieved for functions involving exponential and double-exponential structures. Therefore, the study is expected to contribute to the development of local approximation techniques in non-Newtonian analysis.