Generating functions and applications of multi variable Fibonacci type polynomials

One of the main purpose of this thesis is to construct a new generating function family for the Fibonacci-type polynomials and the Jacobsthal-type polynomials. By using this generating function family and its functional equations, fundamental properties of the Fibonacci-type polynomials and the Jacobsthal-type polynomials are examined. At the same time identities, relations and other formulas that include these polynomial families and Apostol-type and Humbert-type numbers and polynomials are obtained. Primarily, some basic properties of these sequences of numbers and polynomial families are introduced, in general terms, with their generating functions and the Binet formulas. Then, the structure of the newly generated generating function and the relationship with other polynomial families such as Fibonacci numbers and polynomials, Lucas numbers and polynomials, Jacobsthal numbers and polynomials, Vieta Fibonacci polynomials, Vieta-Lucas polynomials, Legendre polynomials, Chebyshev polynomials, Gegenbauer polynomials, Humbert polynomials, Apostol-Bernoulli polynomials, Apostol-Euler polynomials, Genocchi numbers and polynomials, Stirling numbers, Dickson polynomials and other well known sequences of numbers and polynomial families are examined. Various formulas, equations and correlations are given, which are known before and new. Infinite series applications, probability and combinatorial applications and algebraic applications are mentioned.