Applications of generalized Guglielmo numbers

In this thesis, we define generalized Gaussian Guglielmo numbers, dual generalized Guglielmo numbers, hyperbolic generalized Guglielmo numbers, and dual hyperbolic generalized Guglielmo numbers. Moreover, we define recurrence relations, generating function, sum formulas, matrix formulation, Simpson's formula, and some identities on these numbers. We summarize the chapters, given in the thesis, as follows. In Chapter 1, we present some basic definitions, recurrence relations, Binet's formulas, matrix formulation, and sum formulas related to generalized Guglielmo numbers. In addition, we give some properties about Gaussian numbers, dual numbers, dual hyperbolic numbers and make a literature search about Gaussian numbers, dual numbers, hyperbolic and dual hyperbolic numbers. In Chapter 2, we define Gaussian generalized Guglielmo numbers in detail, and focus on four specific cases: Gaussian triangular numbers, Gaussian triangular-Lucas numbers, Gaussian oblong numbers, and Gaussian pentagonal numbers. In addition, we present some identities and matrices related to these sequences, as well as recurrence relations, Binet's formulas, generating functions, Simpson's formulas, and summation formulas. This chapter includes our original study. In Chapter 3, we introduce the generalized hyperbolic Guglielmo numbers. We delve into various specific instances, including hyperbolic triangular numbers, hyperbolic triangular- Lucas numbers, hyperbolic oblong numbers, and hyperbolic pentagonal numbers. We present Binet's formulas, generating functions and summation formulas for these numbers. Furthermore, we provide Catalan's and Cassini's identities and matrices associated with these sequences. This chapter consists our original study. In Chapter 4, we investigate the generalized dual hyperbolic Guglielmo numbers and then various special cases are explored (including dual triangular numbers, dual triangular-Lucas numbers, dual oblong numbers, and dual pentagonal numbers). Binet's formulas, generating functions, and summation formulas for these numbers are presented. Additionally, Catalan's and Cassini's identities are provided, along with matrices associated with these sequences. This chapter includes our original study. In Chapter 5, the generalized dual hyperbolic Guglielmo numbers are introduced. Various special cases are explored (including dual hyperbolic triangular numbers, dual hyperbolic triangular-Lucas numbers, dual hyperbolic oblong numbers, and dual hyperbolic pentagonal numbers). Binet's formulas, generating functions and summation formulas for these numbers are presented. Moreover, Catalan's and Cassini's identities are provided, along with matrices associated with these sequences. This chapter includes our original study.