On some generalized sequence spaces based on Lucas band matrix andmodulus functions

In this thesis, we first mention some fundamental definitions and basic theorems, then we give and examine some basic concepts such as 𝑔-density, 𝑔-statistical convergence, and 𝑔-statistical boundedness, where 𝑔 is a modulus function. Later, we establish the concept of 𝑔𝑘-density of a natural numbers subset 𝐵, then by using a lacunary sequence 𝜃 = (𝑘𝑟 ), we extend it to put the definition of 𝑆𝜃 (𝑔𝑘 )-statistical convergence, where 𝑔𝑘 is the composition of 𝑔 in 𝑘 times, also we give some inclusion relationships between the sets of 𝑆𝜃 (𝑔𝑘 )-statistically convergent sequences. After that, we focus on 𝑔-strong Cesaro summability, then by using 𝜃 = (𝑘𝑟 ) we introduce the concept of 𝑁𝜃 (𝑔𝑘 )-strong convergence and determine the connections between the sets of 𝑁𝜃 (𝑔𝑘 )-strongly convergent sequences. Moreover, by using a sequence of modulus functions 𝐺, we establish the concept of 𝐺𝑘-density of 𝐵, where 𝐺𝑘 is a sequence consisting of the compositions of every modulus of 𝐺 in 𝑘 times. Then by using 𝜃 = (𝑘𝑟 ) and with the help of the Lucas transform 𝐸 ̂(𝑟, 𝑡), we introduce the concepts of 𝑆𝜃 (𝐸 ̂ (𝑟, 𝑡), 𝐺𝑘 )-statistical convergence and 𝑁𝜃 (𝐸 ̂ (𝑟, 𝑡), 𝐺𝑘 )-strong convergence including the relation between the sets of 𝑁𝜃 (𝐸 ̂ (𝑟, 𝑡), 𝐺𝑘 )-strongly convergent sequences and 𝑆𝜃 (𝐸 ̂ (𝑟, 𝑡), 𝐺𝑘 )-statistically convergent sequences. In addition, we generalize two sequence spaces based on the Lucas band matrix and a sequence of modulus functions such as ℓ𝑝(𝐺𝑘 , 𝑣,𝐸 ̂ (𝑟, 𝑡)) and ℓ∞(𝐺𝑘 , 𝑣,𝐸 ̂(𝑟, 𝑡)), then we discuss a geometrical property such as the Gurarii's modulus of convexity of ℓ𝑝(𝐺𝑘 , 𝑣,𝐸 ̂ (𝑟, 𝑡)).