T-balancers and t-balancing numbers

In this thesis, the general terms of t-balancing numbers, t-balancers and Lucas t-balancing numbers for an integer t≥1 are determined in terms of balancing and Lucas-balancing numbers. In the first section, some notations and definitions on Fibonacci, Lucas, Pell, Pell-Lucas, balancing, cobalancing and generalized balancing numbers are given. In the second section, the material and method are given. In the third section, which is the original part of the thesis, the general terms of t-balancing numbers, t-balancers and Lucas t-balancing numbers are given. For this reason, first the set of all positive integer solutions (x_n,y_n) of the Pell equation 2x^2-y^2=2t^2+4t+1 is determined by using its set of representatives. Later the general terms of t-balancing numbers, t-balancers and Lucas t-balancing numbers are determined in terms of balancing and Lucas-balancing numbers by using the set of integer solutions. Here the problem is considered in two cases: 2t^2+4t+1 is a perfect square or not for an integer t≥2. In the last section, result is given.