Generalized Technique for constructing the closed form of the generating functions of certain classes of sequences

We develop generalized techniques for constructing the closed form of the generating functions of sequences of the form𝑓𝑚+1 = 𝑘(𝑓𝑚 + 𝑚𝑘 ) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑚 ≥ 0, 𝑘 > 0 𝑎𝑛𝑑 𝑓0 = 0 𝑎𝑛𝑑 𝑓𝑚+1 = 𝑘(𝑓𝑚 + (𝑚 − 1) 𝑘 )𝑓𝑜𝑟 𝑎𝑙𝑙 𝑚 ≥ 1, 𝑘 > 0 𝑎𝑛𝑑 𝑓1 = 0 using method of differencing in generating function. Also, the techniques of generating functions were applied to solve some important problems of recurrence relations. The findings of this study provide generalized technique and fast method of obtaining the closed form of the generating functions of sequences of the form 𝑓𝑚+1 = 𝑘(𝑓𝑚 + 𝑚𝑘 ) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑚 ≥ 0, 𝑘 > 0 𝑎𝑛𝑑 𝑓0 = 0 and 𝑓𝑚+1 = 𝑘(𝑓𝑚 + (𝑚 − 1) 𝑘 ) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑚 ≥ 1, 𝑘 > 0 𝑎𝑛𝑑 𝑓1 = 0.