On the Summations of Some Special Sequences

It is well known that many number sequences have important parts in mathematics. Especially in the fields of combinatorics and number theory [15, 16]. Recently, interest has been shown in summing infinite series of reciprocals of some special numbers, for example Fibonacci numbers [5], [6] and [7]. It is not easy, in general, to derive the sum of a series whose terms are reciprocals of Fibonacci, Pell, etc. Numbers such that the subscripts are terms of geometric progressions. It seems even more difficult if the subscripts are in arithmetic progression [13]. Many problems related to the sum of the terms of the Fibonacci series were first proposed in [3]. After that Ken Siler proved the ∑_(k=1)^n▒F_(ak-b) summations provided that alarger thanb in [9]. Single-indexed or double-indexed ones could be summed, for example, some summations of the Pell Quaternions and the Pell-Lucas Quaternions were studied in [2], some summations of the generalized Pell sequence were studied in [10], some summations of the generalized dual Pell and some summations of the generalized dual Pell Quaternions were also studied in [14], but no generalizations were made. In this publication, both these generalizations are made for Pell, Lucas, Pell-Lucas, Jacopsthal, Jacopsthal-Lucas, generalized Pell, generalized dual Pell, and generalized dual Pell quaternion sequences and summing infinite series of reciprocals of the Pell numbers are calculated.