A Generalization of Ibn al-Haytham Recursive Formula for Sums of Powers

Abstract

In this paper, we give a sequential generalization of Ibn al-Haytham recursive formula for sums ofpowers of any integer sequence. Then, we obtain a higher order dimensional generalization of the generalizedIbn al-Haytham formula. As by-products, we also show that how our recursive formulas imply other interestinginteger sequences identities like Karaji L-summing equation and Abel’s summation by parts lemma. Finally, as anapplication, we prove several identities related to Fibonnaci and harmonic numbers.