Abstract
The generalization of sum of integral powers of first n-natural numbers has been an interesting problem among the researchers in Analytical Number Theory for decades. This research article mainly focuses on the derivation of generalized result of this sum. More explicit formula has been derived in order to get the sum of any arbitrary integral powers of first n-natural numbers. Furthermore by using the fundamental principles of Combinatorics and Linear Algebra an attempt has been made to answer an interesting question namely: Is the sum of integral powers of natural numbers a unique polynomial? As a result it is depicted that this sum always equals a unique polynomial over natural numbers. Moreover some properties of the coefficients of this polynomial are derived.More importantly a recurrence relation which can give the formulas for sum of any positive integral powers of first n-natural numbers has been proposed and it is strongly believed that this recurrence relation is the most significant thing in this entire discussion
Highlights
Formulas of sum of integers were first given in generalizable from in west by Thomas Harrot (1560-1621) of England
From the above observations one can assume that and this polynomial is free of constant terms
One can make the assumption as follows is a polynomial of degree p+1 in n over N with no constants That is where are unknown coefficients
Summary
Formulas of sum of integers were first given in generalizable from in west by Thomas Harrot (1560-1621) of England. At about the same time Johann Faulhaber (1580-1635) of Germany gave formulas for these sums upto the 17th power for higher than anyone before him, but hid did not make clear to generalize them. In this article an attempt has been made to give the generalized result. First the mathematical modeling to the evaluation of where p=1,2,3,... From the above observations one can assume that and this polynomial is free of constant terms
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