Recurrence relation

Pure mathematics

Discrete mathematics

Archaeology

Economics

Computer science

Polynomial

Relation (database)

Database

Mathematics

Natural number

Natural (archaeology)

Sums of powers

Mathematical analysis

Combinatorics

Finance

Generalization

Algebra over a field

Order (exchange)

History

AIP conference proceedings

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.

A new vision on sum of powers of integers