(Discrete Mathematics Techniques III)
Abstract
We find a direct closed-form solution, i.e. one that does not require iteration, for the general case of the finite-summation-of-integer-powers problem . Having established in Part 2 that the closed-form solution is a polynomial, the summation is here rewritten as the sum of the independent monomials (), where the [...]
The question of what value 0^0 should evaluate to has been discussed since the time of Euler (1700s). There are three candidate choices: 1,0, or “indeterminate” (i.e., throw an error).
In this article, I argue that the only reasonable choice (for discrete mathematics) is 0^0=1 (), and I’ll give a tangible, feel-the-grit-in-your-palms reason [...]
By
Assad Ebrahim, on February 8th, 2010
Topic: Mathematics-Technical(Discrete Mathematics Techniques II)
Abstract
We solve the general case of the finite-summation-of-integer-powers problem for arbitrary , and obtain a -th order recurrence relation that can be used to iteratively obtain the closed form polynomial for for any given . Source code is given for computing these polynomials using Maxima, an open-source (free) symbolic [...]
By
Assad Ebrahim, on February 8th, 2010
Topic: Mathematics-Technical(Discrete Mathematics Techniques I)
Abstract
We motivate an approach that uses recurrence relations to find closed form solutions to the finite-summation-of-integer-powers problem for any individual . The approach is illustrated for small : . Maxima, an open-source (free) software package, is used to demonstrate how a symbolic computation platform can speed up the accurate [...]
