In number theory, difference tables can be constructed to develop and visualize patterns within a given sequence that can sometimes lead to constructing a closed form representation of the sequence. Analyzing patterns in difference tables and corresponding diagonals for various classes of functions, including exponential functions and combinations of these exponentials, reveals a significant amount of structure. This paper serves to explain some of the most notable patterns along diagonals of difference tables and difference tables formed from the diagonals of the previous difference tables of a given sequence of numbers. These diagonals, which can be manipulated to find closed forms for functions of the form f(n)=nkjn where k and j are natural numbers by using a corresponding leading coefficient on a polynomial equivalent to the (j-1)th diagonal, can be found with a simple algorithm.
"Pascal's Triangle in Difference Tables and an Alternate Approach to Exponential Functions,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 18:
2, Article 4.
Available at: https://scholar.rose-hulman.edu/rhumj/vol18/iss2/4