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Is there a write up on the following somewhere? I don't know what to call it and have not found anything online for it. I had worked this out in school as an attempt on Fermat's Last Theorem. I revisited it recently to see if it had other uses.

$$ \begin{array}{c|ccc|c|cccc|c|ccccc} x^3&&&&x^4&&&&&x^5&&&&&&\\ 0&&&&0&&&&&0&&&&&\\ 1&1&&&1&1&&&&1&1&&&&\\ 8&7&6&&16&15&14&&&32&31&30\\ 27&19&12&6&81&65&50&36&&243&211&180&150\\ 64&37&18&6&256&175&110&60&24&1024&781&570&390&240\\ 125&61&24&6&625&369&194&84&24&3125&2101&1320&750&360&120\\ 216&91&30&6&1296&671&302&108&24&7776&4651&2550&1230&480&120\\ 343&127&36&6&2401&1105&434&132&24&16807&9031&4380&1830&600&120\\ \end{array} $$

This is how its constructed, (quite easy to do on a spreadsheet),

  1. The first columns, under $x^n$, are just that for $x = 0 ... 7$
  2. Subsequent columns hold the difference, from the previous column, between the values in the current row and row above it

The leading edge, last column values, ($k_x$), where $x \ge n$, works out to, $k_x = n! = {n \choose 0}x^n - {n \choose 1}(x-1)^n + {n \choose 2}(x-2)^n … - {n \choose n-1}(x - (n-1))^n + {n \choose n}(x-n)^n$

$k_x = n! = \sum_{y=0}^n (-1)^y {n\choose y} (x-y)^n$

where $x < n$,
$k_x = {x \choose 0}x^n - {x \choose 1}(x-1)^n + {x \choose 2}(x-2)^n … - {x \choose x-1}(x - (n-1))^n + {x \choose x}(x-n)^n$

$k_x = \sum_{y=0}^x (-1)^y {x\choose y} (x-y)^n$

The above holds for all $x$ and $n$, even $n=2$, i.e. all of them form a ladder till $x=n$ and then the subsequent $x^n$ are formed by adding the sum of an arithmetic sequence, with difference $n!$ to $x^{n-1}$

The case for $x \ge n$ is known and proven. It is covered in other questions on this forum so I won't repeat it here. The case for $x<n$ follows once $x \ge n$ is established.

I'd like to know what this is called and if there is any analysis on this I can read up. I maybe wrong but to me it has a few possibilities, e.g.

  1. For equations of degree 3, 4 or higher it one could reduce the terms to a quadratic or cubic equation, as these are essentially sums of series
  2. The above could also simplify higher degree residues/reciprocity equations/calculations.
Community
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gautam
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2 Answers2

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For a sequence $a_n$, the values $s_n = a_{n+1} - a_n$ are commonly called the (forward) differences, and $s_n = a_{n} - a_{n-1}$ are the (backward) differences. Another keyphrase: "finite difference".

The terminology and theory of differences has a lot of parallels with the theory of differentiation. E.g.

  • One derives algebraic rules for computing differences
  • One talks of solving difference equations (or the closely related recurrence relations)
  • One exploits interpretation of summation as the anti-difference
  • There is a Newton series for reconstructing the value of a function from its differences

As a guess, I googled the term difference table and got hits, so I guess that's a standard term for the entire table of differences you are constructing.

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Mean value theorem gives you exactly what you are observing.

Corey
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