## Archive for the ‘Research’ Category

### Fractional Fibonacci Numbers

Thursday, 7 June 2018

#### Traditional Formula

There is a well-known formula for the Fibonacci numbers

$\displaystyle F_n = \dfrac{\sqrt{5}}{5} \left( \varphi^n - (-\varphi)^{-n} \right)$

where

$\displaystyle \varphi = \dfrac{\sqrt{5} + 1}{2} \approx 1.618^{+}$

is the golden ratio.

It turns out that there is a way to find $F_x$ for when $x$ is not an integer, but the values are complex rather than real.

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### “New Approach to Sums of Powers” — Headlines and Examples

Thursday, 7 June 2018

As the article on sums of powers was rather long and dense, I thought that it would be worth giving a summary of the main results separately.

I will also show the formulae in action with a worked example.

#### Indirect, Simple Formulae

In the main article, I show that

$\displaystyle {\bigoplus_{m=1}^{n}}{}^{(t)} \; m^k = \sum_{j=0}^{k} \left< \begin{array}{c} k \\ j \\ \end{array} \right> \triangle_{k+t}(n-j)$

This formula is essentially a polynomial of rising factorial powers.

##### Special Cases

Perhaps the most important and useful formulae from the main article are

$\displaystyle n^k = \sum_{j=0}^{k} \left< \begin{array}{c} k \\ j \\ \end{array} \right> \triangle_{k}(n-j)$

and

$\displaystyle \sum_{m=1}^{n} m^k = \sum_{j=0}^{k} \left< \begin{array}{c} k \\ j \\ \end{array} \right> \triangle_{k+1}(n-j)$

### A New Approach to the Sums of Powers

Thursday, 10 May 2018

In the conventional approach to summing powers, that is, finding a polynomial expression for $\sum_{h=1}^{n} h^k$, the coefficients that arise seem to have no pattern. It had always seemed to me that it ought not to be hard to find such expressions with an elementary approach.

### Product Formulae for the Fibonacci Numbers

Monday, 7 May 2018

There is a well-known formula for the Fibonacci numbers

$\displaystyle F_n = \dfrac{\varphi^n - (-\varphi)^{-n}}{\sqrt{5}}$

where

$\displaystyle \varphi = \dfrac{1-\sqrt{5}}{2} \approx 1.618^{+}$

However, I was surprised to find that there are also product formulae involving trigonometric functions.

### Surface Areas and Volumes of Hyperspheres

Thursday, 3 October 2013

I noticed that the formulae for the sizes of the surface (or rather, boundary) and volume (interior) of circles, spheres and hyperspheres seem to have discontinuities (or ‘jumps’) in the index of $\pi$.

However, this is only a result of simplifying a uniform formula for particular cases.

### The Amazing ‘fusc’ Function

Tuesday, 16 April 2013

I recently encountered a collection of related articles regarding an amazing explicit enumeration of the non-negative rational numbers. Whilst experimenting with the mathematical constructs described, I made some discoveries of my own (which I doubt are new, but I have not found them described elsewhere).

Below, are the references and links to those articles. They’re all worth a read.

### Subversion Configuration Suite

Wednesday, 4 July 2012

I’ve now uploaded my Subversion Configuration Suite (svn-meta) repository to Google Code so that the files are publicly available for anyone to use.

### Some Simple New Elementary Functions

Friday, 22 January 2010

In this post I introduce some new functions:

• $\mathrm{exph}(x)$
• $\mathrm{lnh}(x)$
• $\mathrm{cish}(x)$

### Generalised Means

Tuesday, 2 June 2009

There are many notions of average in mathematics and statistics. Well-known are the mean, median and mode.

Also well-known, amongst the means, are the arithmetic mean (A), geometric mean (G), harmonic mean (H) and quadratic mean or  root mean squared (Q or RMS). Recently, I have become interested in the notion of a generalised mean (or generalized mean) over positive (non-negative and non-zero) real numbers $\in \mathbb{R}^+$.

### Falling and Rising Roots

Friday, 8 May 2009

In my post on hypertetrahedral polytopic roots, I gave a few formulae for functions such as the triangular root.

Here I give some very similar formulae for some falling and rising roots.