We investigate a function defined by a generalised continued fraction and find it to be closely related to a generating function of series OEIS:A000698. We then alter the function to contain the set of primes rather than the set of real numbers. This continued fraction function also has a similar form with coefficients \(2,6,48,594,10212,230796,\cdots\). We also transform the sequence \(1,1,1,1,1\cdots\) and gain the Catalan numbers as coefficients of the corresponding Laurent Series. We then provide examples which transform into various OEIS sequences.
Define the function \[f(x)=\frac{1}{x+\frac{2}{x+\frac{3}{x+\cdots}}} = \underset{k=1}{\overset{\infty}{\mathrm \large K \normalsize}} \frac{k}{x}\] evaluating this function until convergence for 16 decimal places, gives something that looks like \(1/x\), however, after diagnosing the coefficients of the Laurent series, by subtracting likely integer terms we very easily find \[\lim_{x\to \infty}f(x)=\frac{1}{x}-\frac{2}{x^3}+\frac{10}{x^5}-\frac{74}{x^7}+\frac{706}{x^9}-\frac{8162}{x^{11}}+\frac{110410}{x^{13}}-\cdots,\] with a search on OEIS giving sequence A000698 which we believe to be plausible. We can easily imagine a similar function \[g(x)=\frac{2}{x+\frac{3}{x+\frac{5}{x+\frac{7}{x+\cdots}}}} = \underset{k=1}{\overset{\infty}{\mathrm \large K \normalsize}} \frac{p_k}{x}\] where the top row of numbers are the prime numbers, \(p_k\). Using the first \(9999\) primes in the continued fraction, this also converges to 16 decimal places for large enough \(x\), and appears to be described by an integral coefficient Laurent series \[g(x)=\frac{2}{x}-\frac{6}{x^3}+\frac{48}{x^5}-\frac{594}{x^7}+\frac{10212}{x^9}-\frac{230796}{x^{11}}+\frac{6569268}{x^{13}}-\cdots\]
That would allow conjecture for a very interesting relationship \[\underset{i=1}{\overset{\infty}{\mathrm \large K \normalsize}} \frac{p_i}{x} = \frac{2}{x} - \frac{6}{x^3} + \frac{48}{x^5} -\frac{594}{x^7} + \frac{10520}{x^9} -\cdots\] where \(p_1=2\),\(p_2=3\) and so one for primes, and the capital \(K\) notation is that for the continued fraction.
A further assessment of many integer sequences inside the continued fraction should be made, and a check of corresponding Laurent series undertaken. It may be common place to find integral coefficients.
The first function investigated in this document is also of interest, in OEIS the Laurent coefficient sequence A000698 is commented “Number of nonisomorphic unlabeled connected Feynman diagrams of order 2n-2 for the electron propagator of quantum electrodynamics (QED), including vanishing diagrams.” The continued fraction clearly has the potential to capture the recursive aspect of the theory, and this may be why the series align.
Many thanks to OEIS for providing their invaluable service. Thanks to Wolfram|Alpha for plots as below, and Mathematica for calculations. This work was undertaken in my spare time while being funded by the EPSRC.