﻿ Generalized expansions of real numbers

Generalized expansions of real numbers
by Simon Plouffe

april 2006

I present here a collection of algorithms that permits the expansion into a finite series or sequence from a real number x(0)  R, the precision used is 64 decimal digits. The collection of mathematical constants was taken from my own collection and theses sources [1]-[6][9][10]. The first and simplest algorithm is the continued fraction algorithm, see Wikipedia for an introduction to continued fractions.

Most of the algorithms will produce a sequence of integers when x  R and can be written as a 2 terms recurrence. If x(0) is the initial value then y(n) will be the terms of the sequence. If x is rational the sequence is finite and if x is irrational the the sequence is infinite but with 64 decimal digits enough terms are computed for detecting simple patterns as with some quadratic irrational like 2 or . Other numbers like exp(1) : 2.71828… do have a pattern which is easily recognizable but most real numbers do not. The goal of this computation of sequences from real numbers using different algorithms is to discover or find if there could be any patterns at all with other algorithms.

The natural question that comes to mind is : is there any closed formula or generating function for those sequences?. For this I can use GFUN [8] which is a Maple package or with Mathematica as well to answer the question [7]. A known example is sinh(1) = 1.175201193643801456… which is : 1, 6, 20, 42, 72, 110, 156, 210, 272, 342, … when expanded into the Engel expansion. That sequence appears to be the coefficients of the series expansion of this rational polynomial :

So the coefficients are given by the polynomial : 4n2+2n, n > 1 by using Montmort formula.

Unfortunately this is a lucky example because for tanh(1) = 0.761… with the Engel expansion we  have : 2, 2, 22, 50, 70, 29091, 49606, 174594, 260086, … which does not coerrespond to any known closed formula where the continued fraction expansion of that same numbers is : 0, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ….

The main algorithms

Where [x] denotes the integer part function and {x} the fractional part, y(n) are renamed a(n).

To see each equation bigger just click over with the mouse.

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Here is a list of 188 mathematical constants and their expansions. The naming convention used here reflects the fact that these constants are now included in my new database of 1.099 billion constants and not yet implemented in my Inverter on the web [9].

References

[1] Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, 1995.
[2] Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 59-65, 2003.
[3] Simon Plouffe :  Plouffe Inverter : a collection of 215 million constants (1998-2006).
[4] Sloane, N. J. A.,The OEIS The On-Line Encyclopedia of Integer Sequences.
[5] Simon Plouffe :  Inverse Symbolic Calculator : a collection of 58 million constants (1995).
[6] Eric Weisstein : Mathworld
[7] François Bergeron, Simon Plouffe, Computing the Generating Function of a Series Given Its First Few Terms, Experimental Mathematics, Vol. 1 (1992), No. 4.

[8] Bruno Salvy, Paul Zimmermann, GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software (TOMS) , Volume 20 ,  Issue 2  (June 1994), pp. 163  177.

[9] Plouffe Inverter, Version of 2005, 1.099 billion constants @ 64 digits. Not yet on the web.

[10] Xavier Gourdon and Pascal Sebah, Numbers, constants and computation.

[11] Simon Plouffe, Approximations de Séries Génératrices et quelques conjectures. Montréal, Canada: Université du Québec à Montréal, Mémoire de Maîtrise, UQAM, 1992.