Problems and heuristics using infinite matrices

Bernoulli-numbers and the Beauty of the Pascal-matrix

The binomial- or Pascal-matrix occurs as very special entity in the zoo of number-theoretic creatures. It has a very basic matrix-logarithm; its signed version has an eigensystem which exhibits the bernoulli-numbers in a very harmonic way.

bernoulli_en.pdf

A small collection about relation of stirling-numbers and Bernoulli-numbers

stirling2.pdf

Numbertheoretical matrices

A collection of articles about Pascal/Binomial-matrices.

This collection was first induced by some nice observations concerning Pascal-matrices and its eigensystems, the "bernoullian" matrices Gp and Gm.

When I evaluated these matrices, more and more interesting properties popped up and very interesting connections to fields of number-theory occured. This introduced then a search for systematics by exploring matrix relations and reading literature.

The main collection of articles is now a compilation of these properties (some are "exotic") as manuscripts, which I am refining step by step with references, proofs and links as they'll occur to me.

The whole project is called

"Identities involving binomial-coefficients, Bernoulli- and Stirlingnumbers"

Keywords: Pascalmatrix, Stirlingnumbers, Eulerian numbers, Bernoulli numbers, Zetamatrix, infinite matrices, combinatorical matrices

I'm currently undertaking a complete restructuring and editing of all articles.

The new version may be found here. Note, that the project-related links in the articles partly still point to the old versions, so you should always return to the html-projectindex. Also they are not intensely reviewed, so for confusion, which possibly may have occured in the restructuring and may have survived I have to plea for patience...

… unfortunately the study of the iterated exponentials (see tetration) consumes almost all of my time, so I couldn't update and finish this project… Hope I can come back to this soon (Jan 2009)

Keywords: Pascalmatrix, Stirlingnumbers, Eulerian numbers, Bernoulli numbers, Zetamatrix, infinite matrices, combinatorical matrices

Procedures for summation of divergent series using matrices

The analysis of the binomial matrix is extended, and a matrix-representation of common summation procedures is investigated. Since I mainly deal with the binomial-matrix, this is almost all about variants of the Euler-summation. The article is in the state of a draft; I'm currently studying the heuristics and assumtions in more details and shall provide more links to literature and online-resources. See pmatrix.pdf

Keywords: Euler summation, divergent series, matrix summation method

Tetration and iteration of functions
using matrix-operators

Some technical articles, mostly workshop-like material about tetration.

When studying the "numbertheoretical matrices" I came across the matrices of Stirlingnumbers and asked, what would I find, if I use their powers (and series of powers) like I did with the binomial matrix. which led to some nice insights into bernoulli-numbers, zeta/eta-function and even series of zetas/etas. The result is iterating powerseries and the generalizing to fractional iteration for maps of x->b^x , x->b^x - 1. This was already seriously considered by Goldbach, Lambert and Euler, and in the 20^th century developed this up to the problem of fractional iteration, where some aspects are still unsolved.

Enter the "hellgate to the powerseries of tetration "

Keywords: Tetration, Powertower, functional iteration, powertowerseries, halfiterate

Exponential diophantine problems

Mersenne numbers, cyclotomic polynomials and primefactorization

By analysis of the problem of Mersenne-primes it is natural to generalize the question to the cyclotomic functions/polynomials. What I did here was analyzing the primefactorization of Mersenne-numbers and cyclotomic functions, finding this way an heuristic approach to the concept of cyclic multiplicative (sub)groups modulo a prime.

This concept and an adapted way of analytical expression allows then to express and analyze more general exponential diophantine problems like catalans, fermats problems, and focuses the question of solvability in terms of a generalized Wieferich-prime-concept. See cyclic subgroups

Collatz-(3x+1)-problem

The Collatz-problem was a main subject for me for several years ago. Sometimes I felt to succeed, but was always lacking the final proof.

Thus I reduced my study to the smaller question of existence of cycles in the collatz-transformation. I decided for a special notational framework focusing the exponents/the structure of a transformation rather than the elements and found some nice properties. However, although this approach allowed to disprove some cycles, the general cycle was still too complex for a definitive result.

Then the most simple form of a cycle ("primitive loop") was investigated in detail. This lead to an intimate relation of approximability of powers of 2 to powers of 3, so this was the main, and crucial problem at the heart of the collatz-loop-question. For instance, by this property one finds a connection to the problem of z-numbers (studied by Kurt Mahler) and even (according to E. Weissstein at mathworld) to an unsolved detail in the Waring-problem – a connection that seems not to be known widely.

The "primitive collatz-loop" was already disproved by Ray Steiner in 1978, and some concatenations (on the way to the general cycle) are already disproven by Simons/de Weger in 2000 and 2002. Not knowing their results beforehand I came to very similar formulae, but stuck with proving them. See for an intro for amateurs: Collatz loops (A more concise  and partly updated version in pdf-format is here )

Derived from the Collatz-loop-discussion there is a small discussion of approximation of 2^S to 3^N, or of the next perfect power of 2 to a power of 3.

Keywords: Collatz-problem, Collatz-conjecture, 3x+1, 3n+1, Collatz cycles

Wieferich-primes and Fermat-quotients

Originally I was interested in the so-called Wieferich-primes w, where 2^w-1 = 1 (mod w²), which means the exceptional cases, that –recalling Fermat's little theorem– not only 2^p-1 = 1 (mod p) for a prime p, but even (mod p²) or modulo higher powers of p. The concept of fermat-quotients generalize this to other bases than 2 and to other exponents of the base of the modulus. I found some discussions about fermat-quotients in internet-articles; here is my view on it. I found some system for fermat-quotients and also provide some "exotic" examples.

For more information I prepared a "directory" for further references. See there

Keywords: Wieferich primenumbers, Fermatquotients

Other musings and mathematical recreations

Factorizing of exp-function: the "Dream of a Sequence"

"Dream of a sequence" – when I played with the question, whether it would be possible to express powerseries, for instance the exponential-function, by an infinite product I found a nice solution some years ago. Recently a correspondent, who has been stumbled into the same question, asked me for a comment, and I looked again at the problem. But this time I recognized that the found coefficients contain a beautiful sequence of numbers – just "a dream of a sequence".

Keywords: Exponential function, factorizing

"Uncompleting" the Gamma-function

I was surprised by the seemingly simple pattern of the taylor-series for the gamma-function: the coefficients tend to 1 very fast. Well – if it is partly so simple: why not invent a separation into one series, which has a closed-form expression (like the geometric series) and into another, residual series, but which is better converging? A short discussion using two different replacements. The latter rediscovers the incomplete gamma and gives thus a nice heuristic interpretation. A not-yet-finished contemplation is that of understanding the Gamma as iteration of another function and that of relating this to known concepts of fractional iterates.

See "uncompleting the gamma" (current version: 08'2012)

Keywords: Gamma function, incomplete gamma, zeta-function, Stieltjes numbers, gamma as functional iteration

The inverse of this matrix M is the Null-matrix?

In a discussion in the mathematical exchange on MSE I was triggered to look at the problem, whether a tpe of infinite matrices can be inverted, when the inverse of the infinite size was assumed to be the limit of the inverses of the finite size n*n where n is thought to increase without bound. I tried a special, but simple (Vandermonde-like) matrix M and the formal reciprocal W seem to come out to be completely zero. I discussed this with the help of the L D U-matrix-decomposition, and arrived at a (nontrivial) infinite set of identities involving the Stirling numbers 1^st kind where the first dozens of empirically tested formulae indeed evaluated to zero. A general formula for each entry of the resulting infinite matrix W is given, but I did not get a general proof yet, that indeed all formulae come out to be zero.

See "infinite Nullmatrix as inverse of M"

Continued fractions with powers of e

Triggered by a question in a mathematical discussion group concerning continued fractions of powers of e (euler constant, base of natural logarithm) I deduce a pretty general formula for expressing powers of e in terms of very regular non-regular continued fractions. The result came out to be known in terms of continued fractions of tan(x)

(article not yet ready)

For e-1 there is a nice "generalized continued fraction" with a very simple pattern in its coefficients. I asked: what if the pattern remains, and only the offset is changed? We find rational compositions of e but surprisingly for some general relation of the offsets we seem to get rational values for that infinite continued fractions! See: GenContFrac RationalValues

Continued fractions – generalization

Three ideas composed together: the well known matrix-multiplication scheme for convergents of continued fractions leads to an eigenvalue-view into that matter. That eigenvalue-view reflects then the occurence of periodic continued-fraction-parameters when square roots of rational numbers are irrational. Now why can't we find periodic parameters for higher roots?

The eigenvalue-idea is then applied to approriate dimensional matrices, for cube-roots use 3x3-matrices and so on. Now, if one has a representation of a periodic integer matrix-multiplication - can the rational approximation of an algebraic number be better than 1/q² ?

(article not yet ready)

Sums of like powers

an elementary approach discussing properties of sums of like powers, which *should* be reworked. But since they were my first number-theoretical experiences in that area, I keep them alive... :-)

You'll find the use of the Euler-triangle here

potenzsummen

hypergeometrische Reihen

An unknown document of P.d.Fermat discovered

(03'2002) The usenet-newsgroup sci.math is well known for the many enthusiastic Fermatists; one of the greatest contributors among them being J.S.H., who recently promoted his revolutionary concept of math-of-objects (or "object orientated math"). So it was obvious this newsgroup would be the best place to announce the sensational finding of an unknown document, most likely a formula with which the great french mathematical amateur Pierre de Fermat set himself a stone. I announced this finding in march, 2002. Enthusiastic reactions in the readers community showed that this was really a hit.

Discovery!