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Wolfdieter Lang

Publications and source records attributed to Wolfdieter Lang.

14 recordsLinked to original sources
arXiv2025

On Positive Integer Descartes-Steiner Curvature Quintuplets

In Descartes' five circle problem integer curvatures (inverse radii) are considered. The positive integer curvature triple [c_1, c_2, c_3] (dimensionless), with non-decreasing entries for three given mutually touching circles, leading to integer curvatures [c_{4,-}, c_{4,+}] for the two circles touching the given ones is called a Descartes-Steiner triple. They come in two types: [c, c, d] (or [c, ,d, d]) and triples with distinct entries. The first case is related to Pythagorean triples. The distinct curvature case is more involved and needs a combined representations of certain binary quadratic forms of the indefinite and definite type. The degenerate case when a straight line touches the three given touching circles can also be characterized completely.

math.NT↗
arXiv2023

Four Sequences of Length 28 and the Gregorian Calendar

It is shown that each sequence giving the number of times a given day of the month falls on a certain day of the week for $400$ successive years of the Gregorian cycle can be composed of various pieces of various length of one of 4 sequences of length 28, used periodically.

math.HO↗
arXiv2023

Cantor's List of Real Algebraic Numbers of Heights 1 to 7

Cantor gave in his fundamental article an elegant proof of the countability of real algebraic numbers based on a positive integer height, denoted by him as N, of integer and irreducible polynomials of given degree (denoted by him as n) with relative prime coefficients. The finite number of real algebraic numbers with given height he called phi(N), and gave the first three instances.\pn Here we give a systematic list for the real algebraic numbers of height, which we denote by n, for n from 1 to 7 and polynomials of degree k.

math.NT↗
arXiv2023

On the Equivalence of Three Complete Cyclic Systems of Integers

The system of coaches by Hilton and Pedersen, the system of cyclic sequences of Schick, and Braendli-Bayne, related to diagonals in regular (2 n)-gons, and the system of modified modular doubling sequences elaborated in this paper are proved to be equivalent. The latter system employs the modified modular equivalence used by Braendli-Bayne. A sequence of Euler tours related on Schick's cycles of diagonals is also presented.

math.NT↗
arXiv2020

The Tribonacci and ABC Representations of Numbers are Equivalent

It is shown that the unique representation of positive integers in terms of tribonacci numbers and the unique representation in terms of iterated A, B and C sequences defined from the tribonacci word are equivalent. Two auxiliary representations are introduced to prove this bijection. It will be established directly on a node and edge labeled tribonacci tree as well as formally. A systematic study of the A, B and C sequences in terms of the tribonacci word is also presented.

math.NT↗
arXiv2017

On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles

The exponential generating function of ordinary generating functions of diagonal sequences of general Sheffer triangles is computed by an application of Lagrange's theorem. For the special Jabotinsky type this is already known. An analogous computation for general Riordan number triangles leads to a formula for the logarithmic generating function of the ordinary generating functions of the product of the entries of the diagonal sequence of Pascal's triangle and those of the {Riordan triangle. For some examples these ordinary generating functions yield in both cases coefficient triangles of certain numerator polynomials.

math.NT↗
arXiv2017

On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers

For finite sums of non-negative powers of arithmetic progressions the generating functions (ordinary and exponential ones) for given powers are computed. This leads to a two parameter generalization of Stirling and Eulerian numbers. A direct generalization of Bernoulli numbers and their polynomials follows. On the way to find the Faulhaber formula for these sums of powers in terms of generalized Bernoulli polynomials one is led to a one parameter generalization of Bernoulli numbers and their polynomials. Generalized Lah numbers are also considered.

math.NT↗
arXiv2017

The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon

The normal field extension Q(rho(n)), with the algebraic number rho(n) = 2 cos(pi/n) for natural n, is related to ratios of the lengths between diagonals and the side of a regular n-gon. This has been considered in a paper by P. Steinbach. These ratios are given by Chebyshev S-polynomials. The product formula for these ratios was found by Steinbach, and is re-derived here from a known formula for the product of Chebyshev S-polynomials. It is shown that it follows also from the S-polynomial recurrence and certain rules following from the trigonometric nature of the argument x = rho(n). The minimal integer polynomial C(n,x) for rho(n) is presented, and its simple zeros are expressed in the power-basis of Q(rho(n)). Also the positive zeros of the Chebyshev polynomial S(k-1,rho(n)) are rewritten in this basis. The number of positive and negative zeros of C(n,x) is determined. The coefficient C(n,0) is computed for special classes of n values. Theorems on C(n,x) in terms of monic integer Chebyshev polynomials of the first kind (called here t-hat) are given. These polynomials can be factorized in terms of the minimal C-polynomials. A conjecture on the discriminant of these polynomials is made. In order to determine the cycle structure of the (Abelian) Galois group a novel modular multiplication, called Modd n is introduced. On the reduced odd residue system Modd n this furnishes a group which is isomorphic to this Galois group.

math.GR↗
arXiv2014

Notes on Some Geometric and Algebraic Problems Solved by Origami

Details for known solutions of some geometric and algebraic problems with the help of origami are presented: two theorems of Haga, the general cubic equation, especially the heptagon equation, doubling the cube as well as the trisection of angles $α$, $π- α$ and $π+ α$.

math.MG↗
arXiv2014

On Collatz' Words, Sequences and Trees

Motivated by a recent work of Trümper we consider the general Collatz word (up-down pattern) and the sequences following this pattern. The recurrences for the first and last sequence entries are given, obtained from repeated application of the general solution of a binary linear inhomogeneous Diophantine equation. These recurrences are then solved. The Collatz tree is also discussed.

math.NT↗
arXiv1997

On sums of powers of zeros of polynomials

Due to Girard's (sometimes called Waring's) formula the sum of the $r-$th power of the zeros of every one variable polynomial of degree $N$, $P_{N}(x)$, can be given explicitly in terms of the coefficients of the monic ${\tilde P}_{N}(x)$ polynomial. This formula is closely related to a known \par \noindent $N-1$ variable generalization of Chebyshev's polynomials of the first kind, $T_{r}^{(N-1)}$. The generating function of these power sums (or moments) is known to involve the logarithmic derivative of the considered polynomial. This entails a simple formula for the Stieltjes transform of the distribution of zeros. Perron-Stieltjes inversion can be used to find this distribution, {\it e.g.} for $N\to \infty$.\par Classical orthogonal polynomials are taken as examples. The results for ordinary Chebyshev $T_{N}(x)$ and $U_{N}(x)$ polynomials are presented in detail. This will correct a statement about power sums of zeros of Chebyshev's $T-$polynomials found in the literature. For the various cases (Jacobi, Laguerre, Hermite) these moment generating functions provide solutions to certain Riccati equations.

math.CA↗
arXiv1996

The Measure of the Orthogonal Polynomials Related to Fibonacci Chains: The Periodic Case

The spectral measure for the two families of orthogonal polynomial systems related to periodic chains with N-particle elementary unit and nearest neighbour harmonic interaction is computed using two different methods. The interest is in the orthogonal polynomials related to Fibonacci chains in the periodic approximation. The relation of the measure to appropriately defined Green's functions is established.

cond-mat↗
arXiv1994

Fibonacci Chain Polynomials: Identities from Self-Similarity

Fibonacci chains are special diatomic, harmonic chains with uniform nearest neighbour interaction and two kinds of atoms (mass-ratio $r$) arranged according to the self-similar binary Fibonacci sequence $ABAABABA...$, which is obtained by repeated substitution of $A \to AB$ and $B \to A$. The implications of the self-similarity of this sequence for the associated orthogonal polynomial system which govern these Fibonacci chains with fixed mass-ratio $r$ are studied.

cond-mat↗