Robert F. Tichy

Mellin transforms and asymptotics: digital sums

Abstract Arithmetic functions related to number representation systems exhibit various periodicity phenomena. For instance, a well-known theorem of Delange expresses the total number of ones in the binary representations of the first n integers in terms of a periodic fractal function. We show that such periodicity phenomena can be analyzed rather systematically using classical tools from analytic number theory, namely the Mellin—Perron formulae. This approach yields naturally the Fourier series involved in the expansions of a variety of digital sums related to number representation systems.

Contributions to digit expansions with respect to linear recurrences

Abstract Extensions and improvements of a recent paper, “On Digit Expansions with Respect to Linear Recurrences” by A. Petho and R. F. Tichy (J. Number Theory 33 (1989), 243–256) are established. Furthermore distribution properties mod 1 of the sequence (xsG(n)) are investigated, where sG(n) denotes the sum-of-digits function with respect to the linear recurrence G.

On the distribution of dividend payments and the discounted penalty function in a risk model with linear dividend barrier

In the framework of classical risk theory we investigate a model that allows for dividend payments according to a time-dependent linear barrier strategy. Partial integro-differential equations for Gerber and Shius discounted penalty function and for the moment generating function of the discounted sum of dividend payments are derived, which generalizes several recent results. Explicit expressions for the nth moment of the discounted sum of dividend payments and for the joint Laplace transform of the time to ruin and the surplus prior to ruin are derived for exponentially distributed claim amounts.

Generalized Zeckendorf expansions

Abstract Digital expansions with respect to linear recurring sequences are considered. Using a general result due to Frougny and Solomyak, finite representations are investigated and a quantitative refinement is established.

Diophantine Equations and Bernoulli Polynomials

Given m, n ≥ 2, we prove that, for sufficiently large y, the sum 1n +···+ yn is not a product of m consecutive integers. We also prove that for m ≠ n we have 1m +···+ xm ≠ 1n +···+ yn, provided x, y are sufficiently large. Among other auxiliary facts, we show that Bernoulli polynomials of odd index are indecomposable, and those of even index are ‘almost’ indecomposable, a result of independent interest.

Deviations from uniformity in random strings

SummaryWe show that almost all binary strings of length n contain all blocks of size (1-ε)log2n a close to uniform number of times. From this, we derive tight bounds on the discrepancy of random infinite strings. Our results are obtained through explicit generating function expressions and contour integration estimates.

An extension of Minkowski's singular function

Abstract Generalizing the so-called Minkowski function, a family of continuous, strictly increasing singular functions on the unit interval is introduced. We investigate the Hausdorff dimension of the set of points where the singular functions have nonvanishing derivatives.

Combinatorial and Arithmetical Properties of Linear Numeration Systems

Dedicated to the memory of Paul ErdősWe extend a result of J. Alexander and D. Zagier on the Garsia entropy of the Erdős measure. Our investigation heavily relies on methods from combinatorics on words. Furthermore, we introduce a new singular measure related to the Farey tree.

ON THE MOMENTS OF THE SUM-OF-DIGITS FUNCTION

In a recent paper in The Fibonacci Quarterly R.E. Kennedy and C.N. Cooper [12] dealt with the second moment of the sum-of-digits function in the decimal number system, explicitly stating the question for higher moments as an open problem. We consider this question in the sequel.

Graphs, partitions and Fibonacci numbers

The Fibonacci number of a graph is the number of independent vertex subsets. In this paper, we investigate trees with large Fibonacci number. In particular, we show that all trees with n edges and Fibonacci number >2^n^-^1+5 have diameter = ~. This is proved by using a natural correspondence between partitions of integers and star-like trees.