Related papers: On One Function Defined on the Cartesian Product a…
A solution is proposed for the problem of composition of ordinary generating functions. A new class of functions that provides a composition of ordinary generating functions is introduced; main theorems are presented; compositae are written…
Adding a column of numbers produces "carries" along the way. We show that random digits produce a pattern of carries with a neat probabilistic description: the carries form a one-dependent determinantal point process. This makes it easy to…
The paper introduces the notion of the size of countable sets that preserves the Part-Whole Principle and generalizes the notion of the cardinality of finite sets. The sizes of natural numbers, integers, rational numbers, and all their…
In this technical report, certain interesting classification of arithmetical functions is proposed. The notion of additively decomposable and multiplicatively decomposable arithmetical functions is proposed. The concepts of arithmetical…
In this paper, we present a new approach to the convolved Fibonacci numbers arising from the generating function of them and give some new and explicit identities for the convolved Fibonacci numbers.
In this paper, we will define general Eulerian numbers and Eulerian polynomials based on general arithmetic progressions. Under the new definitions, we have been successful in extending several well-known properties of traditional Eulerian…
Counting functions are constructed for sums of integers raised to a fixed positive rational power. That is, given values formed by $u_1^{j/k} + u_2^{j/k} + ... + u_l^{j/k}$, $u_i \in \mathbb{Z}^+$, the number of values less than or equal to…
In this paper, we introduce two differential equations arising from the generating function of the Catalan numbers which are `inverses' to each other in some sense. From these differential equations, we obtain some new and explicit…
The generating function and an explicit expression is derived for the (colored) Motzkin numbers of higher rank introduced recently. Considering the special case of rank one yields the corresponding results for the conventional colored…
Starting from a small number of well-motivated axioms, we derive a unique definition of sums with a noninteger number of addends. These "fractional sums" have properties that generalize well-known classical sum identities in a natural way.…
We introduce a non-wellfounded proof system for intuitionistic logic extended with inductive and co-inductive definitions, based on a syntax in which fixpoint formulas are annotated with explicit variables for ordinals. We explore the…
A new number system, the set of the non-Dedekindian numbers, is introduced and characterized axiomatically. It is then proved that any hypercontinous hyperreal number system is strictly included in the set of the Non-Dedekindian Numbers.…
Functionals are an important research subject in Mathematics and Computer Science as well as a challenge in Information Technologies where the current programming paradigm states that only symbolic computations are possible on higher order…
Goodstein's principle is arguably the first purely number-theoretic statement known to be independent of Peano arithmetic. It involves sequences of natural numbers which at first appear to grow very quickly, but eventually decrease to zero.…
A set of functions is defined which is indexed by a positive integer $n$ and partitions of integers. The case $n=1$ reproduces the standard Schur polynomials. These functions are seen to arise naturally as a determinant of an action on the…
In this paper, we introduce a new type degenerate Simsek numbers and their generating function, which are different from degenerate Simsek number studied so far. We establish the explicit formula, recurrence relation and other identities…
In the article 'Ordinal Logics and the Characterizations of the Informal Concept of Proof', Georg Kreisel poses the problem of assigning unique notations to recursive ordinals, and additionally suggests that the methods which are developed…
Sequences are often conveniently encoded in the form of a generating function depending on a formal variable. This note presents two observations that allow one to draw conclusions about the generated sequence from the generating function.…
Let $g \geq 2$. A real number is said to be g-normal if its base g expansion contains every finite sequence of digits with the expected limiting frequency. Let \phi denote Euler's totient function, let \sigma be the sum-of-divisors…
Finite hamiltonian groups are counted. The sequence of numbers of all groups of order $n$ all whose subgroups are normal and the sequence of numbers of all groups of order less or equal to $n$ all whose subgroups are normal are presented.