Related papers: Various Arithmetic Functions and their Application…
By introducing a kind of special functions namely exponent-like function, cosine-like function and sine-like function, we obtain explicitly the basic structures of solutions of initial value problem at the original point for this kind of…
Some identities are presented that generalize the formula x^3 = 3x floor(x floor(x)) - 3 floor(x) floor(x floor(x)) + floor(x)^3 + 3 frac(x) frac(x floor(x)) + frac(x)^3 to a representation of the product x_0x_1 ... x_{n-1}.
Convex solutions $A,B,I,J$ of four Abel equations are numerically studied. We do not know exact formulas for any of these functions, but conjecture that $A,B$ and $I,J$ are closely related. [Corrigendum at end.]
We use the rationality of the generalized $h^{th}$ convergent functions, $Conv_h(\alpha, R; z)$, to the infinite J-fraction expansions enumerating the generalized factorial product sequences, $p_n(\alpha, R) =…
We introduce new refinements of the Bell, factorial, and unsigned Stirling numbers of the first and second kind that unite the derangement, involution, associated factorial, associated Bell, incomplete Stirling, restricted factorial,…
Several conjectural continued fractions found with the help of various algorithms are published in this paper.
We briefly introduce several problems: (1) a generalization of the convex fair partition conjecture, (2) on non-trivial invariants among polyhedrons that can be formed from the same set of face polygons, (3) two questions on assembling…
An integer sequence that is defined by initial values and a linear recurrence with constant integer coefficients, can be represented by the difference of two arithmetic terms containing exponentiation. All constants occuring in the term are…
Challenging the standard notion of totality in computable functions, one has that, given any sufficiently expressive formal axiomatic system, there are total functions that, although computable and "intuitively" understood as being total,…
This paper describes infinite sets of polynomial equations in infinitely many variables with the property that the existence of a solution or even an approximate solution for every finite subset of the equations implies the existence of a…
In studying the complexity of iterative processes it is usually assumed that the arithmetic operations of addition, multiplication, and division can be performed in certain constant times. This assumption is invalid if the precision…
In this paper we define infinite-dimensional algebra and its representation, whose basis is naturally identified with semi-infinite configurations of the square ladder model. We also extrapolate the ideas for the cyclic 3-leg triangular…
We introduce and consider a certain probability question involving elementary number theory and the likelihood that a fixed prime will appear in a certain recursively defined factorization of an integer. We derive several convergent…
The theory of the functional sequences and series is presented; uniformly convergent, convergent in the sense of a mean square and weakly convergent sequences and series are considered. Sequential approach to constructing generalized…
We deal with various Diophantine equations involving the Euler totient function and various sequences of numbers, including factorials, powers, and Fibonacci sequences.
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.…
Some facts about 4-spinors listed and discussed. None, well perhaps some, of the work is original. However, locating formulas in other places has proved a time-consuming process in which one must always worry that the formulas found in any…
This is a survey of results on definability and undefinability in models of arithmetic. The goal is to present a stark difference between undefinability results in the standard model and much stronger versions about expansions of…
What is computable with limited resources? How can we verify the correctness of computations? How to measure computational power with precision? Despite the immense scientific and engineering progress in computing, we still have only…
We have discovered three non-power infinite series representations for Bessel functions of the first kind of integer orders and real arguments. These series contain only elementary functions and are remarkably simple. Each series was…