Related papers: Number of ordered factorizations and recursive div…
This short note provides a sharper upper bound of a well known inequality for the sum of divisors function. This is a problem in pure mathematics related to the distribution of prime numbers. Furthermore, the technique is completely…
Using a specific form of the triple product identity, polygonal number identities are stated. Further number identities are examined that can be considered identities related to modular sets of numbers. The identities can be used to give…
We define two natural classes of functions, called 2-open and 2-closed, that are closest to open and closed functions. We show that they have the following property: there are $X_i \subset X$ $ (i=1,2,...$) such that $f|X_i$ are open or…
This paper introduces a new generalized superfactorial function (referable to as $n^{th}$- degree superfactorial: $sf^{(n)}(x)$) and a generalized hyperfactorial function (referable to as $n^{th}$- degree hyperfactorial: $H^{(n)}(x)$), and…
Three different characterizations of one-component bounded analytic functions are provided. The first one is related to the the inner-outer factorization, the second one is in terms of the size of the reproducing kernels in the…
A new mathematical notation is proposed for the iteration of functions. It facilitates the application of the iteration of functions in mathematical and logical expressions, definitions of sets, and formulations of algorithms. Illustrations…
The details for the construction of an explicit formula for the divisors function d(n) = #{d | n} are formalized in this article. This formula facilitates a unified approach to the investigation of the error terms of the divisor problem and…
The number of parts in the partitions (resp. distinct partitions) of $n$ with parts from a set were considered. Its generating functions were obtained. Consequently, we derive several recurrence identities for the following functions: the…
We introduce the notion of the generalized-analytical function of the poly-number variable, which is a non-trivial generalization of the notion of analytical function of the complex variable and, therefore, may turn out to be fundamental in…
It is well known that the composition of a D-finite function with an algebraic function is again D-finite. We give the first estimates for the orders and the degrees of annihilating operators for the compositions. We find that the analysis…
A vector partition function is the number of ways to write a vector as a non-negative integer-coefficient sum of the elements of a finite set of vectors $\Delta$. We present a new algorithm for computing closed-form formulas for vector…
Two factorizations of a permutation into products of cycles are equivalent if one can be obtained from the other by repeatedly interchanging adjacent disjoint factors. This paper studies the enumeration of equivalence classes under this…
Let L be a bounded distributive lattice. We give several characterizations of those L^n --> L mappings that are polynomial functions, i.e., functions which can be obtained from projections and constant functions using binary joins and…
The prime-counting function $\pi(x)$ which returns the number of primes smaller or equal to a given number is a topic of interest in number theory. An algorithm based on a cyclic group isomorphic to $Z/nZ$, the so-called $Z$-functions, was…
We lay out an infinity categorical interpretation of reconstruction theorems which are germane to the symmetric monoidal perspective of noncommutative algebraic geometry, present sufficient conditions which allow for the factorization of…
The arithmetic function of two variables is defined. Some properties of the function are given along with the formula that is an analog of the so-called Mobius' inversion formula. A heuristic statement is suggested.
It is well-known that differentiation of hypergeometric function multiplied by a certain power function yields another hypergeometric function with a different set of parameters. Such differentiation identities for hypergeometric functions…
In this work, we study vector-valued functional equations with multiple recursive terms that arise naturally when we are dealing with vector-valued multiplicative Lindley-type recursions. We provide a detailed framework for the solution of…
Fractional calculus is the calculus of differentiation and integration of non-integer orders. In a recently paper (Annals of Physics 323 (2008) 2756-2778), the Fundamental Theorem of Fractional Calculus is highlighted. Based on this…
This is an overview of higher structural constructions in physics. The main motivations of our current attempt are as follows: (i) to provide a brief introduction to derived algebraic geometry, (ii) to understand how derived objects…