相关论文: Partial-Fraction Decompositions and Harmonic Numbe…
MacMahon showed that the generating function for partitions into at most $k$ parts can be decomposed into a partial fractions-type sum indexed by the partitions of $k$. In this present work, a generalization of MacMahon's result is given,…
We produce algorithms to detect whether a complex affine variety computed and presented numerically by the machinery of numerical algebraic geometry corresponds to an associated component of a polynomial ideal.
We consider binomial and inverse binomial sums at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi$ or $\log(2)$. In order to perform these simplifications, we view the series as specializations of…
Utilizing spectral residues of parameterized, recursively defined sequences, we develop a general method for generating identities of composition sums. Specific results are obtained by focusing on coefficient sequences of solutions of first…
This paper introduces the bicomplex Prabhakar derivative, extending fractional calculus to four-dimensional bicomplex spaces. Using the generalized kernel involving bicomplex Prabhakar function, we construct the bicomplex Prabhakar…
By applying the derivative operators to Chu-Vandermonde convolution, several general harmonic number identities are established.
We introduce partial differential encodings of Boolean functions as a way of measuring the complexity of Boolean functions. These encodings enable us to derive from group actions non-trivial bounds on the Chow-Rank of polynomials used to…
A mathematical method for constructing fractal curves and surfaces, termed the $p\lambda n$ fractal decomposition, is presented. It allows any function to be split into a finite set of fractal discontinuous functions whose sum is equal…
A generalization of the Chu-Vandermonde convolution is presented and proved with the integral representation method. This identity can be transformed into another identity, which has as special cases two known identities. Another identity…
Continued fractions have been long studied due to their strong properties, such as rational approximation. In this extent, their arithmetic over real numbers has represented an intriguing problem throughout the years. In this paper, we…
In this paper we introduce and study three classes of fractional periodic processes. An application to ring polymers is investigated. We obtain a closed analytic expressions for the form factors, the Debye functions and their asymptotic…
We evaluate the moments of some functions composed with the fractional part of $1/x$. We name them fractional moments. In particular, we obtain expressions for the fractional moments of some trigonometric functions, the Bernoulli…
This paper addresses the approximation of fractional harmonic maps. Besides a unit-length constraint, one has to tackle the difficulty of nonlocality. We establish weak compactness results for critical points of the fractional Dirichlet…
In the present paper, we investigate some interesting properties including several special polynomials arising from Caputo-fractional derivative. From our investigation, we derive a lot of interesting identities of several special…
The partition functions $P(n,m,p)$, the number of integer partitions of $n$ into exactly $m$ parts with each part at most $p$, and $Q(n,m,p)$, the number of integer partitons of $n$ into exactly $m$ distinct parts with each part at most…
This paper studies the problem of decomposing a low-rank matrix into a factor with binary entries, either from $\{\pm 1\}$ or from $\{0,1\}$, and an unconstrained factor. The research answers fundamental questions about the existence and…
We investigate compositions of a positive integer with a fixed number of parts, when there are several types of each natural number. These compositions produce new relationships among binomial coefficients, Catalan numbers, and numbers of…
By establishing an interesting connection between ordinary Bell polynomials and rational convolution powers, some composition and inverse relations of Bell polynomials as well as explicit expressions for convolution roots of sequences are…
Consider the representation of a rational number as a continued fraction, associated with "odd" Euclidean algorithm. In this paper we prove certain properties for the limit distribution function for sequences of rationals with bounded sum…
A symbolic method is used to establish some properties of the Bernoulli-Barnes polynomials.