Related papers: On Infinite Product Identities Generating Solution…
We count the number of irreducible polynomials in several variables of a given degree over a finite field. The results are expressed in terms of a generating series, an exact formula and an asymptotic approximation. We also consider the…
A simple method called symbolic representation for piecewise linear functions on the real line is introduced and used to compute the numbers of periodic points of all periods for some such functions. Since, for every positive integer m, the…
Based on an interesting identity of Bat{\i}r we derive new identities for double sums involving famous number sequences. We also prove some double sum identities for binomial transform pairs.
In this paper, we identify a class of solutions to multidimensional difference equation with rational generating function.
We give the generating function for the index of integer lattice points, relative to a finite order ideal. The index is an important concept in the theory of border bases, an alternative to Gr\"obner bases. Equivalently, we explicitly solve…
We present several identities with a form of polynomials or rational functions that involve Pochhammer and q-Pochhammer symbols and q-binomials (i.e. Gauss polynomials). All these identities were obtained by some analytical methods based on…
In this paper, we clarified the relationship between continued fractions, determinants, and identities, making it easier to apply these methods systematically in other settings. In particular, we studied finite continued fractions from the…
The celebrated quintuple product identity follows surprisingly from an almost-trivial algebraic identity, which is the limiting case of the terminating q-Dixon formula.
Generating identity-preserving faces aims to generate various face images keeping the same identity given a target face image. Although considerable generative models have been developed in recent years, it is still challenging to…
We consider sequences of integers defined by a system of linear inequalities with integer coefficients. We show that when the constraints are strong enough to guarantee that all the entries are nonnegative, the generating function for the…
A general method for solving linear differential equations of arbitrary order, is used to arrive at new representations for the solutions of the known differential equations, both without and with a source term. A new quasi-solvable…
We prove and conjecture some new symmetric function identities, which equate the generating series of 1. Plane partitions, subject to certain restrictions and weightings, and 2. Alternating sign matrices, subject to certain symmetry…
The main result of this paper is two infinity classes of series-product identities which is based on classical Gauss identity and two different interpretations of character formula for irreducible highest weight modules of affine Lie…
We prove a new polynomial refinement of the Capparelli's identities. Using a special case of Bailey's lemma we prove many infinite families of sum-product identities that root from our finite analogues of Capparelli's identities. We also…
We consider nested sums involving the Pochhammer symbol at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi,$ $\log(2)$ or zeta values. In order to perform these simplifications, we view the series as…
We show that the Identity Problem is decidable in polynomial time for finitely generated sub-semigroups of the group $\mathsf{UT}(4, \mathbb{Z})$ of $4 \times 4$ unitriangular integer matrices. As a byproduct of our proof, we also show the…
We provide explicit identity bases for finite cyclic semigroups.
The Tits alternative states that a finitely generated matrix group either contains a nonabelian free subgroup $F_2$, or it is virtually solvable. This paper considers two decision problems in virtually solvable matrix groups: the Identity…
We propose a recursive algorithm for identifying all finite sequences of positive integers whose product equals their sum. Our method uses solutions of strictly shorter length that are iteratively extended in pursuit of a valid solution.…
Spectral decomposition of dynamical equations using curl-eigenfunctions has been extensively used in fluid and plasma dynamics problems using their orthogonality and completeness properties for both linear and non-linear cases. Coefficients…