Related papers: Generating Function and a Rodrigues Formula for th…
We consider the tiling generating functions of semi-hexagons and quartered hexagons with dents on their sides. In general, there are no simple product formulas for these generating functions. However, we show that the modification in the…
A polynomial threshold function (PTF) $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is a function of the form $f(x) = \mathsf{sign}(p(x))$ where $p$ is a polynomial of degree at most $d$. PTFs are a classical and well-studied complexity class…
We derive the P-finite recurrences for classes of sequences with ordinary generating function containing roots of polynomials. The focus is on establishing the D-finite differential equations such that the familiar steps of reducing their…
The main purpose of this paper is to construct not only generating functions of the new approach Genocchi type numbers and polynomials but also interpolation function of these numbers and polynomials which are related to a, b, c arbitrary…
The multiplicative (or geometric) calculus is a non-Newtonian calculus derived from an arithmetic in which the operations of addition/subtraction/multiplication are replaced by multiplication/division/exponentiation. A major difference…
A tropical polynomial in nr variables divided into blocks of r variables each, is r-symmetric, if it is invariant under the action of Sn that permutes the blocks. For r=1 we call these tropical polynomials symmetric. We can define…
We relate a one-parametric generating function for the squares of Legendre polynomials to an arithmetic hypergeometric series whose parametrisation by a level 7 modular function was recently given by Shaun Cooper. By using this modular…
We present a number of identities involving standard and associated Laguerre polynomials. They include double-, and triple-lacunary, ordinary and exponential generating functions of certain classes of Laguerre polynomials.
In this paper we deal with a family of non--standard polynomials orthogonal with respect to an inner product involving differences. This type of inner product is the so--called $\Delta$--Sobolev inner product. Concretely, we consider the…
In this article, the 2-iterated Sheffer polynomials are introduced by means of generating function and operational representation. Using the theory of Riordan arrays and relations between the Sheffer sequences and Riordan arrays, a…
A simple way to generate a Boolean function is to take the sign of a real polynomial in $n$ variables. Such Boolean functions are called polynomial threshold functions. How many low-degree polynomial threshold functions are there? The…
We use a path integral formalism to derive the semiclassical series for the partition function of a particle in D dimensions. We analyze in particular the case of attractive central potentials, obtaining explicit expressions for the…
The main result of the article says that the formal power series equal to the ratio of two neighboring Chebyshev polynomials, after some renormalization, approximates the generating function of the Catalan numbers. We present a proof of…
Our present investigation is motivated essentially by the fact that, in Geometric Function Theory, one can find many interesting and fruitful usages of a wide variety of special functions and special polynomials. The main purpose of this…
We study a generating function for the sum over fatgraphs with specified valences of vertices and faces, inversely weighted by the order of their symmetry group. A compact expression is found for general (i.e. non necessarily connected)…
This paper puts forward a new generalized polynomial dimensional decomposition (PDD), referred to as GPDD, comprising hierarchically ordered measure-consistent multivariate orthogonal polynomials in dependent random variables. Unlike the…
The aim of this paper is to construct generating functions for new families of special polynomials including the Appel polynomials, the Hermite-Kamp\`e de F\`eriet polynomials, the Milne-Thomson type polynomials, parametric kinds of Apostol…
We present a new semiclassical technique which relies on replacing complicated classical manifold structure with simpler manifolds, which are then evaluated by the usual semiclassical rules. Under circumstances where the original manifold…
Explicit expressions for restricted partition function $W(s,{\bf d}^m)$ and its quasiperiodic components $W_j(s,{\bf d}^m)$ (called {\em Sylvester waves}) for a set of positive integers ${\bf d}^m = \{d_1, d_2, ..., d_m\}$ are derived. The…
We define an overpartition analogue of Gaussian polynomials (also known as $q$-binomial coefficients) as a generating function for the number of overpartitions fitting inside the $M \times N$ rectangle. We call these new polynomials over…