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We consider series expansions in bases of classical orthogonal polynomials. When such a series solves a linear differential equation with polynomial coefficients, its coefficients satisfy a linear recurrence equation. We interpret this…
The paper considers urn schemes in which several urns can be involved. Simplified formulas are proposed that allow direct calculation of probabilities without the use of elements combinatorics.
I explain a direct approach to differentiation and integration. Instead of relying on the general notions of real numbers, limits and continuity, we treat functions as the primary objects of our theory, and view differentiation as division…
To make a joint decision, agents (or voters) are often required to provide their preferences as linear orders. To determine a winner, the given linear orders can be aggregated according to a voting protocol. However, in realistic settings,…
We consider the class of counting problems,i.e. functions in $\#$P, which are self reducible, and have easy decision version, i.e. for every input it is easy to decide if the value of the function $f(x)$ is zero. For example,…
Polynomial multiplication is a fundamental problem in symbolic computation. There are efficient methods for the multiplication of two univariate polynomials. However, there is rarely efficiently nontrivial method for the multiplication of…
We derive analytic formulas to reconstruct particle-averaged quantities from experimental results that suffer from the efficiency loss of particle measurements. These formulas are derived under the assumption that the probabilities of…
We consider the class $\mathcal{E}_t(Y)$ of Appell polynomials whose generating function is given by means of a real power $t$ of the moment generating function of a certain random variable $Y$. For such polynomials, we obtain explicit…
We develop a toolbox for the error analysis of linear recurrences with constant or polynomial coefficients, based on generating series, Cauchy's method of majorants, and simple results from analytic combinatorics. We illustrate the power of…
Incorporation of expert information in inference or decision settings is often important, especially in cases where data are unavailable, costly or unreliable. One approach is to elicit prior quantiles from an expert and then to fit these…
Power series in which the summand satisfies a linear recurrence relation with polynomial coefficients are shown to be the solution of a linear differential or algebraic equation. Solving the associated differential or algebraic equation…
In this paper we use computational method based on operational point of view to prove a new generating function of exponential polynomials. We give its applications involving geometric polynomials, Bernoulli and Euler numbers.
The analytical approximations for the moderating neutrons flux density like Fermi spectra, widely used in reactor physics, involve the probability function for moderating neutron to avoid the resonant absorption obtained using some…
The compound decision problem for a vector of independent Poisson random variables with possibly different means has half a century old solution. However, it appears that the classical solution needs smoothing adjustment even when there are…
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…
By using the Poisson formula for resultants and the variants of chip-firing game on graphs, we provide a combinatorial method for computing a class of of resultants, i.e. the characteristic polynomials of the adjacency tensors of starlike…
This document presents a combinatorial framework for analyzing assembly systems using generating functions. We explore the theory through concrete examples, such as linear polymers, and develop recursive equations to characterize valid…
Mutually uncorrelated random discrete events, manifesting a common basic process, are examined often in terms of their occurrence rate as a function of one or more of their distinguishing attributes, such as measurements of photon spectrum…
We describe a general strategy for sampling configurations from a given distribution, NOT based on the standard Metropolis (Markov chain) strategy. It uses the fact that nontrivial problems in statistical physics are high dimensional and…
Voting is a very general method of preference aggregation. A voting rule takes as input every voter's vote (typically, a ranking of the alternatives), and produces as output either just the winning alternative or a ranking of the…