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Loeb showed that a natural extension of the usual binomial coefficient to negative (integer) entries continues to satisfy many of the fundamental properties. In particular, he gave a uniform binomial theorem as well as a combinatorial interpretation in terms of choosing subsets of sets with a negative number of elements. We show that all of this can be extended to the case of Gaussian binomial coefficients. Moreover, we demonstrate that several of the well-known arithmetic properties of binomial coefficients also hold in the case of negative entries. In particular, we show that Lucas' theorem on binomial coefficients modulo p not only extends naturally to the case of negative entries, but even to the Gaussian case. [ABSTRACT FROM AUTHOR]

One is the well-known Vandermonde identity HT ht . At this point, this very clever student invokes Newton's general binomial theorem to produce yet another power series from which emerges, but only after another page of ingenious simplifications, the elegant formula the professor was seeking: HT ht . However, after a few hours (and a half page of calculations), she managed to reduce this sum into a standard form for a general hypergeometric series involving a term HT ht . [Extracted from the article]

In the present study, a simple and efficient expression for the accurate and quick calculation of the specific heat capacity of semiconductor compounds is presented on the basis of n-dimensional Debye functions using binomial coefficients. As will be seen, the present formulation yields compact, closed-form expressions which enable the straightforward calculation of the heat capacity of solids for arbitrary temperature values. As an example of the application, the calculation is performed for the specific heat capacity of semiconductor compounds. The calculated results have been compared with those reported in the literature and are found to be in good agreement with those of other studies. [ABSTRACT FROM AUTHOR]

The article explores the applications of the approach used in the study of the local geometry of Carnot-Carathéodory spaces (and Carnot manifolds) based on the convergence of scaled vector fields and local approximation theorem. It discusses the result of the study which is considered as a description of the coefficients of nilpotentized fields in relation to the basis of a vector field. It mentions that this result served as a key point in obtaining proof for Gromov's theorem on the convergence of scaled vector fields to nilpotentized ones and is used as a proof for the existence of a local approximation theorem for Carnot.

We consider the problem of computing the coefficients of the reliability polynomial (RP) for a random graph with reliable vertices and unreliable edges. To speed up the computation, we use the meaning of RP coefficients in one of its representations and prove vector relations over vectors of binomial coefficients. [ABSTRACT FROM AUTHOR]

k-Adic formulations (for groups of objects of size k) of a variety of 2-adic similarity coefficients (for pairs of objects) for binary (presence/absence) data are presented. The formulations are not functions of 2-adic similarity coefficients. Instead, the main objective of the the paper is to present k-adic formulations that reflect certain basic characteristics of, and have a similar interpretation as, their 2-adic versions. Two major classes are distinguished. The first class is referred to as Bennani-Heiser similarity coefficients, which contains all coefficients that can be defined using just the matches, the number of attributes that are present and that are absent in k objects, and the total number of attributes. The coefficients in the second class can be formulated as functions of Dice’s association indices. [ABSTRACT FROM AUTHOR]

Abstract  Recursion and analytical relations for the evaluation of integer and noninteger n-dimensional Debye functions have been derived. Using the binomial expansion theorem, these functions are expressed through the binomial coefficients and familiar incomplete gamma functions. This simplification and the use of the memory of the computer for calculation of binomial coefficients may extend the limits to large arguments for users and result in speedier calculation, should such limits be required in practice. Comparison of numerical results shows that analytical solutions are accurate almost from the beginning of the calculation time. The series expansion relations obtained is sufficiently accurate over the entire range of parameters. The convergence rate of the series is estimated and discussed. [ABSTRACT FROM AUTHOR]

The article presents information on the observable Linear Nonstationary systems of total Differential Equations. The article describes the class k of a system (the order of differentiability of the output vector) and derive an effective (coefficient) criterion for a system to belong to a given class.

The article presents information on the stability of the stability of a Class of Finite-Difference Equations with slowly varying Coefficients. Linear equations with constant coefficients are one of the few classes (and probably the only one) in which the stability problem can be solved completely. The article analyzes the stability of finite-difference equations whose coefficients vary slowly in the course of time. If all equations with frozen coefficients are stable and the coefficients vary sufficiently slowly, then the original equation is also stable.