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In this paper, we established an effective formula for Dnestrovskii functions based on the binomial expansion theorem which frequently appears in a weakly relativistic loss cone distribution, relativistic, magnetized, and thermal plasmas. The obtained formulas are expressed in terms of binomial coefficients and incomplete Gamma functions. Therefore, the proposed approach is a fully useful method to evaluate Dnestrovskii functions and related physical properties of plasmas. Calculation results of the Dnestrovskii functions are compared with other numerical and theoretical models and indicated acceptable agreement. The results of the calculations demonstrate that the derived formula is valid for a wide range of parameters. [ABSTRACT FROM AUTHOR]

We evaluate, in closed forms, twelve infinite series that contain a free variable x and binomial coefficients ( 3 n + ϵ n) in the denominators. Several summation formulae resembling Gosper's sum are derived as consequences. [ABSTRACT FROM AUTHOR]

Motivated by a problem proposed by Seiffert a quarter of century ago, we explicitly evaluate binomial sums with Pell and Lucas polynomials as weight functions. Their special cases result in several interesting identities concerning Fibonacci and Lucas numbers. [ABSTRACT FROM AUTHOR]

We consider the binomial decomposition of ordered weighted averaging (OWA) functions proposed by Calvo and De Baets (1998) in the framework of Choquet integration. Our aim in the paper is to further investigate the equivalence between the two representations of OWA functions involved in the binomial decomposition: the usual canonical representation in terms of the order statistics, and the binomial representation in terms of the binomial OWA functions. We describe and discuss in detail the linear transformations that relate the coefficients of these two equivalent representations: the original expression of the weights in terms of the coefficients of the binomial representation, and its inverse, the expression of those coefficients in terms of the weights. In both cases simple and direct proofs are presented. Moreover, we consider the linear transformations between the two representations in the general linear algebra framework of unconstrained linear combinations of binomial OWA functions. In this perspective we obtain compact matrix expressions for the linear transformations, which also offer new insight on the geometry of the coefficient constraints in the binomial representation of OWA functions. [ABSTRACT FROM AUTHOR]

In order to accurately and flexibly capture the correlation structure between two random coefficients in the binomial AR(1) process, we propose a new class of models with copula. We derive some basic properties of the process. Then we discuss the conditional least squares (CLS) and conditional maximum likelihood (CML) estimators, as well as their asymptotic properties. We also investigate the finite-sample performance of the proposed method in simulation studies. Finally, a real data example is provided to illustrate the model. • A flexible binomial AR(1) (FBAR(1)) model based on copulas. • Basic probabilistic and statistical properties. • The CLS and CML estimators and the related asymptotic properties. • Real data applications in the fields of meteorology and finance. [ABSTRACT FROM AUTHOR]

We study the distribution of values of the Riemann zeta function ζ (s) on vertical lines ℜ s + i R , by using the theory of Hilbert space. We show among other things, that, ζ (s) has a Fourier expansion in the half-plane ℜ s ≥ 1 / 2 and its Fourier coefficients are the binomial transform involving the Stieltjes constants. As an application, we show explicit computation of the Poisson integral associated with the logarithm of ζ (s) − s / (s − 1). Moreover, we discuss our results with respect to the Riemann and Lindelöf hypotheses on the growth of the Fourier coefficients. For a video summary of this paper, please visit https://youtu.be/wI5fIJMeqp4. [ABSTRACT FROM AUTHOR]

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]

Abstract Following recent paper of the author about polynomial expressions with respect to binomial ideals, the current paper is devoted to the non-binomial case. Beside the established facts in the underlying theory, the correctness and termination of the algorithms are addressed. From the collection of examples, the main large-scale non-binomial one is of particular importance. Application of the new algorithms for the cases of moderately large binomial like cyclic 16-roots and cyclic 18-roots is discussed. A combinatorial analysis for complexity of the proposed algorithms is shown to be at least O (log ⁡ (log ⁡ n) n 2 (log ⁡ n) 2). [ABSTRACT FROM AUTHOR]

We consider Pascal's Triangle r mod p to be the entries of Pascal's Triangle that are congruent to r mod p. Such a representation of Pascal's Triangle exhibits fractal-like structures. When the Triangle is mapped to a subset of the unit square, we show that such a set is nonempty and exists as a limit of a sequence of coarse approximations. We then show that for any given prime p , any such sequence converges to the same set, regardless of the residue(s) considered. As an obvious consequence, this allows us to conclude that the fractal (box-counting) dimension of this nonempty, compact representation of Pascal's Triangle r mod p is independent of r. [ABSTRACT FROM AUTHOR]

In a previous paper, we studied an overpartition analogue of Gaussian polynomials as the generating function for overpartitions fitting inside an m × n rectangle. Here, we add one more parameter counting the number of overlined parts, obtaining a two-parameter generalization [ m + n n ] ‾ q , t of Gaussian polynomials, which is also a ( q , t ) -analogue of Delannoy numbers. First we obtain finite versions of classical q -series identities such as the q -binomial theorem and the Lebesgue identity, as well as two-variable generalizations of classical identities involving Gaussian polynomials. Then, by constructing involutions, we obtain an identity involving a finite theta function and prove the ( q , t ) -log concavity of [ m + n n ] ‾ q , t . We particularly emphasize the role of combinatorial proofs and the consequences of our results on Delannoy numbers. We conclude with some conjectures about the unimodality of [ m + n n ] ‾ q , t . [ABSTRACT FROM AUTHOR]

We develop new closed form representations of sums of cubic alternating harmonic numbers and reciprocal binomial squared coefficients. [ABSTRACT FROM AUTHOR]

By using the Rodriguez-Villegas–Mortenson supercongruences, we prove four supercongruences on sums involving binomial coefficients, which were originally conjectured by Sun. We also confirm a related conjecture of Guo on integer-valued polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we consider two particular binomial sums and which are inspired by two series for obtained by Guillera. We consider their divisibility properties and prove that they are divisible by for all integers . These divisibility properties are stronger than those divisibility results found by He, who proved the above two sums are divisible by with the WZ-method. [ABSTRACT FROM AUTHOR]

Sums of products of two Gaussian q-binomial coefficients with a parametric rational weight function are considered. The partial fraction decomposition technique is used to evaluate the sums in closed form. Interesting applications of these results to certain generalized Fibonomial and Lucanomial sums are provided. [ABSTRACT FROM AUTHOR]

In this paper, we confirm several conjectured congruences of Sun concerning the divisibility of binomial sums. For example, with help of a quadratic hypergeometric transformation, we prove that ∑ k = 0 p − 1 ( p − 1 k ) ( 2 k k ) 2 P k 8 k ≡ 0 ( mod p 2 ) for any prime p ≡ 7 ( mod 8 ) , where P k is the k -th Pell number. Further, we also propose three new congruences of the same type. [ABSTRACT FROM AUTHOR]

The most general form of q-binomial inverse formula was given by Carlitz. In this paper, the author first gives a new form of q-binomial inverse formula. The aim of giving this new form is that a q-analogue of combinatorial identity can be derived. With the aid of the result by Carlitz, the author gives a representation for the q-Catalan-Qi numbers Cn(a; b; q). [ABSTRACT FROM AUTHOR]

We give necessary and sufficient conditions for a given polynomial to be a plucking polynomial of a rooted tree. We discuss the fact that different rooted trees can have the same polynomial. [ABSTRACT FROM AUTHOR]

In this paper, we study q-extension of Mittag-Leffler operators. We establish moments of these operators and estimate convergence results with the help of classical modulus of continuity. Also, we give weighted approximation property and A-statistically convergence of the operators L(β)n,q. [ABSTRACT FROM AUTHOR]

Let F be a finite extension of ℚp. We show that every Schwartz function on F, with values in ℚp, is the p-adic uniform limit of a sequence of Schwartz functions, whose Fourier transforms tend uniformly to 0. The proof uses the notion of q-binomial coefficients and some classical identities between these polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we study the nonautonomous characteristics of the W-shaped solitons on constant backgrounds for a variable-coefficient higher-order nonlinear Schrödinger (vc-HNLS) equation. Several types of solitons are obtained in exponentially, linearly and periodically fluctuating decreasing dispersion profiles, respectively. We also show the compression effect of the third-order dispersion (TOD) coefficient on the W-shaped soliton. We further analyze the formation and spatiotemporal characteristics of the Peregrine combs (PCs) when TOD coefficient is of the periodic form. Moreover, the PC can be converted into the Peregrine wall if the modulation amplitude is equal to 1. [ABSTRACT FROM AUTHOR]

A new generalization of Pascal's triangle, the so-called hyperbolic Pascal triangles were introduced in [1]. The mathematical background goes back to the regular mosaics in the hyperbolic plane. The alternating sum of elements in the rows was given in the special case {4;5} of the hyperbolic Pascal triangles. In this article, we determine the alternating sum generally in the hyperbolic Pascal triangle corresponding to {4;q} with q ≥ 5. [ABSTRACT FROM AUTHOR]

Sun introduced three kinds of numbers: and In this paper, we mainly prove that by establishing some binomial coefficient identities, such as This confirms several recent conjectures of Sun. [ABSTRACT FROM AUTHOR]

Let T = [ t n , k ] n , k ≥ 0 be an infinite lower triangular matrix defined by t 0 , 0 = 1 , t n + 1 , 0 = ∑ j = 0 n z j t n , j , t n + 1 , k + 1 = ∑ j = k n a j , k t n , j for n , k ≥ 0 , where all z j , a j , k are nonnegative and a j , k = 0 unless j ≥ k ≥ 0 . We show that the sequence ( t n , 0 ) n ≥ 0 is log-convex if the coefficient matrix [ ζ , A ] is TP 2 , where ζ = [ z 0 , z 1 , z 2 , … ] ′ and A = [ a i , j ] i , j ≥ 0 . This gives a unified proof of the log-convexity of many well-known combinatorial sequences, including the Catalan numbers, the Motzkin numbers, the central binomial coefficients, the Schröder numbers, the Bell numbers, and so on. [ABSTRACT FROM AUTHOR]

Highlights: [•] We introduce an alternative analytical formula to calculate the Franck–Condon (FC) factors. [•] The main advantage of the base method is its general analytic results. [•] The obtained results show a good agreement with the results reported in the literature. [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 this paper we are interested in two problems stated in the book of Erdős and Graham. The first problem was stated by Erdős and Straus in the following way: Let n ∈ ℕ+ be fixed. Does there exist a positive integer k such that The second problem is similar and was formulated by Erdős and Graham. It can be stated as follows: Can one show that for every nonnegative integer n there is an integer k such that The aim of this paper is to give some computational results related to these problems. In particular we show that the first problem has positive answer for each n ≤ 20. Similarly, we show the existence of desired n in the second problem for all n ≤ 9. We also note some interesting connections between these two problems. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we derive a general expression for the entries of the qth power ( ) of the complex tridiagonal matrix for all , in terms of binomial coefficients. [Copyright &y& Elsevier]

Abstract: Text: We prove congruences, modulo a power of a prime p, for certain finite sums involving central binomial coefficients , partly motivated by analogies with the well-known power series for and . The right-hand sides of those congruences involve values of the finite polylogarithms . Exploiting the available functional equations for the latter we compute those values, modulo the required powers of p, in terms of familiar quantities such as Fermat quotients and Bernoulli numbers. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=W54Ad0YFr8A. [Copyright &y& Elsevier]

We prove a weighted version of the sum of binomial coefficients in arithmetic progression. As an application, we deduce a random walk theorem which generalizes a result of Aldous and Diaconis. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study complex q-Szász-Mirakjan operators attached to analytic functions satisfying a suitable exponential type growth condition. We give a Voronovskaja-type theorem in compact disks for these new operators. Note that our results are different from the results given for other type complex q-Szász-Mirakjan operators in [8]. [ABSTRACT FROM AUTHOR]

We use the notion of subprime factorization to establish recurrence relations for the number of binomial coefficients in a given row of Pascal's triangle that are divisible by and not divisible by , where is a prime. Using these relations to compute this number can provide significant savings in the number of computational steps. [ABSTRACT FROM AUTHOR]

We consider the problem of finding an exact confidence interval for a proportion that is estimated using randomized response. For many randomized response schemes, this is equivalent to finding an exact confidence interval for a bounded binomial proportion. Such intervals can be obtained by truncating standard exact binomial confidence intervals, but the truncated intervals may be empty or misleadingly short. We address this problem by using exact confidence intervals obtained by inverting a likelihood ratio test that takes into account that the proportion is bounded. A simple adjustment is made to keep the intervals from being excessively conservative. An R function for computing the intervals is available as online supplementary material. [ABSTRACT FROM AUTHOR]

Abstract: Let be a fixed positive integer and let be a certain type of binomial sum. In this paper, we show that for most n the sum of the digits of in base b is at least , where is some positive constant depending on b and on the sequence of binomial sums. Our results include middle binomial coefficients and Apéry numbers . The proof uses a result of McIntosh regarding the asymptotic expansions of such binomial sums as well as Bakerʼs theorem on lower bounds for nonzero linear forms in logarithms of algebraic numbers. [Copyright &y& Elsevier]

We answer a question on the conditioned binomial coefficients raised in an article of Barlotti and Pannone, thus giving an alternative proof of an extension of Frobenius' generalization of Sylow's theorem. [ABSTRACT FROM AUTHOR]

We establish two binomial coefficient-generalized harmonic sum identities using the partial fraction decomposition method. These identities are a key ingredient in the proofs of numerous supercongruences. In particular, in other works of the author, they are used to establish modulo pk ( k > 1) congruences between truncated generalized hypergeometric series, and a function which extends Greene's hypergeometric function over finite fields to the p-adic setting. A specialization of one of these congruences is used to prove an outstanding conjecture of Rodriguez-Villegas which relates a truncated generalized hypergeometric series to the p-th Fourier coefficient of a particular modular form. [ABSTRACT FROM AUTHOR]

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]

We provide a probabilistic comparison of split and type 1 triangle cuts for mixed-integer programs with two rows and two integer variables in terms of cut coefficients and volume cutoff. Under a specific probabilistic model of the problem parameters, we show that for the above measure, the probability that a split cut is better than a type 1 triangle cut is higher than the probability that a type 1 triangle cut is better than a split cut. [ABSTRACT FROM AUTHOR]

We introduce new q-tangent numbers based on the Carlitz q-analog of the Eulerian polynomial and the so-called doubloon combinatorial set-up. Those new q-tangent numbers are polynomials with positive integral coefficients. They are divisible by products of binomials of the form 1 + qi, the quotients being q-analogs of the reduced tangent numbers having an explicit combinatorial interpretation. [ABSTRACT FROM PUBLISHER]

In this paper, we consider translation and multiplication operators acting on the rings of symmetric and nonsymmetric polynomials and study their matrix coefficients with respect to the bases of Jack polynomials and interpolation polynomials. The main new insight is that the symmetric and nonsymmetric cases share a key combinatorial feature, that of a locally finite graded poset with a minimum element. This allows us to treat both cases in a simple and unified manner. [ABSTRACT FROM PUBLISHER]

Let pn be the n-th prime number. We prove the following double-inequality. For all integers k ≥ 5 we have exp[k(c0 - log log k)] ≤ (k2 k)/ p1 . p2 … pk ≤ exp[k(c1 - log log k)] with the best possible constants c0 = 1/5 log 23 + log log 5 = 1.10298 … and c1 =1/192 log ( 36864 192) + log log 192 - 1/192 log (p1 . p2 … p192) = 2.04287 …. This refines a result published by Gupta and Khare in 1977. [ABSTRACT FROM AUTHOR]

Let s and t be variables. Define polynomials { n} in s, t by {0} = 0, {1} = 1, and { n} = s { n - 1} + t { n - 2} for n ≥ 2. If s, t are integers then the corresponding sequence of integers is called a Lucas sequence. Define an analogue of the binomial coefficients by where { n}! = {1} {2} ⋯ { n}. It is easy to see that is a polynomial in s and t. The purpose of this note is to give two combinatorial interpretations for this polynomial in terms of statistics on integer partitions inside a k × ( n - k) rectangle. When s = t = 1 we obtain combinatorial interpretations of the fibonomial coefficients which are simpler than any that have previously appeared in the literature. [ABSTRACT FROM AUTHOR]

In this short paper we establish identities involving sums of products of binomial coefficients and coefficients that satisfy the general second-order linear recurrence. We obtain generalizations of identities of Carlitz, Prodinger and Haukkanen. [ABSTRACT FROM AUTHOR]

In this note, we introduce M-bonomial coefficients or (M-bonacci binomial coefficients). These are similar to the binomial and the Fibonomial (or Fibonacci-binomial) coefficients and can be displayed in a triangle similar to Pascal's triangle from which some identities become obvious. [ABSTRACT FROM AUTHOR]