相关论文: Sur une generalisation des coefficients binomiaux

Here Euler notes the recursive relation for the general binomial coefficients, by assuming that (1+x)^a can be expanded in a power series.

We examine connections between rationality of certain indefinite integrals and equilibrium of Coulomb charges in the complex plane.

The causal set theory d'Alembertian has rational coefficients for which alternating expressions are known. Here, a combinatorial interpretation of these numbers is given.

Considered two linear regression models of a given response variable with some predictor set and its subset. It is shown that there is a linear relationship between coefficients of these models. Some corollaries of the proved theorem is…

In this study, the new algebraic properties related to bivariate Fibonacci polynomials has been given. We present the partial derivatives of these polynomials in the form of convolution of bivariate Fibonacci polynomials. Also, we define a…

Transformation formulas for four-parameter refinements of the q-trinomial coefficients are proven. The iterative nature of these transformations allows for the easy derivation of several infinite series of q-trinomial identities, and can be…

We study Jack characters, which are the coefficients of the power-sum expansion of Jack symmetric functions with a suitable normalization. These quantities have been introduced by Lassalle who formulated some challenging conjectures about…

The definition of the binomial coefficient in terms of gamma functions also allows non-integer arguments. For nonnegative integer arguments the gamma functions reduce to factorials, leading to the well-known Pascal triangle. Using a…

We explicitly determine the Ap\'ery limits for the sums of powers of binomial coefficients. As an application, we prove a weak version of Franel's conjecture on the order of the recurrences for these sequences. Namely, we prove the…

We deliver here second new $\textit{H(x)}-binomials'$ recurrence formula, were $H(x)-binomials' $ array is appointed by $Ward-Horadam$ sequence of functions which in predominantly considered cases where chosen to be polynomials . Secondly,…

We give a number theoretic proof of the integrality of certain BPS invariants of knots. The formulas for these numbers are sums involving binomial coefficients and the M\"obius function. We also prove a conjecture about further divisibility…

Abstract. We generalize the $b$-ary binomial coefficients with negative entries, which is based on the generating function obtained in early work. Besides an explicit expression involving the restricted partition, several properties such as…

We interpret the coefficients of the cyclotomic polynomial in terms of simplicial homology.

We evaluate binomial series with harmonic number coefficients, providing recursion relations, integral representations, and several examples. The results are of interest to analytic number theory, the analysis of algorithms, and…

We define generalized bivariate polynomials, from which upon specification of initial conditions the bivariate Fibonacci and Lucas polynomials are obtained. Using essentially a matrix approach we derive identities and inequalities that in…

A combinatorial proof of the unimodality of the generalized q-Gaussian coefficients based on the explicit formula for Kostka-Foulkes polynomials is given.

We reinterpret a conjecture of Breuil on the locally analytic $\mathrm{Ext}^1$ in a functorial way using $(\varphi,\Gamma)$-modules (possibly with $t$-torsion) over the Robba ring, making it more accurate. Then we prove several special or…

A generalized central trinomial coefficient $T_n(b,c)$ is the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$ with $b,c\in\mathbb Z$. In this paper we investigate congruences and series for sums of terms related to central binomial…

We classify the polynomials with integral coefficients that, when evaluated on a group element of finite order $n$, define a unit in the integral group ring for infinitely many positive integers $n$. We show that this happens if and only if…

In what follows, we pose two general conjectures about decompositions of homogeneous polynomials as sums of powers. The first one (suggested by G. Ottaviani) deals with the generic k-rank of complex-valued forms of any degree divisible by k…