Letting B n (x) the n-th Bell polynomial, it is well known that B n admit specific integer coordinates in the two following bases x i numbers and binomial coefficients. Our aim is to prove that, for r + s = n, the sequence x j B k (x) is a family of bases of the Q-vectorial space formed by polynomials of Q [X ] for which B n admits a Binomial Recurrence Coefficient.

Letting Bn,r be the n-th r-Bell polynomial, it is well known that Bn(x) admits specific integer coordinates in the two bases {x}i and {xBi(x)}i according to, respectively, the Stirling numbers and the binomial coefficients. Our aim is to prove that the sequences Bn+m,r(x) and Bn,r+s(x) admit a binomial recurrence coefficient in different bases of the Q-vector space formed by polynomials of Q[X].

X iv :m at h/ 00 04 18 7v 1 [ m at h. H O ] 1 A pr 2 00 0 Journal of Nonlinear Mathematical Physics 2000, V.7, N 2, 244–262. (Re)-Examining the Past q-Newton Binomial: From Euler To Gauss Boris A. KUPERSHMIDT The University of Tennessee Space Institute, Tullahoma, TN 37388, USA E-mail: [email protected] Received March 6, 2000; Accepted April 7, 2000 Abstract A counter-intuitive result of Gauss ...

Partial fraction decomposition method is applied to evaluate a general determinant of shifted factorial fractions, which contains several Gaussian binomial determinant identities .

The classical identities between the q-binomial coefficients and factorials can be generalized to a context where numbers are replaced by braids. More precisely, for every pair i, n of natural numbers, there is defined an element b (n) i of the braid group algebra kBn, and these satisfy analogs of the classical identities for the binomial coefficients. By choosing representations of the braid g...

Recently, Ni and Pan proved a q-congruence on certain sums involving central q-binomial coefficients, which was conjectured by Guo. In this paper, we give generalization of confirm another q-congruence, also Our proof uses Pan’s technique simple observed Guo Schlosser.

The focus of this paper is the study of generalized Fibonacci polynomials and Fibonomial coefficients. The former are polynomials {n} in variables s, t given by {0} = 0, {1} = 1, and {n} = s{n−1}+t{n−2} for n ≥ 2. The latter are defined by { n k } = {n}!/({k}!{n−k}!) where {n}! = {1}{2} . . . {n}. These quotients are also polynomials in s, t and specializations give the ordinary binomial coeffi...