We give combinatorial proofs of some binomial and q-binomial identities in the literature, such as ∞ ∑ k=−∞ (−1)kq(9k2+3k)/2 [ 2n n + 3k ] = (1 + q) n−1 ∏ k=1 (1 + q + q) (n ≥ 1),

Abstract. The q-binomial coefficient is a polynomial in q. Given an integer t and a residue class r modulo t, a sieved q-binomial coefficient is the sum of those terms whose exponents are congruent to r modulo t. In this paper explicit polynomial identities in q are given for sieved q-binomial coefficients. As a limiting case, generating functions for the sieved partition function are found as ...

An integer composition of a nonnegative integer n is a tuple (π1, . . . , πk) of nonnegative integers whose sum is n; the πi’s are called the parts of the composition. For fixed number k of parts, the number of f -weighted integer compositions (also called f -colored integer compositions in the literature), in which each part size s may occur in f(s) different colors, is given by the extended b...

If Fq is a finite field and G is a subgroup of the linear automorphisms of F q , a solution to the problem of finding all the G-invariant linear codes C of F q (i.e. such that g(C) = C for all g ∈ G) is offered. This will be referred as the invariance problem. When n = |G|t, we determine conditions for the existance of an isomorphism of Fq[G]modules between F q and Fq[G]× · · · × Fq[G] (t-times...

In this paper, we prove some identities for the alternating sums of squares and cubes of the partial sum of the q-binomial coefficients. Our proof also leads to a q-analogue of the sum of the first n squares due to Schlosser.

q-binomial coefficient identity Victor J. W. Guo and Jiang Zeng Department of Mathematics, East China Normal University, Shanghai 200062, People’s Republic of China [email protected], http://math.ecnu.edu.cn/~jwguo Université de Lyon; Université Lyon 1; Institut Camille Jordan, UMR 5208 du CNRS; 43, boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France [email protected], ...

In this paper we prove the fo rmula ~o w ~-dk (Pn-bkk)(q +k bk) ( _ 1 ) k (p --kbW/d) (k q-kW/d) -I (p + q -k =kffio n k / and ob ta in a more general t r ans fo rma t ion for s u m s of the fo rm ~ (a +kbk) (c + b(n-k))f(k)g(n -k ). k=o n k These sums con ta in m a n y special cases in the l i terature of statistics, probabili ty, combinator ia l analysis, special funct ions , and o ther areas...

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 p (k > 1) congruences between truncated generalized hypergeometric series, and a function which extends Greene’s ...