‎the number of factorizations of a finite abelian group as the product of two subgroups is computed in two different ways and a combinatorial identity involving gaussian binomial coefficients is presented‎.

We first extend the digital binomial identity as given by Nguyen et al. to an identity in an arbitrary base b, by introducing the b-ary binomial coefficients. Then, we study the properties of these coefficients such as their orthogonality, their link with Lucas’ theorem and their extension to multinomial coefficients. Finally, we analyze the structure of the corresponding b-ary Pascal-like tria...

3. On binomial coefficients 13 3.1. Definitions and basic properties . . . . . . . . . . . . . . . . . . . . . 14 3.2. Binomial coefficients and polynomials . . . . . . . . . . . . . . . . . . 21 3.3. The Chu-Vandermonde identity . . . . . . . . . . . . . . . . . . . . . 24 3.4. Further results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.5. Additional exercises . . . . ...

By considering the famous identity on the convolution of the central binomial coefficients

Dominique Foata [2] [6] gave a beautiful combinatonal proof or the following binomial coefficients identity, that is trivially equivalent to the famous PfaCSaalschutz identity: a + b a + e b + c (a + b + r-n)! (a + k) (e + k) (h + k)

In this note we shall prove the following curious identity of sums of powers of the partial sum of binomial coefficients.

A binomial identity ((1) below), which relates the famous Apéry numbers and the sums of cubes of binomial coefficients (for which Franel has established a recurrence relation almost 100 years ago), can be seen as a particular instance of a Legendre transform between sequences. A proof of this identity can be based on the more general fact that the Apéry and Franel recurrence relations themselve...

1.1 q -binomial coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 1.2 Unimodality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 1.3 Congruences for the partition function . . . . . . . . . . . . . . . . . . . . . . . . . 143 1.4 The Jacobi triple product identity . . . . . . . . . . . . . . . . . ...

for any a, b, c ∈ N = {0, 1, 2, . . .}. During the second author’s visit (January–March, 2005) to the Institute of Camille Jordan at Univ. Lyon-I, Dr. Victor Jun Wei Guo told Sun his following conjecture involving sums of products of three binomial coefficients and said that he failed to prove this “difficult conjecture” during the past two years though he had tried to work it out again and again.