Binomial coefficients have been used for centuries in a variety of fields and accumulated numerous definitions. In this paper, we introduce new way defining binomial as repeated sums ones. A multitude coefficient identities will be shown order to prove definition. Using definition, simplify some particular such the Harmonic sum Binomial-Harmonic sum. We derive formulae simplifying general <i...

Binomial coefficients arise naturally in combinatorics. Recently the speaker initiated the study of certain divisibility properties of binomial coefficients, and products or sums of binomial coefficients. In this talk we introduce the speaker’s related results and various conjectures. The materials come from the author’s two preprints: 1. Z. W. Sun, Products and sums divisible by central binomi...

Let (b n) n≥0 be the binomial transform of (a n) n≥0. We show how a binomial transformation identity of Chen proves a symmetrical Bernoulli number identity attributed to Carlitz. We then modify Chen's identity to prove a new binomial transformation identity.

Ismail et al. (Constr. Approx. 15 (1999) 69–81) proved the positivity of some trigonometric polynomials with single binomial coefficients. In this paper, we prove some similar results by replacing the binomial coefficients with products of two binomial coefficients.