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),

recently, zhang and song [q. zhang, y. song, fixed point theory forgeneralized $varphi$-weak contractions,appl. math. lett. 22(2009) 75-78] proved a common fixed point theorem for two mapssatisfying generalized $varphi$-weak contractions. in this paper, we prove a common fixed point theorem fora family of compatible maps. in fact, a new generalization of zhangand song's theorem is given.

The generalized binomial distribution in Tsallis statistics (power-law system) is explicitly formulated from the precise q-Stirling’s formula. The α-divergence (or q-divergence) is uniquely derived from the generalized binomial distribution in the sense that when α → −1 (i.e., q → 1) it recovers KL divergence obtained from the standard binomial distribution. Based on these combinatorial conside...

Recently, Zhang and Song [Q. Zhang, Y. Song, Fixed point theory forgeneralized $varphi$-weak contractions,Appl. Math. Lett. 22(2009) 75-78] proved a common fixed point theorem for two mapssatisfying generalized $varphi$-weak contractions. In this paper, we prove a common fixed point theorem fora family of compatible maps. In fact, a new generalization of Zhangand Song's theorem is given.

The main results of this paper are computations of a few determinants related to binomial coefficients. The most interesting example, discovered after browsing through [5], is obtained by considering binomial coefficients modulo 4. We include related results contained in [2] and [3]. The next section recalls mostly well-known facts concerning binomial and q−binomial coefficients and states the ...

This document provides proofs of the following: • The binomial distribution can be approximated with a Gaussian distribution at large values of N . • The arcsin square-root transform is the variance stabilising transform for the binomial distribution. • The Gaussian approximation for the binomial distribution is more accurate at smaller values of N after the variance stabilising transform has b...