We define an overpartition analogue of Gaussian polynomials (also known as q-binomial coefficients) as a generating function for the number of overpartitions fitting inside the M ×N rectangle. We call these new polynomials over Gaussian polynomials or over q-binomial coefficients. We investigate basic properties and applications of over q-binomial coefficients. In particular, via the recurrence...

In this paper, a new class of a weighted quadrature rule is represented as --------------------------------------------  where  is a weight function,  are interpolation nodes,  are the corresponding weight coefficients and denotes the error term. The general form of interpolation nodes are considered as   that  and we obtain the explicit expressions of the coefficients  using the q-...

Probabilistic proofs and interpretations are given for the q-binomial theorem, q-binomial series, two of Euler's fundamental partition identities, and for q-analogs of product expansions for the Riemann zeta and Euler phi functions. The underlying processes involve Bernoulli trials with variable probabilities. Also presented are several variations on the classical derangement problem inherent i...

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 . . . . . . . . . . . . . . . . . ...

We investigate the divisibility of q-trinomial coefficients introduced by Andrews and Baxter, which is analogous to q-Wolstenholme theorem regarding q-binomial coefficients. A congruence for sums central also established.

We notice two symmetric q-identities, which are special cases of the transformations of 2φ1 series in Gasper and Rahman’s book (Basic Hypergeometric Series, Cambridge University Press, 1990, p. 241). In this paper, we give combinatorial proofs of these two identities and the q-binomial theorem by using conjugation of 2-modular diagrams.

A weight-dependent generalization of the binomial theorem for noncommuting variables is presented. This result extends the well-known binomial theorem for q-commuting variables by a generic weight function depending on two integers. For two special cases of the weight function, in both cases restricting it to depend only on a single integer, the noncommutative binomial theorem involves an expan...