Using functional equations, we derive noncommutative extensions of Ramanujan's 1 ψ 1 summation. 1. Introduction. Hypergeometric series with noncommutative parameters and argument, in the special case involving square matrices, have been the subject of recent study, see e.g. the papers by Duval and Ovsienko [DO], Grünbaum [G], Tirao [T], and some of the references mentioned therein. Of course, t...

Following [9] some properties of Q-probability and Q-states are studied. Representation theorem of Q-probabilities and Q-states, the existence of the joint observable and The central limit theorem are proved. Keywords—Q-probability, Q-state, representation theorem, intuitionistic fuzzy events, joint Q-observable, Central limit theorem.

We determine all integers n such that n2 has at most three base-q digits for q ∈ {2, 3, 4, 5, 8, 16}. More generally, we show that all solutions to equations of the shape Y 2 = t + M · q + N · q, where q is an odd prime, n > m > 0 and t2, |M |, N < q, either arise from “obvious” polynomial families or satisfy m ≤ 3. Our arguments rely upon Padé approximants to the binomial function, considered ...

– We provide numerical indications of the q-generalised central limit theorem that has been conjectured (Tsallis 2004) in nonextensive statistical mechanics. We focus on N binary random variables correlated in a scale-invariant way. The correlations are introduced by imposing the Leibnitz rule on a probability set based on the so-called q-product with q ≤ 1. We show that, in the large N limit (...