Non-Commutative Q-Binomial Formula
Sengul Nalci, Oktay Pashaev
TL;DR
This paper introduces a new q-binomial formula for Q-commutative operators, generalizing classical binomial formulas and enabling the study of non-commutative q-analytic functions and waves.
Contribution
It presents a novel q-binomial formula for Q-commutative operators with coefficients based on a two-base q-binomial, extending existing binomial identities.
Findings
Generalizes classical binomial formulas including Newton and Fibonacci.
Defines a q-analogue of functions of two non-commutative variables.
Potential applications in non-commutative q-analytic functions and wave studies.
Abstract
In this paper, we found new q-binomial formula for Q-commutative operators. Expansion coefficients in this formula are given by q-binomial coefficients with two bases (q,Q), determined by Q-commutative q-Pascal triangle. Our formula generalizes all well-known binomial formulas in the form of Newton, Gauss, symmetrical, non-commutative and Binet-Fibonacci binomials. By our non-commutative q-binomial, we introduce q-analogue of function of two non-commutative variables, which could be used in study of non-commutative q-analytic functions and non- commutative q-traveling waves.
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Taxonomy
TopicsMathematical functions and polynomials · Nonlinear Waves and Solitons · Quantum Mechanics and Non-Hermitian Physics