Quantum Calculus

Given the inverse of the Golden Mean $ \tau^{ -1} = \phi  = { 1\over 2} (\sqrt 5  - 1)$, 
it is known that the continuous fraction expansion of $ \phi^{ -1} = 1 + \phi = \tau$ is  $ ( 1, 1, 1, \cdots )$.  Integer solutions
for the pairs of numbers $ ( d_i, n_i ), i = 1, 2, 3, \cdots $ are found obeying the equation
$ ( 1 + \phi)^n = d + \phi^n$.  The latter equation was inspired from El Naschie's formulation
of fractal Cantorian space time $ {\cal E}_\infty$,  and such that it furnishes the continuous fraction expansion of $  ( 1 + \phi )^n ~= ~  (d, d, d, d, \cdots  )$, generalizing the original expression for the Golden mean.    Hardy showed that is possible to demonstrate nonlocality without using Bell inequalities for two particles prepared in $nonmaximally$ entangled states.  The maximal probability of obtaining his nonlocality proof was found to be precisely $\phi^5$.  Zheng showed that three-particle nonmaximally entangled states revealed quantum nonlocality without using inequalities, and the maximal probability of obtaining the nonlocality proof was found to be $ 0.25 \sim \phi^3 = 0.236$.  Given that the two-parameter $ p, q$ quantum-calculus deformations of the integers $ [ n ]_{ p, q}  = F_n $  $coincide$ precisely with the Fibonacci numbers, as a result of Binet's formula when
$ p = ( 1 + \phi) = \tau,  q  =  - \phi =  - \tau^{ -1} $,  we explore further the implications of these results in the quantum entanglement of two-particle spin-$s$ states.

The one-dimensional infinitely deep square quantum well inherited its name from an ordinary rigid box (the properties of the box), and is considered to be one of the most elementary problems in non-relativistic quantum mechanics due to its simplicity. Given the nature of the problem, we discuss several methods and their solutions which are limited to the one-dimensional time-independent Schrödinger equation-1D TISE. We will consider: (i) non-relativistic quantum mechanics, (ii) supersymmetric quantum mechanics (SUSYQM), (iii) fractional quantum mechanics, (iv) q-quantum mechanics, and (v) the q-hypergeometric differential equation. Ultimately, we show by using an appropriate substitution that it is possible to transform the 1D TISE, such that it becomes the q-hypergeometric differential equation. To sum up, the non-compact q-deformed Lie group-SU q (1, 1)-and Lie algebra-su q (1, 1)-are described since the former gives the dynamical group or the spectrum generating algebra of the system.

Two pairs of generalized q-factorial moments involving the Heine and the Euler distributions, respectively, are established. Moreover, these pairs of q-factorial moments are shown to be proper q-analogues of the generalized factorial moments.

Two q-analogues of the Elzaki transform, called Mangontarum q-transforms, are introduced in this paper. Properties such as the transforms of q-trigonometric functions, transform of q-derivatives, duality relation, convolution identity, q-derivative of transforms and transform of the Heaviside function are derived and presented. Moreover, we will consider applications of the Mangontarum q-transform of the first kind to some ordinary q-differential equations with initial values.

"""The fundamental aim of this paper is to consider some applications of umbral calculus by utilizing from the extended p-adic q-invariant integral on Zp . From those consideration, we derive some new interesting properties on the extended p-adic q-Bernoulli numbers and polynomials. That is, a systemic study of the class of Sheffer sequences in connection with generating function of the p-adic q-Bernoulli polynomials are given in the present work.
"""

This work aims to examine a boundary value problem which consists of a second order q-differential equation and eigenvalue dependent boundary conditions. We introduce a modified inner product in a suitable direct sum space and define a symmetric operator. We investigate the properties of eigenvalues and eigenfunctions and construct the Green's function.

We introduce the Hahn quantum variational calculus. Necessary and sufficient optimality conditions for the basic, isoperimetric, and Hahn quantum Lagrange problems, are studied. We also show the validity of Leitmann's direct method for the Hahn quantum variational calculus, and give explicit solutions to some concrete problems. To illustrate the results, we provide several examples and discuss a quantum version of the well known Ramsey model of economics.