Related papers: A note on the q-derivative operator

In this letter, the (q,h)-analogue of Newton's binomial formula is obtained in the (q,h)-deformed quantum plane which reduces for h=0 to the q-analogue. For (q=1,h=0), this is just the usual one as it should be. Moreover, the h-analogue is…

Let $w_{n+2}=pw_{n+1}+qw_{n}$ for $n\geq0$ with $w_0=a$ and $w_1=b$. In this paper we find an explicit expression, in terms of determinants, for $\sum_{n\geq0} w_n^kx^n$ for any $k\geq1$. As a consequence, we derive all the previously known…

Motivated by a quaternionic formulation of quantum mechanics, we discuss quaternionic and complex linear differential equations. We touch only a few aspects of the mathematical theory, namely the resolution of the second order differential…

Although being powerful, the differential transform method yet suffers from a drawback which is how to compute the differential transform of nonlinear non-autonomous functions that can limit its applicability. In order to overcome this…

We prove that the derivative of a non-linear entire function is unbounded on the preimage of an unbounded set.

Recently, some problems have been found in the definition of the partial derivative in the case of the presence of both explicit and implicit functional dependencies in the classical analysis. In this talk we investigate the influence of…

We give yet another proof for Fa\`{a} di Bruno's formula for higher derivatives of composite functions. Our proof technique relies on reinterpreting the composition of two power series as the generating function for weighted integer…

In this note, we shall give an improved lower bound for the argument of a power of a given algebraic number which has absolute value one but is not a root of unity.

We prove a uniqueness theorem for an entire function, which shares certain values with its higher order derivatives.

We extend to several variables an earlier result of ours, according to which an entire function of one variable of sufficiently small exponential type, having all derivatives of even order taking integer values at two points, is a…

We study gradient estimates of $q$-harmonic functions $u$ of the fractional Schr{\"o}dinger operator $\Delta^{\alpha/2} + q$, $\alpha \in (0,1]$ in bounded domains $D \subset \R^d$. For nonnegative $u$ we show that if $q$ is H{\"o}lder…

In recent papers, R. Bhatia, T. Jain and P. Grover obtained formulas for directional derivatives, of all orders, of the determinant, the permanent, the $m$-th compound map and the $m$-th induced power map. In this paper we generalize these…

The primary purpose of this paper is to show the existence of normal square and nth roots of some classes of bounded operators on Hilbert spaces. Two interesting simple results hold. Namely, under simple conditions we show that if any…

We prove that the play and the stop operator possess Newton and Bouligand derivatives, and exhibit formulas for those derivatives. The remainder estimate is given in a strenghtened form, and a corresponding chain rule is developed. The…

A number of results are proved concerning non-real zeros of derivatives of real and strictly non-real meromorphic functions in the plane

In this paper, we study the unicity of entire functions concerning their $q-$shifts and $k-$th derivatives and prove: Let $f(z)$ be a transcendental entire function of zero-order, and $g(z)$ define as in (1.1). Let $a(z), b(z)$ be two…

In this article we discuss an important students' misconception about derivatives, that the expression of the derivative of the function contains the information as to whether the function is differentiable or not where the expression is…

This article will prove a theorem for the existence of k-factor for k>1 ,and present an efficient algorithm for computing k-factor for all values of k based on this theorem.

An algebraic $q$-difference equation is considered. A sufficient condition for the existence of a formal power-logarithmic expansion of a solution to such an equation in the neighborhood of zero is proposed. An example of applying this…

Motivated by the recent work of Hirschhorn on vanishing coefficients of the arithmetic progressions in certain $q$-series expansions, we study some variants of these $q$-series and prove some comparable results. For instance, let…