The presentation of a new type of quantum calculus Abdolali Neamaty a - PDF document

DOI 10.1515/tmj-2017-00 22 The presentation of a new type of quantum calculus Abdolali Neamaty a ∗ and Mehdi Tourani b Department of Mathematics, University of Mazandaran, Babolsar, Iran E-mail: namaty@umz.ac.ir a , mehdi.tourani1@gmail.com b Abstract In this paper we introduce a new type of quantum calculus, the p -calculus involving two concepts of p -derivative and p -integral. After familiarity with them some results are given. 2010 Mathematics Subject Classification. 05A30 . 34A25 Keywords. p -derivative, p -antiderivative, p -integral. 1 Introduction Simply put, quantum calculus is ordinary calculus without taking limit. In ordinary calculus, the f ( y ) − f ( x ) derivative of a function f ( x ) is defined as f ′ ( x ) = lim . However, if we avoid taking the y − x y → x limit and also take y = x p , where p is a fixed number different from 1, i.e., by considering the following expression: f ( x p ) − f ( x ) (1.1) , x p − x then, we create a new type of quantum calculus, the p -calculus, and the corresponding express is the definition of the p -derivative. The formula (1 . 1) and several of the results derived from it which will be mentioned in the next sections, appear to be new. In [8] the authors developed two types of quantum calculus, the q -calculus and the h -calculus. If in the definition of f ′ ( x ), as has been stated above, we do not take limit and also take y = qx or y = x + h , where q is a fixed number different from 1, and h a fixed number different from 0, the q -derivative and the h -derivative of f ( x ) are defined. For more details, we refer the readers to [1, 2, 4, 7]. Generally, in the last decades the q -calculus has developed into an interdisciplinary subject, which is briefly discussed in chapters 3 and 7 of [3] and also has interesting applications in various sciences such as physics, chemistry, etc [5, 6]. A history of the q -calculus was given by T.Ernst [3]. The purpose of this paper is to introduce another type of quantum calculus, the p -calculus, also we’re going to give some results by it. The paper has been organized as follows. In section 2, we define the p -derivative, also some of its properties will be expressed. In section 3, we introduce the p -integral, including a sufficient condition for its convergence is given. In section 4, we will define the definite p -integral, followed by the definition of the improper p -integral. Finally, we will conclude our discussion by fundamental theorem of p -calculus. 2 p -Derivative Throughout this section, we assume that p is a fixed number different from 1 and domain of function f ( x ) is [0 , + ∞ ). ∗ Corresponding author Tbilisi Mathematical Journal 10(2) (2017), pp. 15–28. Tbilisi Centre for Mathematical Sciences. Unauthenticated Received by the editors: 14 May 2016. Download Date | 2/17/17 6:16 AM Accepted for publication: 25 December 2016.

16 A. Neamaty, M. Tourani Definition 2.1. Let f ( x ) be an arbitrary function. We define its p -differential to be d p f ( x ) = f ( x p ) − f ( x ) . In particular, d p x = x p − x . By the p -differential we can define p -derivative of a function. Definition 2.2. Let f ( x ) be an arbitrary function. We define its p -derivative to be D p f ( x ) = f ( x p ) − f ( x ) ifx � = 0 , 1 , x p − x and D p f (0) = lim x → 0 + D p f ( x ) , D p f (1) = lim x → 1 D p f ( x ) . Remark 2.3. If f ( x ) is differentiable, then lim p → 1 D p f ( x ) = f ′ ( x ), and also if f ′ ( x ) exists in a neighborhood of x = 0, x = 1 and is continuous at x = 0 and x = 1, then we have D p f (0) = f ′ + (0) , D p f (1) = f ′ (1) . Definition 2.4. The p -derivative of higher order of function f is defined by ( D 0 ( D n p f )( x ) = D p ( D n − 1 p f )( x ) = f ( x ) , f )( x ) , n ∈ N. p Example 2.5. Let f ( x ) = c , g ( x ) = x n and h ( x ) = ln( x ) where c is constant and n ∈ N . Then we have ( i ) D p f ( x ) = 0 , = x pn − x n = x ( p − 1) n − 1 ( ii ) D p g ( x ) = g ( x p ) − g ( x ) x p − 1 − 1 x n − 1 , x p − x x p − x ( iii ) D p h ( x ) = h ( x p ) − h ( x ) = ( p − 1) ln( x ) = ( p − 1) ln( x ) 1 x . x p − x x p − x x p − 1 − 1 Notice that the p -derivative is a linear operator, i.e., for any constants a and b , and arbitrary functions f ( x ) and g ( x ), we have D p ( af ( x ) + bg ( x )) = aD p f ( x ) + bD p g ( x ) . We want now to compute the p -derivative of the product and the quotient of f ( x ) and g ( x ). f ( x p ) g ( x p ) − f ( x ) g ( x ) D p ( f ( x ) g ( x )) = x p − x f ( x p ) g ( x p ) − f ( x ) g ( x p ) + f ( x ) g ( x p ) − f ( x ) g ( x ) = x p − x ( f ( x p ) − f ( x )) g ( x p ) + f ( x )( g ( x p ) − g ( x )) = . x p − x Thus D p ( f ( x ) g ( x )) = g ( x p ) D p f ( x ) + f ( x ) D p g ( x ) . (2.1) Unauthenticated Download Date | 2/17/17 6:16 AM

The presentation of a new type of quantum calculus 17 Similarly, we can interchange f and g , and obtain D p ( f ( x ) g ( x )) = g ( x ) D p f ( x ) + f ( x p ) D p g ( x ) , (2.2) which both of (2 . 1) and (2 . 2) are valid and equivalent. Here let us prove quotient rule. By changing f ( x ) to f ( x ) g ( x ) in (2 . 1), we have D p f ( x ) = D p ( f ( x ) g ( x ) g ( x )) = g ( x p ) D p ( f ( x ) g ( x ) ) + f ( x ) g ( x ) D p g ( x ) , and thus D p ( f ( x ) g ( x ) ) = g ( x ) D p f ( x ) − f ( x ) D p g ( x ) . (2.3) g ( x ) g ( x p ) Using (2 . 2) with functions f ( x ) g ( x ) and g ( x ), we obtain g ( x ) ) = g ( x p ) D p f ( x ) − f ( x p ) D p g ( x ) D p ( f ( x ) . (2.4) g ( x ) g ( x p ) Both of (2 . 3) and (2 . 4) are valid. Note 2.6. We do not have a general chain rule for p -derivatives, but in most cases we can have the following rule: D p [ f ( u ( x ))] = D p u ( x ) D ln( u ( x )) f ( u ( x )) , h ( x ) where h ( x ) is depended on u ( x ). Example 2.7. If α > 0 and u ( x ) = αx β , then f ( αx pβ ) − f ( αx β ) D p [ f ( u ( x ))] = x p − x · αx pβ − αx β f ( αx pβ ) − f ( αx β ) = αx pβ − αx β x p − x = D ln( α ) + pβ ln( x ) f ( u ( x )) D p u ( x ) , ln( u ( x )) ln( α ) + pβ ln( x ) ln( u ( x )) = αx pβ . because, u ( x ) Example 2.8. If α > 0 and u ( x ) = αe x , then f ( αe x p ) − f ( αe x ) D p [ f ( u ( x ))] = x p − x · αe x p − αe x f ( αe x p ) − f ( αe x ) = αe x p − αe x x p − x = D ln( α ) + x p f ( u ( x )) D p u ( x ) , ln( u ( x )) Unauthenticated Download Date | 2/17/17 6:16 AM

18 A. Neamaty, M. Tourani ln( α ) + x p ln( u ( x )) = αe x p . because, u ( x ) 3 p -Integral The first thing that comes to our mind after studying the derivative of a function is its integral topic. Before investigating it, let us define p -antiderivative of a function. Definition 3.1. A function F ( x ) is a p -antiderivative of f ( x ) if D p F ( x ) = f ( x ). It is denoted by � F ( x ) = f ( x ) d p x. Notice that as in ordinary calculus, the p -antiderivative of a function might not be unique. We can prove the uniqueness by some restrictions on the p -antiderivative and on p . Theorem 3.2. Suppose 0 < p < 1. Then, up to adding a constant, any function f ( x ) has at most one p -antiderivative that is continuous at x = 1. Proof. Suppose F 1 and F 2 are two p -antiderivative of f ( x ) that are continuous at x = 1. Let Φ( x ) = F 1 ( x ) − F 2 ( x ). Since F 1 and F 2 are continuous at x = 1 and also by the definition of p -derivative that lead to D p Φ( x ) = 0, we have Φ is continuous at x = 1 and Φ( x p ) = Φ( x ) for any x . Since for some sufficiently large N > 0, Φ( x p N ) = Φ( x p N +1 ) = ... = Φ( x ) and also by the continuity Φ at x = 1, it follows that Φ( x ) = Φ(1). � As was mentioned we denote the p -antiderivative of f ( x ) by function F ( x ) such that D p F ( x ) = f ( x ). Here we’re going to construct the p -antiderivative. For this purpose, we use of an operator. We define an operator ˆ M p , by ˆ M p ( F ( x )) = F ( x p ). Then we have: M p − 1) F ( x ) = F ( x p ) − F ( x ) 1 x p − x ( ˆ = D p F ( x ) = f ( x ) . x p − x ˆ p ( F ( x )) = F ( x p j ) for j ∈ { 0 , 1 , 2 , 3 , ... } and also by the geometric series expansion, we M j Since formally have ∞ ∞ 1 ( x p j − x p j +1 ) f ( x p j ) . ˆ (( x − x p ) f ( x )) = � M j p (( x − x p ) f ( x )) = � F ( x ) = (3.1) 1 − ˆ M p j =0 j =0 It is worth mentioning that we say that (3.1) is formal because the series does not always converge. Definition 3.3. The p -integral of f ( x ) is defined to be the series ∞ ( x p j − x p j +1 ) f ( x p j ) . � (3.2) j =0 Remark 3.4. Generally, the p -integral does not always converge to a p -antiderivative. Here we want to give a sufficient condition for convergence the p -integral to a p -antiderivative. Unauthenticated Download Date | 2/17/17 6:16 AM