Analysis and Algebraic Structures of q-Analysis and its Generalizations

School of Education, Culture and Communication

Division of Applied Mathematics

MASTER THESIS IN MATHEMATICS / APPLIED MATHEMATICS

Analysis and Algebraic Structures of q-Analysis and its Generalizations

by

Olle Karlsson

Master thesis in Mathematics / Applied Mathematics

DIVISION OF APPLIED MATHEMATICS

MÄLARDALEN UNIVERSITY SE-721 23 VÄSTERÅS, SWEDEN

School of Education, Culture and Communication

Division of Applied Mathematics

Master thesis in mathematics / applied mathematics

Date:

2020-06-11

Project name:

Analysis and Algebraic Structures of q-analysis and its Generalizations

Author: Olle Karlsson Supervisor(s): Sergei Silvestrov Reviewer: Masood Aryapoor Examiner: Lars Hellström Comprising: 30 ECTS credits

Abstract

In this thesis we explore the concept of q-calculus and its generalisation. We begin by defining q-combinatorics which uses a real number to define a new set of numbers and then use these numbers to get classic combinatoric elements. These results have use when we work on our algebra that are related with this specific real number. We then work out some results involving one of the operators in the algebra. This operator together with a similar operator produces some special differential equations that we explore. Then we go on to define integrals as the inverse operator to the one used for our differential equations. In the last chapter we try to generalise everything we have explored until then.

Contents

1 Introduction 4 1.1 Approach . . . 4 1.2 Motivation . . . 5 1.3 Previous Publications . . . 5 2 q-Combinatorics 6 2.1 q-numbers . . . 6 2.2 q-binomials . . . 7 3 Algebra 9 3.1 q-deformed algebras . . . 9

3.2 The algebra for q-calculus . . . 10

4 q-Derivative 12 4.1 Rules for the q-differentiation operator . . . 12

4.2 Some results from q-differential equations . . . 16

5 q-Antiderivative and q-Integration 21 5.1 q-Antiderivative . . . 21

5.2 A return to q-differential equations . . . 22

5.3 q-Integration . . . 23

5.4 Fundamental theorem of q-calculus . . . 25

6 g-Calculus 27 6.1 Introduction . . . 27

6.2 Algebra of g-calculus . . . 27

6.3 Differentiation in g-calculus . . . 29

6.4 An attempt at g-combinatorics . . . 32

6.5 Antiderivative and Integration in g-calculus . . . 32

6.6 g-differential equations . . . 33

7 Conclusions 35 7.1 Results . . . 35

Chapter 1

Introduction

1.1

Approach

In this thesis we explore the Analysis and Algebraic structure of the concept of q-calculus and its generalisation. We begin in Chapter 2 by defining what a q-number is. Then we use this definition to develop the theory of q-combinatorics in general with q-factorials and q-binomials in particular.

In Chapter 3 we delve into the algebra beginning by defining what a q-deformed algebra is and explore how some operators might interact with each other. When we have established the general properties of a q-deformed algebra we specify the operators used in q-calculus and show that they satisfy the requirements for a q-deformed algebra.

In Chapter 4 we go from the algebra of the the q-calculus to looking at the analytics of the algebra. We state and prove properties such as linearity and the existence of a Liebnitz rule in the analysis, then we define a q version of Taylor series and with it q versions of famous functions. Next up we define some relations that the q-differentation operator has with its relative the 1/q-differentiation operator.

In the next section 4.2 we begin looking into differential equations and its solutions. We begin with solving some easy equations explicitly and then we prove a theorem on how solu-tions of a set of equasolu-tions relate to each other.

In Chapter 5 we turn it around and instead of working with a q version of differentiation we develop q-antiderivative and an explicit formula for q-antiderivative of polynomials. We then use this to define integrals in the q sense as well as some results like the Fundamental Theorem of q-calculus.

In Chapter 6 we work on generalising everything we have done previously with mixed suc-cess. First we begin with checking if there is some general behaviour to the algebra. Then we move on to work out the analytics of our general differentiation operator and how it relates to related functions. We also attempt to generalise a version of q-combinatorics. Lastly we define integration and differential equations in the context of a generalised version of q-calculus.

1.2

Motivation

In this thesis we will explore what has been denoted q-calculus as well as its generalisations. This is together with h-calculus sometimes called calculus without limits. From Definition 5.1 in Principle of Mathematics[2] by Walter Rudin, we have the definition for differentiating a function.

Definition 1. Let f be defined and real-valued on [a, b]. For any x ∈ [a, b] define the quotient function φ (t) = f(t) − f (x) t− x (a < t < b,t 6= x) and define d f(x) dx = limt→xφ (t)

given that the limit exists.

This definition allows for t to approach x in a multitude of ways the most common is of course when t = x + h and we let h → 0 but we could also have t = qx and get the limit by letting q → 1. This is the approach we will explore mostly In this thesis. We will also look at some general behaviours of φ (t) where we use a general function g(x) = t.

1.3

Previous Publications

The main source of information from q-calculus In this thesis is the book Quantum Calcu-lus by Victor Kac and Pokman Cheung [1]. It builds up and explains the main points of q-calculus and most of the information from chapters 2,4 and 5 is based on this book. The algebra in Chapter 3 is mostly based on the works of Lars Hellström and Sergei Silvestrov in Two-sided ideals in q-deformed Heisenberg algebrasand Commuting Elements in Deformed Heisenberg Algebras [5], [6]. Chapter 2 with q-combinatorics can be found more indepth in Toufik Mansour and Matthias Schork book Commutation Relations, Normal Ordering, and Stirling Numbers [7]. The exploration off q-differential equation was based on a currently unpublished paper by Sergei Silvestrov and Predrag M Rajkovi´c titled Two approaches to the q-Rodrigues formula [3]. For a published work similar to this we recommend Vicentius Radulescus paper Rodrigues-type formulae for Hermite and Laguerre polynomials [8].

Chapter 2

q-Combinatorics

2.1

q-numbers

Here we will define what is known as q-numbers, this is effectively a translation and rescaling of the real number line by a relation dependent on a fixed real number q. So if we are given a real number q lets try and motivate a function for our rescaling between the reals and the reals. To start off let us consider the limit of the geometric sum

lim q→1 n−1

∑

k=0 qk= n.

Remember that q-calculus was sometimes called the calculus without limits just taking this sum and call it the q-version of the number n.

Definition 2. If q ∈ R \ {0} then n−1

∑

k=0 qk= nq or shorter nq= [n].

This is a start but we want to expand it to cover more than just the positive whole numbers. To do this we begin by using the formula for geometric sums, then we can rewrite the sum if q6= 1 as n−1

∑

k=0 qk= q n_{− 1} q− 1 .

Since both these are the same when q 6= 1 we can just take this as our new and improved definition for [n]. This gives us a more useful definition of a q-analogue number.

Definition 3. Let us define

[n] =q

n_{− 1}

q− 1 .

This new definition gives us a way to extend the result to encompass all real numbers by letting the the variable n be extended to any real number. This is the final form of this definition and it conforms to the previous definitions for when α = n.

Definition 4. For α ∈ R and q ∈ R+\ {1} we set

qα_{− 1}

q− 1 = αq and call it a q-number.

We need to restrict our choices for q because if we have negative q we would get complex numbers for certain α. Now we will show that this definition still keeps the property of having the limit converging to what we would expect. This is not a very important property but it is satisfying to know.

Proposition 1. If α ∈ R then

lim

q→1αq= α.

Proof. Since we have

lim

q→1q

α_{− 1 = lim}

q→1q− 1 = 0

we can use L’Hopitals rule from classic analysis and the fraction becomes lim

q→1

α qα −1 1 = α.

The whole concept of q-numbers was originally[1] derived from its relation with certain algebraic operators that we will define later. For now we will be satisfied with having it rigorously defined and will continue to show some results related to q-numbers.

These results are still satisfied if we allow for complex q and this makes so that we can have more choices for q. If q is negative then [n] alternate sign depending on if n is even or odd and if q ≤ −1 or −1 ≤ q ≤ 0. The special case when q = −1 we get [n] ∈ {0, 1}. We will assume that q is a positive real number that is not 1.

2.2

q-binomials

Now that we have a definition for q-numbers we can start exploring how it can be used to define its own set of combinatorics. Combinatorics has lots of tools and one of the most famous ones is factorials. We modify the definition of factorials and create q-factorials and subsequently q-binomials.

Definition 5. Let us define

[n]! = [n][n − 1][n − 2] . . . [2][1]

for any positive integer n. We also define [0]! = 1. With this we can also define n k  q = [n]! [k]![n − k]!.

One important result for binomial coefficients is the Pascal rule but the original pascal rule does not translate into q-combinatorics but requires some minor modifications. There do exist Pascal rules for q-combinatorics and they are very useful so let us present them.

Theorem 1. For q-combinatorics there are two versions of the q-Pascal rule namely n k  q =n − 1 k− 1  q + qkn − 1 k  q and n k  q = qn−kn − 1 k− 1  q +n − 1 k  q . Proof. We begin by separating whole q-numbers into other q-numbers. For any 1 ≤ k ≤ n − 1, [n] = n−1

∑

j=0 qj = k−1

∑

j=0 qj+ qk n−k−1

∑

j=0 qj = [k] + qk[n − k].

This makes it possible to break our q-binomial apart and reconstruct it into two fractions instead of simply one. With this we get for our q-binomial so that

n k  q = [n]! [k]![n − k]! = [n − 1]![n] [k]![n − k]! = [n − 1]!([k] + q k_{[n − k])} [k]![n − k]! = [n − 1]! [k − 1]![n − k]!+ q k [n − 1]! [k]![n − k − 1]! =n − 1 k− 1  q + qkn − 1 k  q .

This is one of the versions. The other we get because the factorials in the denominator can be exchanged for each other such that

n k  q =  n n− k  q .

This will then give the corresponding other version of the q-pascal rule.

Now we have what we need from q-combinatorics we can move on and use it to develop our algebra that we will be working with.

Chapter 3

Algebra

3.1

q-deformed algebras

First we will just present some general definitions and results from algebra connected to an element q from an underlying associative field[6]. This field is not required to be the reals. But the results and definition from the last chapter still applies except for the ones that require limits.

For general operators A and B let us define some relations.

Definition 6. Operators A and B are said to satisfy the q-deformed Heisenberg relation if AB− qBA = I.

With I being the identity operator.

This definition is based on the literature [5]. We also have the following relation that is very important going forward.

Definition 7. Operators A and B are said to q-commute if AB= qBA.

These two relations often come together in the same algebra with multiple operators. Remark 1. If we have operators A, B and C then maybe A and B satisfy the q-Heisenberg relation and C q-commute with A and B.

This is just an example of how an algebra might behave.

Our first result regarding the relation between operators that q-commute is a version of the binomial formula. Now if we have variables X ,Y such that they q-commute then the following result holds.

Theorem 2. If Y X = qXY and q commute with both X and Y then (X +Y )n= n

∑

j=0 n j  q XjYn− j.

Proof. Let us prove this by induction on n. The statement is obviously true for n = 1. Next assume the relation is satisfied for some n then let us compute the following

(X +Y )n+1= (X +Y )n(X +Y ) = n

∑

j=0 n j  q XjYn− j ! (X +Y ) = n

∑

j=0 n j  q XjYn− jX+ n

∑

j=0 n j  q XjYn− j+1 = n

∑

j=0 n j  q qn− jXj+1Yn− j+ n

∑

j=0 n j  q XjYn− j+1 = n+1

∑

j=1  n j− 1  q qn− j+1XjYn− j+1+ n

∑

j=0 n j  q XjYn− j+1 = Yn+1+ n

∑

j=1 qn− j+1  n j− 1  q +n j  q ! XjYn− j+1+ Xn+1 = n+1

∑

j=0 n + 1 j  q XjYn+1− j.

Here we use the assumption and then we do some manipulation with the index and the fact that X and Y q-commute. Then at the end we use a version of the q-Pascal rule to combine the sums into one and thereby prove the theorem.

Now we have the general definition for q-related operators and how they work then we can begin by specifying our operators.

3.2

The algebra for q-calculus

Now we will expand using our definitions from the last section by building the set we are working on and adding operators to our algebra. Lets assume we start with the field of real numbers and the operator Mx defined as multiplying by variable x. This then expands just the

reals to the set of all polynomials with real coefficients. Now lets define what we will call the q-differential operator.

Definition 8. Given a function f (x) then we define the operator Dq, where q 6= 1, as follows

D_qf(x) = f(qx) − f (x) qx− x =

f(qx) − f (x) (q − 1) x .

The operator Mx together with Dqform a q-deformed Heisenberg algebra[6]. This means

that they satisfy the q-deformed Heisenberg relation from Definition 6 and form an algebra. Let us show that it satisfies the q-deformed Heisenberg relation

This is done by extending the operators by their definitions such that Dq(Mxf(x)) − qMx(Dqf(x)) = D_q(x f (x)) − qx(D_qf(x)) = qx f(qx) − x f (x) (q − 1) x − qx f(qx) − f (x) (q − 1) x = qx f(qx) − x f (x) − qx f (qx) + qx f (x) (q − 1) x = (q − 1)x f (x) (q − 1)x = f (x).

There exist other operators that can be included to extend the algebra one that is useful and often considered is translations

T_q: f (x) → f (qx).

This operator appears often in physics[6]. We can see that for q = 1 this is the identity operator and this is still true even for the next representation. The operator can be represented by compositions of the operators we already have namely

Tq= (q − 1)MxDq+ I.

Translation is also q-commutative with the other two operators. For Mxthis is as simple as

T_qM_xf(x) = T_qx f(x) = qx f (qx) = qM_xf(qx) = qM_xT_qf(x).

Now we have showed that Tqand Mx q-commute. Next we want to show the same for Tqand

D_q.

So for q-derivatives we get

qT_qD_qf(x) =qT_q f (qx) − f (x) (q − 1) x  =qf(q 2_{x) − f (qx)} (q − 1) qx =f(q 2_{x) − f (qx)} (q − 1) x =D_qf(qx) =DqTqf(x).

We can now see that the operators Mx, Dqand Tqare represented in Remark 1. So far they

work on the set of polynomials with real coefficient. Now we will let go of the algebra for a while and focus more on the analytics concerning these operators.

Chapter 4

q-Derivative

4.1

Rules for the q-differentiation operator

Earlier we defined q-combinatorics now we will show how q-numbers are related to the oper-ator Dq. To motivate the use of q-numbers we derive from f (x) = xnthat

D_qf(x) =f(qx) − f (x) qx− x (4.1) =q n_xn_{− x}n (q − 1)x (4.2) =q n_{− 1} q− 1 x n−1 _(4.3) =[n]xn−1. (4.4)

The same is of course true for any real number α not just the whole numbers. From now on we will be considering the results on general functions. We have that the q-differential is a linear operator.

Lemma 1. The operator Dqis linear namely that

D_q(a f (x) + bg(x)) = aDq( f (x)) + bDq(g(x)). (4.5)

Proof. The proof is very simple

D_q(a f (x) + bg(x)) =a f(qx) + bg(qx) − a f (x) − bg(x) qx− x =af(qx) − f (x) qx− x + b g(qx) − g(x) qx− x =aDq( f (x)) + bDq(g(x)).

Another very important rule for the Dq operator is the q-Leibniz rule also known as the

product rule for q-calculus. It is very similar to its classic counterpart but has, as is the case with many results, two versions.

Lemma 2. The q-Leibniz rule is as follows

Dq( f (x)g(x)) = Dq( f (x))g(qx) + f (x)Dq(g(x)) (4.6)

or

D_q( f (x)g(x)) = D_q( f (x))g(x) + f (qx)D_q(g(x)). (4.7) Proof. To find this we simply take

D_q( f (x)g(x)) = f(qx)g(qx) − f (x)g(x) qx− x = f(qx)g(qx) − f (x)g(qx) + f (x)g(qx) − f (x)g(x) qx− x = f(qx)g(qx) − f (x)g(qx) qx− x + f(x)g(qx) − f (x)g(x) qx− x = D_q( f (x))g(qx) + f (x)D_q(g(x)).

The commutative property of multiplication of functions also implies another setup for the product rule in q-algebra namely

D_q( f (x)g(x)) = Dq(g(x) f (x)) =

D_q(g(x)) f (qx) + g(x)Dq( f (x)) =

D_q( f (x))g(x) + f (qx)Dq(g(x)).

This then proves both versions of the product rule for q-calculus.

We can also use the q-Leibniz rule to develop a version of the quotient rule for q-derivatives by applying the q-Leibniz rule to

g(x)f(x)

g(x) = f (x). Then we obtain the following result

g(qx)Dq  f (x) g(x)  + f(x) g(x)Dqg(x) = Dqf(x) ⇐⇒ Dq  f (x) g(x)  = g(x)Dqf(x) − f (x)Dqg(x) g(x)g(qx) .

As these derivations were done with general functions the results continue to be true when we later expand the set we are working on.

But for now we will just try and specify some other special functions that have special properties in q-calculus. We will develop untill we get Taylor polynomials in q-calculus. Obviously we have that x − a is in the space, so we can also assume that we have (x − a)n. If we want to have a q-version of Taylor polynomials we will need a q-version of this function. But we get

Dq((x − a)n) 6= n(x − a)n−1.

This implies that if we want to continue to develop a Taylor polynomial then we need an analogous alternative result for q-calculus.

Definition 9. We can easily define such a polynomial

(x − a)n_q= (x − a) (x − qa) . . . x − qn−1a
. This does satisfy the required result that

Dq (x − a)nq
= [n](x − a)n−1q .

This is easily proven by induction, the case when n = 1 it is obviously true. Assuming it works for some n and using the product rule we get

D_q (x − a)n+1_q
= Dq (x − qna)(x − a)nq
= (x − a)n_q+ q x − qn−1a
Dq (x − a)nq
= (x − a)n_q+ q x − qn−1a
[n] (x − a)n−1_q = (x − a)n_q+ q[n] (x − a)n_q = [n + 1] (x − a)n_q.

Now can we state the definition for a q-Taylor polynomial around a point c.

Definition 10. For any polynomial f (x) of degree N and any number c we have the following q-taylor expansion: g(x) = N

∑

i=0 Di_qf
(c)(x − c) i q [i]!

Now let us prove that this is in fact the same polynomial as the original function. Theorem 3. For any polynomial f (x) with q-taylor expansion g(x) then f (x) = g(x).

Proof. A polynomial f (x) of degree N can be represented by a sum ∑Ni=0ai(x − c)iq. Such that

f(x) =

N

∑

i=0

ai(x − c)iq.

If we use the Dqoperator on f (x) iterative and evaluate it in c then we get that ai=

(Di_qf)(c) [i]! .

We can extend this notion into formal power series, formal power series are generalizations of polynomials that allow for a countably infinite amount of terms. In general when working with formal power series we do not care about whether or not they converges. For any formal power series it can be expressed as a q-Taylor series around some point a

f(x) = ∞

∑

k=0 c_k(x − a)k= ∞

∑

k=0 Dk_qf(a)(x − a)k_q [k]! .

To see this it is the same argument as in classic calculus we just q-differentiate f (x) and set the function to a. Then we end up after doing this k-times with Dk_qf(a) = c_k[k]! and this gives us a formula for c_k. Setting a = 0 gives us the q-version of the Maclaurin series, this is the version we will be using going forward.

From now we will be working in the set of formal power series with real coefficients. We could also expand this into Laurent series with a q-version version of the Cauchy formula[4].

From this we can define other functions outside of simple polynomials, basically every function that has a Taylor expansion in classic calculus has a q-version some of the more interesting are here:

ex_q= ∞

∑

k=0 xk [k]! sinq(x) = ∞

∑

k=0 (−1)k x 2k+1 [2k + 1]! cosq(x) = ∞

∑

k=0 (−1)k x 2k [2k]!.

We can of course see that when q → 1 these converge to the corresponding functions in classic calculus. We have the classic results that Dqexq= exqand

sinq(x) = eix_q − e−ix_q 2i cosq(x) = eix_q + e−ix_q 2

In most literature there is another definition for a version of the exponential function namely ex_1/q this induces its own version of the trigonometric functions and have its own results. We get ex_qe−x_1/q= 1 so maybe 1/q has some more interesting relations. It is important to understand that the classic result ex+y = exeydoes not hold for the q-version of the function. We will also look at the the operator D_1/q as a complement to Dq and see if we get some useful results.

Later on we will see a generalisation of these results. First a result where we change what operator we use. Lemma 3. For a fixed q ∈ R \ {0, 1} we have that

Proof. The proof is as follows Dqf(x) = f(qx) − f (x) qx− x = f((qx)) − f ((qx)/q) (qx) − (qx)/q = D1/qf(qx)

Next is a result on how they behave when we use them together. It turns out that they q-commute.

Lemma 4. For a fixed q ∈ R \ {0, 1} we have that

D_1/qD_qf(x) = qD_qD_1/qf(x), (4.9) Proof. The proof is as follows

D_1/qDqf(x) = D1/q  f (qx) − f (x) qx− x  = = f (qx/q) − f (x/q) x− x/q − f(qx) − f (x) qx− x  1 x/q − x = q  −f(x) − f (x/q) x− x/q + f(qx) − f (x) qx− x  1 qx− x = qDq  f (x/q) − f (x) x/q − x  = qDqD1/qf(x).

These results will be important for the next section where we will use them to find results from a special version of q-differential equations.

4.2

Some results from q-differential equations

If we want to consider the behaviour of the q-differential operator on functions we are effect-ively exploring simple versions of differential equations. The simplest is

D_qf(x) = 0

which can be rewrited as f (qx) = f (x) for all x ∈ R \ {0} and q 6= 1 or −1. The solution to this is for formal power series only the constant function. This is motivated more thoroughly in the next chapter.

Next we consider the differential equation

τ (x)Dqf(x) = λ f (x).

Given a polynomial τ(x) of degree one or zero. Lets begin with when τ is of degree 0 then we take it to be one and

Because we only work with polynomial’s and formal power series and Dqreduces a

polyno-mials degree we can not have that f (x) is a polynomial. Then since we have f (x) = ∑∞

i=0aixiwe get D_qf(x) = ∞

∑

i=0 ai+1[i + 1]xi= λ ∞

∑

i=0 a_ixi.

This implies that ai+1= _[i+1]λ ai. This identifies the function up to multiplication with a

con-stant. If we choose a0= c then ai= cλ

i

[i]!and this is the formula for the function f (x) = c · e λ x q .

Next we have the case when τ(x) has degree one as then we can have f (x) as a polynomial as well. We can consider τ(x) = x + b if the leading coefficient is not one then we can simply divide with it throughout. For a polynomial f (x) = ∑Ni=0aixiwe would get

(x + b)Dq N

∑

i=0 a_ixi ! = λ N

∑

i=0 a_ixi (4.10) (x + b) N−1

∑

i=0 ai+1[i + 1]xi= λ N

∑

i=0 aixi (4.11) N

∑

i=1 a_i[i]xi+ b N−1

∑

i=0 a_i+1[i + 1]xi= λ N

∑

i=0 a_ixi (4.12)

This relation defines the solution to the equation up to multiplication with a constant. We might also have a solution that is a formal power series but we will leave that for now. The next step is to consider q-differential equations of higher degrees but we will use D_1/qtogether with Dqto represent second degree differential equations.

Until now we have been looking at explicit solutions to differential equations. Now we will be looking at relations between differential equations and solutions. So let us now consider the q-differential equation

F_qy(x) ≡ σ (x)DqD1/qy(x) + τ(x)Dqy(x) + λ y(x) = 0, (4.13)

where σ (x) and τ(x) are given polynomials. Then if it has a polynomial solution y(x) = Pn(x)

of degree n > 0, then it follows that σ (x) and τ(x) have to be polynomials of at most the second and first degree respectively. This follows from the fact that Dqand D1/q reduce the

degree of the polynomial y(x) and thus so that there is a possibility of cancellation this has to be true. This following theorem is based of the work in [3].

Theorem 4. Let Pn(x) be a solution to 4.13. Then the polynomial vk(x) = DkqPn(x) for k ∈

(1, . . . , n) solves the q-differential equation

Fq,n(k)vk≡ σ (x)DqD1/qvk(x) + τk(x)Dqvk(x) + µkvk(x) = 0, (4.14)

τk(x) = qkτ (qkx) + k−1

∑

j=0 qj Dqσ
(qjx) = qkτ (qkx) +σ (x) − σ (xq k₎ x(1 − q) , µk(x) = qk µ0+ k

∑

j=1 qj D_qτk− j
(x). (4.15)

Proof. The proof is based on induction. For k = 0 this is stated to be a solution in the present-ation of the theorem. Now suppose it is true for some k: 0 < k < n. By using Lemma 4, we can rewrite equation 4.14 such that

σ (x)D_1/qvk+1(x) + qτk(x)vk+1(x) + qµkvk(x) = 0.

Applying the q-derivative to this equation we get the following setup D_qσ (x)D_1/qvk+1(x)  + qDq  τk(x)vk+1(x)  + qµkDqvk(x) = 0.

Using the q-product rule from Lemma 2 then we get σ (x) · DqD1/qvk+1(x) + Dqσ (x) · D1/qvk+1(qx)

+ qD_qτk(x) · vk+1(x) + τk(qx) · Dqvk+1(x)



+ qµkvk+1(x) = 0.

According to Lemma 3, we have D_1/qvk+1(qx) = Dqvk+1(x). This together with

rearrange-ment of terms gives us

Fq,n(k+1)vk+1≡ σ (x)DqD1/qvk+1(x) + τk+1(x)Dqvk+1(x) + µk+1vk+1(x) = 0,

where τk+1 and µk+1are connected with τk and µkby the relations

τk(x) = qτk−1(qx) + Dqσ (x), µk= q



µk−1+ Dqτk−1(x)



. (4.16)

Using this recursive relation we can by iterating it find a closed formula for τk(x) and µk

for k ∈ (0, . . . n) τk(x) = qkτ (qkx) + k−1

∑

j=0 qj D_qσ
(qjx) = qkτ (qkx) +σ (x) − σ (xq k₎ x(1 − q) , µk(x) = qk µ0+ k

∑

j=1 qj Dqτk− j
(x). (4.17)

We will return to equations of this type later both when talking about Anti-derivative and Integration in q-calculus in Chapter 5 but also when discussing generalisations in Chapter 6.

We will now look at a special case of equation 4.13. Suppose there exists a function ρ(x), defined on I ⊂ R, such that the function y(x), which is a solution of equation 4.13, also satisfies the equation D_q  σ (x)ρ (x)D_1/qy(x)  + λ ρ(x)y(x) = 0. (4.18)

By the product rule for Dq, we get

σ (x)ρ (x)DqD1/qy(x) + D1/qy(qx) · Dq



σ (x)ρ (x) 

+ λ ρ(x)y(x) = 0. Using Lemma 3, we have D_1/qy(qx) = Dqy(x), from which we can write

ρ (x)  σ (x)DqD1/qy(x) + Dqy(x) Dq(σ (x)ρ(x)) ρ (x) + λ y(x)  = 0. (4.19) By comparing equations 4.13 and 4.19, we can see that these equations are equivalent if and only if the q-differential equation for ρ(x) is valid:

Dq



σ (x)ρ (x) 

= τ(x)ρ(x) (4.20)

This q-differential equation is called the Pearson q-differential equation. If we write it in the form

ρ (qx) ρ (x) =

σ (x) − (1 − q)x τ (x)

σ (qx) , (4.21)

we find a relation which determines the function ρ(x).

In the same manner as for Theorem 4, the function ρk(x) can be evaluated by

ρ0= ρ(x),

ρk(qx)

ρk(x)

= σ (x) − (1 − q)x τk(x)

σ (qx) (k ∈ N0), (4.22) such that the following relation is satisfied:

Dq



σ (x)ρk(x)D1/qvk(x)



+ µkρk(x)vk(x) = 0 (k ∈ N0). (4.23)

This can be rewritten as σ (qx)ρk(qx) ρk(x) =σ (x) − (1 − q) x τk(x) =σ (x) − (1 − q) x qτk−1(qx) + Dqσ (x)
=σ (x) − (1 − q) x (qτk−1(qx)) − σ (x) + σ (qx) =σ (qx) − (1 − q) qx τk−1(qx) =σ (q2x)ρk−1(q 2_x) ρk−1(qx) . So we have that, writing it more clearly

κ (x) ≡ ρn(x) ρn−1(qx) σ (qx)

= ρn(qx)

ρn−1(q2x) σ (q2x) (∀x ∈ I ⊂ R).

We can normalize ρ(x) choosing κ(x) ≡ 1, wherefrom

ρn(x) = ρn−1(qx) σ (qx) (n = 1, 2, . . .).

Repeating this relation, we conclude that

ρn(x) = ρ(qnx) n

∏

j=1

σ (qjx). (4.24)

This give us a formula for finding ρk(x) if the original equation has a ρ(x). This can be

Chapter 5

q-Antiderivative and q-Integration

5.1

q-Antiderivative

Since we have defined a q-derivative it would be natural to try and define a q-antiderivative. Definition 11. The function F(x) is a q-antiderivative of f (x) if DqF(x) = f (x) and we

rep-resent it by Z f(x)dqx or more compact Z q f(x).

As it is with normal calculus this q-antiderivative is not unique since if we add a function such that Dqφ (x) = 0 then that is also a q-antiderivative to the same function. Since Dqφ (x) =

0 if and only if φ (qx) = φ (x) and we have only worked with formal power series this implies that qnc_n= cn for each n ≥ 1 and all x in the power series. For q 6= −1 this means that φ

is constant. So working only within power series f (x) has a unique q-antiderivative up to an addition with a constant.

Finding the antiderivative is an operator which maps formal power series to formal power series modulo constants. It preserves the linearity which it inherits from the q-derivative. Lemma 5. If F(x) =R qf(x) and G(x) = R qg(x) then Z q (a f (x) + b f (x)) = a Z q f(x) + b Z q f(x).

Proof. We start with

D_q(aF(x) + bG(x)) = aDqF(x) + bDqG(x) = (a f (x) + bg(x)) .

Now we find the antiderivative for both sides of this equation aF(x) + bG(x) =

Z

q

This makes it so we can take each term of a polynomial separately and take the antideriv-ative. So let us look at the special case xn. We know that Dq(xn) = [n]xn−1 so since [n] is a

constant we get thatR

qxn= 1 [n]x

n+1_{+ c for some constant c.}

These results gives us a closed formula for antiderivative of a polynomial. The polynomial f(x) = ∑n_k=0a_kxkhas the antiderivative

Z q f(x) = c + n+1

∑

k=1 a_k−1 [k] x k_.

This is essential for the next section where we use this to expand on the q-differential equation.

5.2

A return to q-differential equations

If we again return to equation 4.13 we can show, using the results from the previous section, that:

Theorem 5. If y(x) = Pn(x) is a polynomial of degree n such that it is a solution to the equation

F_qy(x) ≡ σ (x)DqD1/qy(x) + τ(x)Dqy(x) + λq,ny(x) = 0, (5.1)

where, as before, σ and τ are polynomials then there exists a constant c such thatR

qy(x) is a

solution to the similar equation F_q? Z q y(x) ≡ σ (x)DqD1/q Z q y(x) + τ?(x)Dq Z q y(x) + λ_q? Z q y(x) = 0. Where τ?(x) = 1/qτ(x/q) − Dqσ (x/q), λ?= 1/qλ − Dqτ?(x). (5.2)

Proof. The proof is the reverse of what we did in the last chapter. Take and represent τ (x) = qτ?(qx) + Dqσ (x), λ = q



λ?+ Dqτ?(x)

 .

Substitute these into equation 5.1 and reversing the steps taken in the proof for Theorem 4. Then we end up with

D_q  σ (x)D_1/qy(x) + qτ?(x)y(x) + qλ? Z q y(x)  = 0. Taking the antiderivative of this we get

1/qσ (x)D_1/qy(x) + τ?(x)y(x) + λ?

Z

q

y(x) + d = 0 with d being an arbitrary constant. Changing y(x) for Dq

R

qy(x) give us the equation

σ (x)DqD1/q Z q y(x) + τ?(x)Dq Z q y(x) + λq? Z q y(x) + d = 0.

Putting d = 0 and using the fact that the constant part of the first two terms are unchanged by our choice of constant forR

qy(x), as it gets removed in the derivation. So then we can choose

our constant forR

qy(x) such that it is the negative of this constant scaled for λq?.

This makes it so that we have given a polynomial solution to a q-differential equation we have an infinite number of solutions to similar q-differential equations.

Since d is an arbitrary constant each choice gives different q-differential equations, but to keep the same similarity as with equation 4.13 we have to choose it equal to zero. As for the antiderivative for y(x) since it is a polynomial we can use our closed formula and it is unique except for the constant part. But as stated in the proof the only term of the differential equation that uses the new constant is the term λ?R

qy(x) and therefore we can use it as a way to correct

the constant part of the whole equation.

5.3

q-Integration

In the following sections we will work with arbitrary functions on R. By now we have used the notion of antiderivatives to prove some results, but in classic calculus the biggest use of antiderivatives is in the use of integrals. So let us move on to define integrals in q-calculus. By definition we have

f(x) = F(qx) − F(x) (q − 1)x =

1

(q − 1)x(Tq− 1)F(x) We can rewrite this such that we get a formula for the q-antiderivative

F(x) = 1 1 − Tq ((1 − q)x f (x)) = (1 − q) ∞

∑

k=0 T_qk(x f (x)) .

This gives after we apply the operator to the function

Z f(x)dqx= (1 − q)x ∞

∑

k=0 qkf(qkx). (5.3)

This is called the Jackson integral[1] and it is related to the Riemann integral in that when q→ 1 they converge but while the Riemann integral has equal width rectangles before taking the limit the Jackson integral has a geometric distribution of the width increasing the further from zero we get.

This series need not always converge to the q-antiderivative since we treated the operator as a number less than one so that we could use the formula for geometric series. The next theorem is a sufficient condition for the convergence of the Jackson integral.

Theorem 6. Suppose 0 < q < 1. If | f (x)xα_{| is bounded on the interval (0, A] for some 0 ≤}

α < 1, then the Jackson integral converges absolutely to a function F (x) on (0, A], which is a q-antiderivative of f(x).

Proof. If | f (x)xα_{| ≤ M on (0, A] then for any 0 < x ≤ A and j ≥ 0}

| f (qjx)| < M(qjx)−α⇒ |qjf(qjx)| < Mx−α q1−α
j.

Since 0 < q1−α< 1 we see that the serie converges by comparing it to a convergent geometric series. This shows that the serie converges pointwise to some function F(x).

The theorem also guarantees that F(x) is continuous at x = 0 with F(0) = 0 as seen in Theorem 19.1 in [1]. Finding a general integral is useful in its own right but as in classic calculus we sometimes want to specify a domain and get a specific result for this we need the definite integral.

Definition 12. Suppose 0 < a < b. Then the definite q-integral is defined as

Z b 0 f(x)dqx= (1 − q)b ∞

∑

j=0 qjf(qjb) Z b a f(x)dqx= Z b 0 f(x)dqx− Z a 0 f(x)dqx

Now that we have the definite integral it of course makes us again think about classic calculus and we wonder what we can do to find an improper q-integral. To define the improper q-integral we begin inspecting the definite integral

Z qj qj+1 f(x)dqx= Z qj 0 f(x)dqx− Z qj+1 0 f(x)dqx = (1 − q) ∞

∑

k=0 qj+kf(qj+k) − (1 − q) ∞

∑

k=0 qj+k+1f(qj+k+1) = (1 − q)qjf(qj).

We need this as just taking b → ∞ in the definite integral makes little sense. Because of this we instead use the previous definite integral to define the improper q-integral with the following formula.

Definition 13. The improper q-integral of f (x) on [0, ∞) is defined as follows

Z ∞ 0 f(x)dqx= ∞

∑

j=−∞ Z qj qj+1 f(x)dqx if 0 < q < 1, or Z ∞ 0 f(x)dqx= ∞

∑

j=−∞ Z qj+1 qj f(x)dqx if q > 1.

As before this makes us wonder when this integral converges as this is not really presented in the definition or the derivation.

Theorem 7. The improper q-integral we defined converges if xα_{f(x) is bounded in a}

neigh-borhood of x= 0 with some α < 1 and for sufficiently large x with some α > 1.

Proof. The proof is similar to that of the q-antiderivative convergence, for a detailed proof see [1] Proposition 19.1.

5.4

Fundamental theorem of q-calculus

All of this theory ultimately ends up in a familiar result from classic calculus namely the Fundamental Theorem of q-calculus.

Theorem 8 (Fundamental theorem of q-calculus). If F(x) is an antiderivative of f (x) and F(x) is continuous at x = 0 and 0 ≤ q ≤ 1, then we have

Z b

a

f(x)dqx= F(b) − F(a),

where0 ≤ a < b ≤ ∞.

Proof. Since F(x) is continuous at x=0 and an antiderivative of f (x) we have that the partial sum of equation 5.3 is (1 − q)x N

∑

j=0 qjf(qjx) = (1 − q)x N

∑

j=0 qjF(q j_{x) − F(q}j+1_x) (1 − q)qj_x = N

∑

j=0 F(qjx) − F(qj+1x) = F(x) − F(qN+1x)

Letting N → ∞ we approaches F(x) − F(0). So F(x) is given by the Jackson formula up to addition of a constant represented as such

F(x) = F(0) + (1 − q)x ∞

∑

j=0 qjf(qjx). Since by definition, Z a 0 f(x)dqx= (1 − q)a ∞

∑

j=0 qjf(qja) we get by substituting Z a 0 f(x)dqx= F(a) − F(0).

Using the same argument we get for finite b,

Z b

0

f(x)dqx= F(b) − F(0),

and because of that we get

Z b a f(x)dqx= Z b 0 f(x)dqx− Z a 0 f(x)dqx= F(b) − F(a).

Assigning a = qj+1and b = qj when 0 < q < 1 and consider the telescoping sum implied by the definition of the improper q-integral we see that the theorem is true for improper integrals if limx→∞F(x) exists.

Chapter 6

g-Calculus

6.1

Introduction

Now that we are familiar with q-calculus and we have h-calculus as the calculus of finite differences, one might wonder what happens if we combine them into qx + h or maybe what happens if instead of qx we have qxα_{. Could we then develop some kind of rules and relations}

for these new algebras? Instead of limiting ourselves to these special cases so let us explore the general case and assume that g(x) is an arbitrary function on R.

6.2

Algebra of g-calculus

In chapter 3 we discussed the algebra for the q-calculus and we discovered that the operators formed a q-deformed Heisenberg algebra. Let us now define a general version of those oper-ators and see if we can find any satisfactory relations. We will continue to work on functions defined on R.

Definition 14. If g(x) is an arbitrary function then we define T_g(x)( f (x)) = f (g(x)) as well as

D_g(x)( f (x)) = f(g(x)) − f (x) g(x) − x .

As mentioned before some special cases that exist include, for some positive whole number n, the following operators

D_q,n( f )(x) = f(qx n_{) − f (x)} qxn− x , T_q,n( f )(x) = f (qxn) Dq1,q2( f )(x) = f(q1x+ q2) − f (x) (q₁x+ q₂) − x T_q1,q2( f )(x) = f (q1x+ q2).

We will see that some of these have more results than others. Returning to the general case we can see, from the definition that we get

T_g(x)= (g(Mx) − Mx)Dg(x)+ I.

From inspection of the relation between Mxand Dg(x) and some manipulations it is possible to

determine that they satisfy a g(x)-deformed Heisenberg relation if g(x) is a polynomial. We will just verify this quickly

(DgMx− g(Mx)Dg)( f ) = Dg(x f (x)) − g(Mx)Dg( f (x)) =g(x) f (g(x)) − x f (x) g(x) − x − g(x) f(g(x)) − f (x) g(x) − x =−x f (x) + g(x) f (x) g(x) − x = f (x). This means that we have that

D_gM_x− g(Mx)Dg= I.

Using this we get a new representation of Tg that comes from substituting the identity in

T_g’s definition with the Heisenberg relation, this give us T_g=(g(M_x) − M_x)D_g(x)+ I

=(g(Mx) − Mx)Dg(x)+ DgMx− g(Mx)Dg

=DgMx− MxDg.

We found out that Tqand Mxq-commuted and using the same approach we can show that

T_gM_x= g(Mx)Tg. Or that Tgand Mxg(x)-commute. This we can easily verify as follows

T_g(M_x( f (x))) = T_g(x f (x)) = g(x) f (g(x)) = g(x)T_g( f (x)) = g(M_x)T_g( f (x)).

Now we might presume that Tgand Dgalso would g(x)-commute but this is not necessarily

the case. The relation between Tgand Dgthat we found was

g(g(x)) − g(x) g(x) − x TgDg= DgTg or equivalently  g2_{(x) − x} g(x) − x − 1  TgDg= DgTg.

This comes from the fact that

DgTgf(x) = Dgf(g(x)) = f(g2(x)) − f (g(x)) g(x) − x T_gD_gf(x) = Tg f(g(x)) − f (x) g(x) − x = f(g2(x)) − f (g(x)) g2_{(x) − g(x)}

here the nominators are the same so we just set them equal. This is how the operators behave with each other. These results are based on g(x) being a polynomial and this is so that putting operators inside of g(x) would make sense.

6.3

Differentiation in g-calculus

For the analytics of the g(x)-differential operator we will explore some results similar to chapter 3. We begin with linearity.

Lemma 6. The operator Dgis linear, that is,

D_g(x)(a f (x) + bh(x)) = aD_g(x)( f (x)) + bD_g(x)(h(x)). Proof. Linearity follows from

D_g(x)(a f (x) + bh(x)) =a f(g(x)) + bh(g(x)) − a f (x) − bh(x) g(x) − x =af(g(x)) − f (x) g(x) − x + b h(g(x)) − h(x) g(x) − x =aD_g(x)( f (x)) + bD_g(x)(h(x)).

We also have a version of the Leibniz rule for Dg.

Lemma 7. The product rule for Dgis

D_g(h(x) f (x)) = h(g(x))D_g( f (x)) + f (x)D_g(h(x)). (6.1) Or the equivalent

D_g(h(x) f (x)) = h(x)Dg( f (x)) + f (g(x))Dg(h(x)).

Proof. The proof follows as such

D_g(h(x) f (x)) =h(g(x)) f (g(x)) − h(x) f (x) g(x) − x =h(g(x)) f (g(x)) − h(g(x)) f (x) + h(g(x)) f (x) − h(x) f (x) g(x) − x =h(g(x)) f (g(x)) − h(g(x)) f (x) g(x) − x + h(g(x)) f (x) − h(x) f (x) g(x) − x =h(g(x))D_g( f (x)) + f (x)D_g(h(x)).

Similarly as for the q-variation the division rule for Dgis then obtained by using _h(x)f(x)h(x)

giving the following result

h(g(x))D_g f (x) h(x)  + f(x) h(x)Dgh(x) = Dgf(x) ⇐⇒ D_g f (x) h(x)  = h(x)Dgf(x) − f (x)Dgh(x) h(x)h(g(x)) .

Proposition 2. If h(g(x)) = β (h(x)) for some polynomial β (x) then D_g(x)( f (h(x))) = D_{β (h)}f(h) · D_g(x)h(x) Proof. The proof is straight forward

D_g(x)( f (h(x))) = f(h(g(x))) − f (h(x)) g(x) − x = f(h(g(x))) − f (h(x)) h(g(x)) − h(x) h(g(x)) − h(x) g(x) − x = f(β (h(x))) − f (h(x)) β (h(x)) − h(x) h(g(x)) − h(x) g(x) − x = D_{β (h)}f(h) · D_g(x)h(x)

So now the problem is to find such a polynomial β given polynomials g and h. We will see how we might find this later. Going forward we will explore some results that require certain restrictions on our functions g(x) and h(x).

Lemma 8. If h ◦ g(x) = g ◦ h(x) = x then D_gD_hf(x) = α(x)DhDgf(x) with α (x) = −h(x) − x g(x) − x. Proof. DgDhf(x) = Dg  f (h(x)) − f (x) h(x) − x  = f (h(g(x))) − f (g(x)) h(g(x)) − g(x) − f(h(x)) − f (x) h(x) − x  1 g(x) − x = f (x) − f (g(x)) x− g(x) − f(h(x)) − f (x) h(x) − x  1 g(x) − x = −h(x) − x g(x) − x  f (h(x)) − f (x) h(x) − x − f(g(x)) − f (x) g(x) − x  1 h(x) − x = −h(x) − x g(x) − xDh  f (g(x)) − f (x) g(x) − x  = −h(x) − x g(x) − xDhDgf(x) = α(x)DhDgf(x)

This Lemma was the first time we required something that will be used frequently going forward namely h ◦ g(x) = g ◦ h(x) = x or that h(x) is the inverse of g(x). If we require both to be polynomials the only time this can happen is if both are linear and then they satisfy if g(x) = ax + b that h(x) =x−b_a and α(x) = a. Another result is that we can change the function we differentiate with by exchanging it with its inverse.

Proposition 3. If h ◦ g(x) = g ◦ h(x) = x then

D_gf(x) = Dhf(g(x)).

Proof. The proof is as follows

D_gf(x) = f(g(x)) − f (x) g(x) − x

= f(g(x)) − f (g(h(x)) g(x) − g(h(x)) = Dhf(g(x)).

Looking only at the analytics of this section we see that all results except Proposition 2 are satisfied for arbitrary functions g(x) and h(x). Going forward we will ignore our requirement that g(x) and h(x) are polynomials and just think of them as arbitrary functions. This is done as to keep the theory interesting. For example a different version of Proposition 2.

Proposition 4. If h(x) is an invertible function on some intervall I and β (x) = h ◦ g ◦ h−1(x) then

D_g(x)( f (h(x))) = D_{β (h)}f(h) · D_g(x)h(x) on that intervall I.

Proof. The proof is straight forward

D_g(x)( f (h(x))) = f(h(g(x))) − f (h(x)) g(x) − x = f(h(g(x))) − f (h(x)) h(g(x)) − h(x) h(g(x)) − h(x) g(x) − x = f(h(g(h −1_{(h(x))))) − f (h(x))} h(g(h−1(h(x)))) − h(x) h(g(x)) − h(x) g(x) − x = f(β (h(x))) − f (h(x)) β (h(x)) − h(x) h(g(x)) − h(x) g(x) − x = D_{β (h)}f(h) · D_g(x)h(x)

6.4

An attempt at g-combinatorics

For a general version of a Taylor series we run into problems early as there are not really any general version of q-numbers and therefore no generalised q-combinatorics and no factorial. We can see why by exploring a motivation we used earlier namely

D_g(xn) = (g(x)) n_{− x}n g(x) − x = (g(x))n xn − 1 g(x) x − 1 xn−1 = _g(x) x n − 1 g(x) x − 1 xn−1 = _g(x) x − 1  ∑n−1_k=0 _g(x) x k g(x) x − 1 xn−1 = n−1

∑

k=0  g(x) x k! xn−1. So then n−1

∑

k=0  g(x) x k

is some kind of equivalent to q-numbers for this operator but it is not generally independent of x. This same problem occurs for h-calculus but when h → 0 it still converges. For finite differences it is often also just assumed that the Taylor series exist and work and therefore no special series was ever developed. From this we can see that as g(x) → x then the sum converges to n.

6.5

Antiderivative and Integration in g-calculus

Now that we have discussed general g-derivatives could we maybe formulate some version of a general integral. We did this for the q-calculus so if we use the same approach we should get some result.

Definition 15. Let us define a g-Antiderivative such that for a F(x) if D_g(F(x)) = f (x) then F(x) is a g-Antiderivative to f (x) denoted

F(x) =

Z

f(x)dgx.

By this definition we have that

f(x) = Dg(F(x)) =

F(g(x)) − F(x) g(x) − x

Let us heuristically just divide by Tg− 1 the same way as we did in chapter 5 and see what we get F(x) = 1 Tg− 1 f(x) (g(x) − x) = ∞

∑

k=0 T_gk( f (x)(x − g(x))) = ∞

∑

k=0 f(gk(x))(gk(x) − gk+1(x)).

We define this as the basis for the general integral. We can not really set it up like we did with the q-integral as the behaviour of gk(x) − gk−1(x) is not predictable. This definition gives us for g(x) = x + h the Riemann Integral and for g(x) = qx the Jackson Integral. The definition for these two integrals is slightly different and made to be more intuitive. This can be seen by the integrals approaching from different sides, as for the Jackson integral we let the sum go towards zero with its terms but for the Riemann integral we go away from zero. The behaviour of g(x) would be interesting in how one would define its integral as it can even jump around on the real line. The same property is what makes general versions of things like the mean value theorems unfeasible without very strict restrictions on g(x)[9].

6.6

g-differential equations

Here we discuss how the results from our q-differential equations translates into a more general setting. Let us consider the g-differential equation where h(x) is such that g ◦ h(x) = x

F_g,hy(x) ≡ σ (x)DgDhy(x) + τ(x)Dgy(x) + λ y(x) = 0, (6.2)

where σ (x) and τ(x) are given functions. We loose the interaction of degrees on σ and τ we have from Theorem 4 by not restricting us to having g and h as polynomials.

Theorem 9. Let f (x) be a solution to 6.2 and τ(x) is a polynomial with deg(τ) = 1. If h◦ g(x) = g ◦ h(x) = x then α(x) = −h(x)−x_g(x)−x and the function v(x) = Dgf(x) satisfies the

gh-differential equation

F_g,h? v(x) ≡ σ?(x)D_gD_hv(x) + τ?(x)D_gv(x) + µv(x) = 0, (6.3) where the leading functions are connected by

σ?(x) = σ (x) · α(x), τ?(x) = τ(g(x)) + Dg(σ (x)α(x)) , µ = λ + Dgτ (x). (6.4)

Proof. We will prove it in a similar way as we did with Theorem 4. We begin with the original differential equation 6.2

Then we get by changing Dgf(x) = v(x) the following equation. We use Lemma 8 to

commute Dhand Dg

σ (x)α (x)Dhv(x) + τ(x)v(x) + λ f (x) = 0.

Applying the g-derivative on this equation next, we get Dg  σ (x)α (x)D_hv(x)  + Dg  τ (x)v(x)  + λ Dgf(x) = 0,

from which we deduce using the product rule for g-differentials σ (x)α (x) · DgDhv(x) + Dg(σ (x)α(x)) · Dhv(g(x))

+D_gτ (x) · v(x) + τk(g(x)) · Dgv(x)



+ λ v(x) = 0. Rearrangement of terms gives

F_g,h? v(x) ≡ σ?(x)DgDhv(x) + τ?(x)Dgv(x) + µv(x) = 0,

If we do decide we require g(x) and h(x) to be a polynomials then we have as stated earlier that g(x) = ax + b and h(x) =x−b_a . It follows from this that

α (x) = −h(x) − x g(x) − x = a.

This also makes it so that τ and σ have to be polynomials of degree 1 and 2 respectively and we have a recursive relation very similar to Theorem 4.

In Theorem 9 we require τ to be of degree 1 otherwise µ is not a constant but a function. For the same reason we have to change σ , in Theorem 4 we divide by α since it is a constant and this makes it so we can keep the same σ even when iterating Theorem 9 multiple times.

Chapter 7

Conclusions

7.1

Results

In this paper we have explored q-calculus by building it from the ground up by using some basic operators. We begin with defining the notion of q-numbers, specific numbers gained from the reals by scaling. We explore how these numbers behave when the scaling is changed and the properties when we go outside the positive reals with q. The numbers are later shown to be integral parts of the relation between certain functions and operators on those functions. We also prove some special versions of combinatoric results for these new real numbers.

When building the setup we start with the reals and use an operator to extend it to the set of polynomials, this is our set of functions we work on with our operators. Later we further extend it to a more general set of functions by first extending the set to all formal power series. We show that these operators when operating on this set of functions satisfy a q-deformed Heisenberg algebra.

Exploring the analytic properties of one of the operators specifically we then develop a q-generalisation of the Taylor series. Therefore we can generalise any analytic function in a neighbourhood. We also defined and proved results about integration among them the Funda-mental Theorem of q-calculus and the explicit formula for Jackson Integrals.

The most interesting part is in chapter 6 where we explore the structure of q-calculus but with general functions g(x). Finding generalisations for many results and presenting what the limitations are.

7.2

Further research

For further research there exist areas that are developed in q-calculus that we have not covered that could be extended into the generalisation of the theory. As an example it could be extended to consider complex valued functions. An area that was considered but not included in this thesis was that of a variant of the mean value theorem. This theorem can be extended into the domain of q-calculus but further extension to the generalised g-calculus makes it so that

strict rules are applied to the function g(x). This is so that g(x) stays in a neighbourhood when iterated on itself. More similar results from q-calculus could be explored in the general context.

Bibliography

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