Theory and Modern Applications

# Generalized q-Taylor formulas

## Abstract

In this paper, new generalized q-Taylor formulas involving both Riemann-Liouville and Caputo q-difference operators are constructed. Some applications with solutions of fractional q-difference equations are also given.

## 1 Introduction

A q-analogue of Taylor series was introduced by Jackson [1]:

$$f(x)=\sum_{n=0}^{\infty}\frac{(1-q)^{n}}{(q;q)_{n}}D_{q}^{n}f(a) [x-a]_{n},$$
(1.1)

where $$0< q<1$$, $$D_{q}$$ is the q-derivative, and

$$[x-a]_{n}:= (x-a) (x-qa)\cdots \bigl(x-q^{n-1}a \bigr),\quad n \geqslant1, [x-a]_{0}:=1.$$

Al-Salam and Verma [2] introduced the following q-interpolation series:

$$f(x)=\sum_{n=0}^{\infty}(-1)^{n}q^{-n(n-1)/2} \frac {(1-q)^{n}}{(q;q)_{n}}D_{q}^{n}f\bigl(aq^{-n}\bigr) [x-a]_{n}.$$
(1.2)

Al-Salam and Verma gave only formal proofs for (1.2); see [1, 2]. Analytic proofs of (1.1) and (1.2) were given in [3].

Results of generalized Taylor formulas involving the classical fractional derivative may be found in [4, 5]. In [5], a generalized Taylor formula involving the classical Riemann-Liouville fractional derivative of order α is deduced, whereas the generalized Taylor formula in [4] contains Caputo fractional derivative of order α, where $$0<\alpha\leq1$$.

In [6], a q-Taylor formula in terms of Riemann-Liouville fractional q-derivative $$D_{q,a}^{\alpha}$$ of order α is obtained. This result can be stated as follows.

### Theorem A

([6])

Let f be a function defined on $$(0,b)$$ and $$\alpha\in(0,1)$$. Then f can be expanded in the form

\begin{aligned} f(x) =&\sum_{k=0}^{n-1} \frac{(D_{q,a}^{\alpha+k}f)(c)}{ \Gamma_{q}(\alpha+k+1)}(x-c)^{(\alpha+k)} \\ &{}+\frac{1}{\Gamma_{q}(\alpha)} \int_{a}^{c} (x-c)^{(\alpha-1)} \bigl(D_{q,a}^{\alpha}f\bigr) (t) \,d_{q}t \\ &{} -K(a) (x-c)^{(\alpha-1)}+\bigl(I_{q,c}^{\alpha+n}D_{q,a}^{\alpha+n}f \bigr) (x), \end{aligned}
(1.3)

where $$0< a< c< x< b$$, and $$K(a)$$ does not depend on x.

Also, in [7], a generalized q-Taylor formula in fractional q-calculus is established and used in deriving certain q-generating functions for the basic hyper-geometric functions.

In this paper, we give generalized Taylor formulas involving Riemann-Liouville fractional q-derivatives of order α and Caputo fractional q-derivatives of order α; see (4.3) and (4.4). We also give sufficient conditions that guarantee that the remainders of these formulas vanish to get infinite expansions.

In the following section, we give a brief account of the q-notations and notions that will be used throughout this paper. In Section 3, we give q-analogues of mean value theorems on $$[0,a]$$. In Section 4, we give generalized q-Taylor formulas involving both Riemann-Liouville fractional q-derivative and Caputo fractional q-derivative. Then conditions for infinite expansion for some functions are given. In the last section, we apply the obtained results in solving certain q-difference equations.

## 2 Notation and preliminaries

In the following, q is a positive number, $$q<1$$. We follow [8] for the definition of the q-shifted factorial, Jackson q-integral, q-derivative, q-gamma function $$\Gamma_{q}(z)$$, and q-beta function $$B_{q}(\alpha,\beta)$$. Also, we follow [9] for the definition of the q-derivative at zero and the q-regular at zero functions.

The following q-integral is useful and will be used in the sequel:

$$\int_{0}^{x}(qt/x;q)_{\beta-1}t^{\alpha-1} \,d_{q}t=x^{\alpha }B_{q}(\alpha,\beta),\quad \alpha, \beta,x>0;$$
(2.1)

it can be proved by setting $$\xi=t/x$$.

By $$L_{q}^{1}(0,a)$$, $$a>0$$, we mean the Banach space of all functions defined on $$(0,a]$$ such that

$$\Vert f\Vert := \int_{0}^{a}\bigl\vert f(t)\bigr\vert \,d_{q}t< \infty,$$
(2.2)

where two functions in $$L_{q}^{1}(0,a)$$ are considered to be the same function if they have the same values at the sequence $$\{aq^{n}\}_{n=0}^{\infty}$$.

Let $$\mathcal{L}_{q}^{1}(0,a)$$ denote the space of all functions f defined on $$(0,a]$$ such that $$f\in L_{q}^{1}(0,x)$$ for all $$x\in(qa,a]$$. The space $$\mathcal{AC}_{q}[0,a]$$ is the space of all functions f defined on $$[0,a]$$ such that f is q-regular at zero and

$$\sum_{j=0}^{\infty}\bigl\vert f \bigl(tq^{j}\bigr)-f\bigl(tq^{j+1}\bigr)\bigr\vert < \infty, \quad t\in (qa,a].$$
(2.3)

A characterization of the space $$\mathcal{AC}_{q}[0,a]$$ is given as follows (see [9]).

### Theorem B

Let f be a function defined on $$[0,a]$$. Then $$f\in\mathcal{AC}_{q}[0,a]$$ if and only if there exist a constant c and a function ϕ in $$\mathcal{L}_{q}^{1}[0,a]$$ such that

$$f\in \mathcal{AC}_{q}[0,a]\quad \Longleftrightarrow\quad f(x)=c+ \int_{0}^{x}\phi(u) \,d_{q}u,\quad x\in[0,a].$$
(2.4)

Moreover, c and ϕ are uniquely determined by $$c=f(0)$$ and $$\phi(x)=D_{q}f(x)$$ for all $$x\in(0,a]$$.

The Riemann-Liouville fractional q-integral operator is introduced in [10] by Al-Salam through

$$I_{q}^{\alpha}f(x):=\frac{x^{\alpha-1}}{\Gamma_{q}(\alpha)} \int _{0}^{x} (qt/x;q)_{\alpha-1}f(t) \,d_{q}t,\quad \alpha\notin \{-1,-2,\ldots\}.$$
(2.5)

In [6], the generalized Riemann-Liouville fractional q-integral operator for $$\alpha\in\mathbb{R^{+}}$$ is given as

$$I_{q,a}^{\alpha}f(x):=\frac{x^{\alpha-1}}{\Gamma_{q}(\alpha)} \int _{a}^{x} (qt/x;q)_{\alpha-1}f(t) \,d_{q}t.$$
(2.6)

Using the definition of the q-integral, (2.5) reduces to

$$I_{q}^{\alpha}f(x)=x^{\alpha}(1-q)^{\alpha} \sum_{n=0}^{\infty }q^{n} \frac{(q^{\alpha};q)_{n}}{(q;q)_{n}}f\bigl(xq^{n}\bigr),$$
(2.7)

which is valid for all α. For example,

$$I_{q}^{\alpha}x^{\beta-1}= \frac{\Gamma_{q}(\beta)}{\Gamma_{q}(\beta +\alpha)}x^{\alpha+\beta-1}.$$
(2.8)

This basic Riemann-Liouville fractional q-integral was also given later by Agarwal [11]. In the same paper, he introduced the following semigroup property:

$$I_{q}^{\alpha}I_{q}^{\beta}f(x)=I_{q}^{\beta}I_{q}^{\alpha}f(x)=I_{q}^{\alpha +\beta}f(x),\quad \alpha, \beta\geq0.$$
(2.9)

The generalized Riemann-Liouville fractional q-derivative is given in [6] by

$$D_{q,a}^{\alpha}f(x)=D_{q}I_{q,a}^{1-\alpha}f(x), \quad a\ge0,$$
(2.10)

and $$D_{q,0}^{\alpha}f(x)=D_{q}^{\alpha}f(x)$$. The Caputo fractional q-derivative of order α, $$0<\alpha\leq1$$, is (see[12])

$${}^{c}D_{q}^{\alpha}f(x):=I_{q}^{1-\alpha}D_{q}f(x).$$
(2.11)

Let $$\mathcal{AC}_{q}^{(k)}[0,a]$$, $$k\in N$$, be the space of all functions f defined on $$[0,a]$$ such that $$f, D_{q} f,\ldots, D^{k-1}_{q} f$$ are q-regular at zero and $$D^{k-1}_{q} f\in \mathcal{AC}_{q}[0,a]$$.

For $$\alpha> 0$$, let $$k = \ulcorner\alpha\urcorner$$, where is the ceiling function. Then the Riemann-Liouville fractional derivative $${D}_{q}^{\alpha}f(x)$$ exists if (see [9])

$$f\in\mathcal{L}_{q}^{1}[0;a],\quad I_{q}^{k-\alpha}D_{q}^{k}f \in\mathcal{AC}_{q}^{(k)}[0,a],$$

and $${}^{c}D_{q}^{\alpha}f(x)$$ exists if $$f\in\mathcal{AC}_{q}^{(k)}[0,a]$$.

The following results are proved in [12] for any $$\alpha>0$$; the result for the case $$0<\alpha<1$$ is introduced in the following theorems without proof.

### Theorem C

Assume that $$f\in\mathcal{L}_{q}^{1}[0;a]$$ and $$I_{q}^{1-\alpha}f\in\mathcal{AC}_{q}[0,a]$$, where $$0<\alpha<1$$. Then the Riemann-Liouville fractional derivative of order α, $$0<\alpha< 1$$, exists, and

$$I_{q}^{\alpha}{D}_{q}^{\alpha }f(x)=f(x)- \frac {I_{q}^{1-\alpha}f(0)}{\Gamma_{q}(\alpha)}x^{\alpha-1.}$$
(2.12)

### Theorem D

If $$f\in\mathcal{AC}_{q}[0,a]$$, then

$$I_{q}^{\alpha}{}^{c}D_{q}^{\alpha }f(x)=I_{q}D_{q}f(x)=f(x)-f(0)$$
(2.13)

for $$0<\alpha<1$$.

It is worth mentioning that the key point in the proofs of Theorems C and D is the q-integration by parts formula:

$$\int_{0}^{b}f(t)D_{q}g(t) \,d_{q}t= (fg) (b)-\lim_{n\to\infty }(fg) \bigl(bq^{n}\bigr)- \int_{0}^{b}D_{q}f(t) g(qt) \,d_{q}t.$$

Hence, if fg is q-regular at zero, then the limit on the right-hand side is nothing but $$(fg)(0)$$.

## 3 Generalized q-mean value theorems on $$[0,a]$$

In this section, we introduce two q-analogues of the mean value theorems. The first one is for q-integrals on an interval of the form $$[0,a]$$, and the second is a mean value theorem with both of Riemann-Liouville fractional q-derivative and Caputo fractional q-derivative on $$[0,a]$$. The first one can be stated as follows.

### Theorem 3.1

(Mean value theorem for q-integrals)

Let g be a continuous function defined on $$[0,a]$$, and h be a nonnegative function defined on $$[0,a]$$ and q-regular at zero. Then

$$\int_{0}^{a}g(t)h(t) \,d_{q}t=g(\xi) \int_{0}^{a}h(t) \,d_{q}t$$
(3.1)

for some $$\xi\in[0,a]$$.

### Proof

The proof is similar to the classical case (see [13], p.139) and is omitted. □

The derivations of the main results of this paper mainly depend on Theorem 3.1.

### Remark 3.2

1. (1)

We cannot replace the lower end point of the q-integrals in (3.1) by arbitrary nonzero number because the inequality

$$\biggl\vert \int_{c}^{a}f(t) \,d_{q}t \biggr\vert \le \int_{c}^{a} \bigl\vert f(t) \bigr\vert \,d_{q}t,$$

holds only for $$c\in\{0,aq^{n}, n\in\mathbb{N}_{0}\}$$. In this case, (3.1) is also true.

2. (2)

There are q-analogues of mean value theorems on $$[a,b]$$ in [14], but all these analogues are valid only for certain values of q. For example, one of the mean value theorems for q-integrals in [14] is the following:

Let f, g be continuous functions on $$[a,b]$$ . Then there exists $$\widehat{q}\in(0,1)$$ such that

$$\bigl(\forall q\in(\widehat{q},1)\bigr)\ \bigl(\exists\xi\in[a,b]\bigr):\quad \int_{a}^{b}g(t)f(t) \,d_{q}t=g(\xi) \int_{a}^{b}f(t) \,d_{q}t.$$

The second theorem is a q-analogue of the mean value theorem for derivative on $$[0,a]$$. Throughout the rest of this article, we assume that $$0<\alpha<1$$.

### Theorem 3.3

1. (1)

If $$f\in \mathcal{L}_{q}^{1}[0;a]$$, $$I_{q}^{1-\alpha}f\in\mathcal{AC}_{q}[0,a]$$, and $$x^{1-\alpha}D^{\alpha}_{q}f\in C[0,a]$$, then

$$f(x)=\frac{I_{q}^{1-\alpha}f(0)}{\Gamma_{q}(\alpha)} x^{\alpha-1} +\frac{{\Gamma_{q}(\alpha)} \xi^{1-\alpha} D^{\alpha}_{q}f(\xi)}{\Gamma_{q}(2\alpha)} x^{2\alpha-1}.$$
(3.2)
2. (2)

If $$f\in\mathcal{AC}_{q}[0,a]$$ and $${}^{c}D_{q}^{\alpha}f\in C[0,a]$$, then

$$f(x)=f(0)+\frac{{}^{c}D_{q}^{\alpha}f(\xi)}{\Gamma _{q}(\alpha)} x^{\alpha}$$
(3.3)

for some ξ lying in the interval $$[0,x]$$ and all $$x\in(0,a]$$.

### Proof

We first prove (3.2). Since (see [15], p.494)

$$B_{q}(\alpha,\beta)=\frac{\Gamma_{q}(\alpha)\Gamma_{q}(\beta)}{\Gamma _{q}(\alpha+\beta)},$$

from (2.5), Theorem 3.1, and (2.1) we get

\begin{aligned} I_{q}^{\alpha}D^{\alpha}_{q}f(x) =&\frac{x^{\alpha-1}}{\Gamma_{q}(\alpha)} \int _{0}^{x}(qt/x;q)_{\alpha-1} t^{\alpha-1} t^{1-\alpha}D^{\alpha}_{q}f(t) \,d_{q}t \\ =&\frac{x^{\alpha-1}}{\Gamma_{q}(\alpha)} \xi^{1-\alpha}D^{\alpha}_{q}(\xi) \int_{0}^{x}(qt/x;q)_{\alpha-1} t^{\alpha-1} \,d_{q}t \\ =&\frac{{\Gamma_{q}(\alpha)} \xi^{1-\alpha} D^{\alpha}_{q}f(\xi)}{\Gamma_{q}(2\alpha)} x^{2\alpha-1} \end{aligned}

for $$0\leq\xi\leq x$$. Hence, (3.2) follows from (2.12). Similarly, using (2.13), we can prove (3.3). □

## 4 Generalized q-Taylor formula

In this section, we introduce generalized q-Taylor formulas for functions in terms of the sequential Riemann-Liouville q-derivative and the sequential Caputo fractional q-derivatives, where the sequential Riemann-Liouville q-derivative $${\mathcal{D}}_{q}^{n\alpha}$$ and Caputo fractional q-derivative $${}^{c}\mathcal{D}_{q}^{n\alpha}$$, $$n\in\mathbb{N}$$, are

$${\mathcal{D}}_{q}^{n\alpha}={D}_{q}^{\alpha} \cdots{D}_{q}^{\alpha } \quad \mbox{and}\quad {}^{c} \mathcal{D}_{q}^{n\alpha}={}^{c}D_{q}^{\alpha} \cdots {}^{c}D_{q}^{\alpha} \quad \mbox{(n times)},$$

respectively. The following lemma is important to get these formulas.

### Lemma 4.1

1. (1)

If $${\mathcal{D}}_{q}^{k\alpha}f\in\mathcal{L}_{q}^{1}[0,a]$$ and $$I_{q}^{1-\alpha}{\mathcal{D}}_{q}^{k\alpha}f\in\mathcal{AC}_{q}[0,a]$$, $$k=0,1,\ldots,n$$, then

$$I_{q}^{n\alpha}{\mathcal{D}}_{q}^{n\alpha}f(x)-I_{q}^{(n+1)\alpha }{ \mathcal{D}}_{q}^{(n+1)\alpha}f(x)=\frac{I_{q}^{1-\alpha}{\mathcal {D}}_{q}^{n\alpha}f(0)}{ \Gamma_{q}((n+1)\alpha)}x^{(n+1)\alpha-1}.$$
(4.1)
2. (2)

If $${}^{c}\mathcal{D}_{q}^{k\alpha}f \in\mathcal{AC}_{q}[0,a]$$, $$k=0,1,\ldots,n$$, then

$$I_{q}^{n\alpha}{}^{c} \mathcal{D}_{q}^{n\alpha}f(x)-I_{q}^{(n+1)\alpha }{}^{c} \mathcal{D}_{q}^{(n+1)\alpha }f(x)=\frac{{}^{c}\mathcal{D}_{q}^{n\alpha}f(0)}{ \Gamma_{q}(n\alpha+1)}x^{n\alpha}.$$
(4.2)

### Proof

We give a proof of (4.1), and the proof of (4.2) can be obtained similarly. Applying (2.12) and (2.8), we obtain

\begin{aligned} I_{q}^{n\alpha}{\mathcal{D}}_{q}^{n\alpha}f(x)-I_{q}^{(n+1)\alpha }{ \mathcal{D}}_{q}^{(n+1)\alpha}f(x) =&I_{q}^{n\alpha} \bigl({\mathcal{D}}_{q}^{n\alpha}f(x)-I_{q}^{\alpha }{D}_{q}^{\alpha} \bigl({\mathcal{D}}_{q}^{n\alpha}f(x)\bigr)\bigr) \\ =&I_{q}^{n\alpha} \biggl(\frac{I_{q}^{1-\alpha}{\mathcal{D}}_{q}^{n\alpha }f(0)}{\Gamma_{q}(\alpha)} x^{\alpha-1} \biggr) \\ =&\frac{I_{q}^{1-\alpha}{\mathcal{D}}_{q}^{n\alpha}f(0)}{\Gamma _{q}(\alpha)} I_{q}^{n\alpha}\bigl(x^{\alpha-1}\bigr) \\ =&\frac{I_{q}^{1-\alpha}{\mathcal{D}}_{q}^{n\alpha}f(0)}{\Gamma _{q}((n+1)\alpha)} x^{(n+1)\alpha-1}, \end{aligned}

and the lemma follows. □

### Theorem 4.2

(Generalized q-Taylor formulas)

1. (1)

Suppose that $${\mathcal{D}}_{q}^{k\alpha}f\in \mathcal{L}_{q}^{1}[0,a]$$, $$I_{q}^{1-\alpha}{\mathcal{D}}_{q}^{k\alpha}f\in \mathcal{AC}_{q}[0,a]$$, $$k=0,1,\ldots,n-1$$, and $$x^{1-\alpha}{\mathcal{D}}_{q}^{n\alpha}f\in C[0,a]$$. Then

$$f(x)=\sum_{k=1}^{n-1} \frac{I_{q}^{1-\alpha}{\mathcal{D}}_{q}^{k\alpha}f(0)}{ \Gamma_{q}((k+1)\alpha)}x^{(k+1)\alpha-1} +\frac{\Gamma_{q}(\alpha) \xi^{1-\alpha} {\mathcal {D}}_{q}^{n\alpha}f(\xi)}{\Gamma_{q}((n+1)\alpha)} x^{(n+1)\alpha-1}.$$
(4.3)
2. (2)

Suppose that $${}^{c}{\mathcal{D}}_{q}^{k\alpha}f \in\mathcal{AC}_{q}[0,a]$$, $$k=0,1,\ldots,n-1$$, and $${}^{c}D^{n\alpha}_{q}f\in C[0,a]$$. Thus,

$$f(x)=\sum_{k=0}^{n-1} \frac{{}^{c}\mathcal{D}_{q}^{k\alpha }f(0)}{\Gamma_{q}(k\alpha +1)}x^{k\alpha} +\frac{{}^{c}\mathcal{D}_{q}^{n\alpha}f(\xi)}{\Gamma_{q}(n\alpha +1)}x^{n\alpha},$$
(4.4)

where $$0\leq\xi\leq x$$.

### Proof

For (4.3), applying (4.1), we obtain

\begin{aligned}& \sum_{k=0}^{n-1} \bigl[I_{q}^{k\alpha}{ \mathcal{D}}_{q}^{k\alpha }f(x)-I_{q}^{(k+1)\alpha}{ \mathcal{D}}_{q}^{(k+1)\alpha}f(x) \bigr] \\& \quad =\sum _{k=0}^{n-1}\frac{I_{q}^{1-\alpha}{\mathcal{D}}_{q}^{k\alpha}f(0)}{ \Gamma_{q}((k+1)\alpha)}x^{(k+1)\alpha-1}, \end{aligned}
(4.5)

that is,

$$f(x)=\sum_{k=0}^{n-1} \frac{I_{q}^{1-\alpha}{\mathcal{D}}_{q}^{k\alpha}f(0)}{ \Gamma_{q}((k+1)\alpha)}x^{(k+1)\alpha-1}+I_{q}^{n\alpha}{\mathcal {D}}_{q}^{n\alpha}f(x).$$
(4.6)

Applying the q-integral mean value theorem and (2.8) yield

\begin{aligned} I_{q}^{n\alpha}{\mathcal{D}}_{q}^{n\alpha}f(x) =& \frac{x^{n\alpha -1}}{\Gamma_{q}(n\alpha)} \int_{0}^{x}(qt/x;q)_{n\alpha-1} t^{\alpha-1} t^{1-\alpha }{\mathcal{D}}_{q}^{n\alpha}f(t) \,d_{q}t \\ =&\frac{x^{n\alpha-1}}{\Gamma_{q}(n\alpha)} \xi^{1-\alpha }{\mathcal{D}}_{q}^{n\alpha}f( \xi) \int_{0}^{x}(qt/x;q)_{n\alpha-1} t^{\alpha-1} \,d_{q}t \\ =&\frac{\Gamma_{q}(\alpha) \xi^{1-\alpha} {\mathcal {D}}_{q}^{n\alpha}f(\xi)}{\Gamma_{q}((n+1)\alpha)} x^{(n+1)\alpha-1} \end{aligned}
(4.7)

for some $$\xi\in[0,x]$$. Combining (4.6) and (4.7) yields (4.3).

By using (4.2), (4.4) can be treated similarly. □

A natural question arises: can we expand a function f in terms of q-fractional derivatives? That is,

$$f(x)=x^{\alpha-1}\sum_{k=0}^{\infty}c_{k}x^{k\alpha} \quad \mbox{or}\quad f(x)=\sum_{k=0}^{\infty}c_{k}x^{k\alpha}?$$

The following theorem gives the answer for such expansions with sufficient conditions for the uniform convergence.

### Theorem 4.3

Assume that $$f\in\mathcal{L}_{q}^{1}[0,a]$$ and $$x^{1-\alpha}{\mathcal{D}}_{q}^{n\alpha}f\in C[0,a]$$ for all $$n\in\mathbb{N}$$. If

$$\bigl\vert x^{1-\alpha}{\mathcal{D}}_{q}^{n\alpha}f(x) \bigr\vert \le c A^{n\alpha}, \quad \forall x\in[0,a], n\in\mathbb{N},$$

where c is a positive constant, and A is a positive number satisfying $$A<\frac{1}{a(1-q)}$$, then f has the expansion

$$f(x)=\sum_{k=0}^{\infty } \frac{I_{q}^{1-\alpha}{\mathcal{D}}_{q}^{k\alpha}f(0)}{ \Gamma_{q}((k+1)\alpha)}x^{(k+1)\alpha-1}.$$
(4.8)

Moreover, the series $$\sum_{k=0}^{\infty}\frac{I_{q}^{1-\alpha}{\mathcal{D}}_{q}^{k\alpha}f(0)}{ \Gamma_{q}((k+1)\alpha)}x^{k\alpha}$$ converges uniformly to $$x^{1-\alpha}f(x)$$ on $$[0,a]$$.

### Proof

Using (4.3), we obtain

\begin{aligned}& \Biggl\vert x^{1-\alpha}f(x)-\sum_{k=0}^{n-1} \frac {I_{q}^{1-\alpha}{\mathcal{D}}_{q}^{k\alpha}f(0)}{ \Gamma_{q}((k+1)\alpha)}x^{k\alpha}\Biggr\vert \\& \quad \le c \Gamma_{q}( \alpha)\frac{ (a A)^{n\alpha}}{\Gamma _{q}((n+1)\alpha)} \\& \quad =\frac{c \Gamma_{q}(\alpha)(q^{(n+1)\alpha };q)_{\infty}}{(q;q)_{\infty}}\frac{ (a A)^{n\alpha }}{(1-q)^{1-(n+1)\alpha}} \\& \quad =\frac{c \Gamma_{q}(\alpha)(q^{(n+1)\alpha };q)_{\infty}}{(q;q)_{\infty}(1-q)^{1-\alpha}} \bigl(a A(1-q) \bigr)^{n\alpha} \longrightarrow0 \quad \mbox{as } n\rightarrow\infty. \end{aligned}

Thus, the result follows. □

### Theorem 4.4

Assume that $${}^{c}\mathcal{D}_{q}^{n\alpha}f\in C[0,a]$$ for $$n\in \mathbb{N}$$. If

$$\bigl\vert {}^{c}\mathcal{D}_{q}^{n\alpha}f(x)\bigr\vert \le c A^{n\alpha},\quad \forall x\in[0,a], n\in\mathbb{N},$$

where c is a positive constant, and A is a positive number satisfying $$A<\frac{1}{a(1-q)}$$, then f has the expansion

$$f(x)=\sum_{k=0}^{\infty} \frac{{}^{c}\mathcal{D}_{q}^{k\alpha }f(0)}{\Gamma _{q}(k\alpha+1)}x^{k\alpha},$$
(4.9)

and the series on the right-hand side of (4.9) converges uniformly to $$f(x)$$ on $$[0,a]$$.

### Proof

The proof is similar to the proof of Theorem 4.3 and is omitted. □

### Remark 4.5

1. (1)

If a function f has the expansion

$$f(x)=\sum_{k=0}^{\infty}a_{k} x^{(k+1)\alpha-1},$$

then we can deduce that

$$a_{k}=\frac{I_{q}^{1-\alpha}{\mathcal{D}}_{q}^{k\alpha}f(0)}{ \Gamma_{q}((k+1)\alpha)}.$$

Also, if a function f has the expansion

$$f(x)=\sum_{k=0}^{\infty}b_{k} x^{k\alpha},$$

then we can deduce that

$$b_{k}=\frac{{}^{c}\mathcal{D}_{q}^{k\alpha}f(0)}{\Gamma_{q}(k\alpha+1)}.$$
2. (2)

The results of this paper are valid if f is a function defined on intervals of the form $$[-a,a]$$ or $$[-a,0]$$, where $$a>0$$. In these two cases, $$\mathcal{L}_{q}^{1}[-a,b]$$, $$b=0$$ or a, is the space of all functions defined on $$[-a,b]$$ such that

$$\sum_{k=0}^{\infty}q^{k}(1-q)\bigl\vert f\bigl(xq^{k}\bigr)\bigr\vert < \infty \quad \mbox{for all } x\in [-a,b].$$

The space $$\mathcal{AC}_{q}[-a,b]$$ is the space of all q-regular at zero functions that satisfy condition (2.3) for all $$t\in[-a,b]$$.

## 5 Examples

In this section, we apply the generalized q-Taylor formula to solve fractional q-difference equations with constant coefficients. A solution to this type of equations is introduced in [12] by using q-Laplace transforms. In the following examples, λ is a real number. We assume that the conditions of Theorems 4.3 and 4.4 are satisfied.

### Example 5.1

Consider the q-initial value problem

$${}^{c}D_{q}^{\alpha}y(x)=\lambda y(x),\quad y(0)=y_{0}, x>0.$$
(5.1)

We assume that $$y\in C[0,a]$$ for some $$a>0$$ to be determined later. By (5.1), $${}^{c}\mathcal{D}_{q}^{n\alpha }y(x)=\lambda^{n} y(x)$$. Consequently,

$$\bigl\vert {}^{c}\mathcal{D}_{q}^{n\alpha}y(x)\bigr\vert \leq c \vert \lambda \vert ^{n},\quad c:=\max_{x\in [0,a]} \bigl\vert y(x)\bigr\vert .$$

Hence, if we assume that $$\vert \lambda a^{\alpha}(1-q)^{\alpha} \vert <1$$, then $$y(x)$$ can be written as

$$y(x)=\sum_{n=0}^{\infty}{}^{c} \mathcal{D}_{q}^{n\alpha}y(0)\frac {x^{n\alpha }}{\Gamma_{q}(n\alpha+1)} =y_{0} e_{\alpha,1}\bigl(\lambda x^{\alpha};q\bigr), \quad x\in[0,a],$$
(5.2)

where $$e_{\nu,\mu}(z;q)$$ is one of the q-Mittag-Leffler function defined by

$$e_{\nu,\mu}(z;q)=\sum_{k=0}^{\infty} \frac{z^{k}}{\Gamma_{q}(\nu k+\mu)},\quad \vert z\vert < (1-q)^{\nu}.$$

### Example 5.2

Consider the q-initial value problem

$${}^{c}\mathcal{D}_{q}^{2\alpha} y(x)=-y(x),\quad y(0)=0, \qquad {}^{c}D_{q}^{\alpha}y(0)=1.$$
(5.3)

We assume that $$y,{}^{c}D_{q}^{\alpha}y\in C[0,a]$$ for some $$a>0$$ to be determined later. From (5.3), we conclude that

$${}^{c}\mathcal{D}_{q}^{(2n+1)\alpha}y(x)=(-1)^{n} {}^{c}D_{q}^{\alpha }y(x),\qquad {}^{c} \mathcal{D}_{q} ^{(2n)\alpha}y(x)=(-1)^{n} y(x),\quad n\in \mathbb{N}.$$

Hence, if $$c=\max{ \{\max_{x\in[0,a]}\vert y(x)\vert , \max_{x\in[0,a]}\vert {}^{c}D_{q}^{\alpha}y(x)\vert \}}$$, then

$$\bigl\vert {}^{c}\mathcal{D}_{q}^{n\alpha}y(x)\bigr\vert \leq c,\quad \forall n\in\mathbb{N}.$$

Therefore, by Theorem 4.3, if a is chosen such that $$a<\frac{1}{(1-q)}$$, then

\begin{aligned} y(x) =&\sum_{n=0}^{\infty}{}^{c} \mathcal{D}_{q}^{n\alpha}y(0) \frac {x^{n\alpha }}{\Gamma_{q}(n\alpha+1)} \\ =&\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{(2n+1)\alpha}}{\Gamma _{q}((2n+1)\alpha+1)}=x^{\alpha} e_{2\alpha,\alpha+1}\bigl(-x^{2\alpha};q \bigr). \end{aligned}
(5.4)

It is worth mentioning that if we set $$\alpha=1$$ in (5.4), then we get the Jackson q-sine function introduced in [16]. Thus, we may consider the function in (5.4) as a fractional analogue of the Jackson q-sine function.

### Example 5.3

Consider the q-initial value problem

$${D}_{q}^{\alpha} y(x)=\lambda y(x),\qquad \bigl[x^{1-\alpha}y \bigr]\bigl(0^{+}\bigr)=\frac{y_{0}}{\Gamma_{q}(\alpha)}.$$
(5.5)

Hence, $${\mathcal{D}}_{q}^{n\alpha}y(x)=\lambda^{n} y(x)$$. We seek a solution y such that $$x^{1-\alpha} y(x)\in C[0,a]$$ for some a. Then

$$\bigl\vert x^{1-\alpha}{\mathcal{D}}_{q}^{n\alpha}y(x) \bigr\vert \leq c \vert \lambda \vert ^{n}, \quad c:=\max _{x\in[0,a]}\bigl\vert x^{1-\alpha}y(x)\bigr\vert .$$

We can show that

$$I_{q}^{1-\alpha}D_{q}^{\alpha}y(0)= \Gamma_{q}(\alpha)\bigl[x^{1-\alpha}y(x)\bigr]\bigl(0^{+}\bigr).$$
(5.6)

Consequently, $$I_{q}^{1-\alpha}{\mathcal{D}}_{q}^{k\alpha}y(0)=\lambda^{n}y_{0}$$. Therefore,

\begin{aligned} y(x) =&\sum_{k=0}^{\infty} \frac{I_{q}^{1-\alpha}{\mathcal {D}}_{q}^{k\alpha}y(0)}{ \Gamma_{q}((k+1)\alpha)}x^{(k+1)\alpha-1} \\ =&y_{0} x^{\alpha-1}\sum_{k=0}^{\infty} \frac{(\lambda x^{\alpha})^{k}}{\Gamma_{q}((k+1)\alpha)}=y_{0} x^{\alpha-1} e_{\alpha,\alpha}\bigl(\lambda x^{\alpha};q\bigr), \end{aligned}

where $$\vert \lambda a^{\alpha}(1-q)^{\alpha} \vert <1$$.

### Example 5.4

Consider the q-initial value problem

$$\mathcal{D}^{2\alpha}_{q} y(x)=-\lambda y(x),\qquad \bigl[x^{1-\alpha}y\bigr]\bigl(0^{+}\bigr)=\frac{y_{1}}{\Gamma_{q}(\alpha)},\qquad \bigl[x^{1-\alpha}{D}_{q}^{\alpha}y\bigr]\bigl(0^{+}\bigr)= \frac{y_{2}}{\Gamma_{q}(\alpha)}.$$
(5.7)

Thus,

$${\mathcal{D}}_{q}^{2n\alpha}y(x)=(-\lambda)^{n} y(x),\qquad { \mathcal {D}}_{q}^{(2n+1)\alpha}y(x)=(-\lambda)^{n} D^{\alpha}_{q}y(x).$$

For a solution y such that $$x^{1-\alpha} y(x), x^{1-\alpha} D^{\alpha}_{q}y(x)\in C[0,a]$$ for some a, we have

$$\bigl\vert x^{1-\alpha}{\mathcal{D}}^{n\alpha}_{q}y(x) \bigr\vert \leq c \vert \lambda \vert ^{n},\quad c:=\max\Bigl\{ \max _{x\in[0,a]}\bigl\vert x^{1-\alpha}y(x)\bigr\vert , \max _{x\in [0,a]}\bigl\vert x^{1-\alpha} D^{\alpha}_{q}y(x) \bigr\vert \Bigr\} .$$

Also,

$$I_{q}^{1-\alpha}{\mathcal{D}}_{q}^{2n\alpha}y(0)=(- \lambda)^{n} y_{1},\qquad I_{q}^{1-\alpha}{ \mathcal{D}}_{q}^{(2n+1)\alpha}y(0)=(-\lambda)^{n}y_{2}.$$

Consequently,

\begin{aligned} y(x) =&x^{\alpha-1}\sum _{k=0}^{\infty}\frac{I_{q}^{1-\alpha}{\mathcal {D}}_{q}^{2k\alpha}y(0)}{ \Gamma_{q}((2k+)\alpha)}x^{2k\alpha}+x^{\alpha-1} \sum_{k=0}^{\infty }\frac{I_{q}^{1-\alpha}{\mathcal{D}}_{q}^{(2k+1)\alpha}y(0)}{ \Gamma_{q}((2k+2)\alpha)}x^{(2k+1)\alpha} \\ =&y_{1} x^{\alpha-1}\sum_{k=0}^{\infty} \frac{(-\lambda)^{k} x^{2k\alpha}}{\Gamma_{q}((2k+1)\alpha)} + y_{2} x^{\alpha-1}\sum _{k=0}^{\infty}\frac{(-\lambda)^{k} x^{(2k+1)\alpha}}{\Gamma_{q}((2k+2)\alpha)} \\ =&y_{1} x^{\alpha-1}e_{2\alpha,\alpha}\bigl(-\lambda x^{2\alpha};q\bigr)+ y_{2} x^{2\alpha-1} e_{2\alpha,2\alpha} \bigl(-\lambda x^{2\alpha};q\bigr), \end{aligned}

where $$\vert \lambda a^{\alpha}(1-q)^{\alpha} \vert <1$$.

### Example 5.5

Consider the initial value problem

$$D_{q}^{\alpha}y(x)=\lambda q^{\alpha(1-\alpha)} y \bigl(q^{\alpha}x\bigr),\qquad \bigl[x^{1-\alpha}y\bigr]\bigl(0^{+}\bigr)= \frac{1}{\Gamma_{q}(\alpha)}.$$
(5.8)

Applying

$$D_{q,x}^{\alpha}y(x\beta)=\beta\bigl(D_{q}^{\alpha}y \bigr) (x\beta),$$
(5.9)

on (5.8) $$n-1$$ times, we obtain

$${\mathcal{D}}_{q}^{n\alpha}y(x)= \bigl(\lambda q^{\alpha(1-\alpha)} \bigr)^{n} q^{\frac{n(n-1)\alpha}{2}}y\bigl(xq^{n\alpha}\bigr).$$
(5.10)

For a solution y such that $$x^{1-\alpha} y(x)\in C[0,a]$$, we have

$$\bigl\vert x^{1-\alpha}{\mathcal{D}}^{n\alpha}_{q}y(x) \bigr\vert \leq c \vert \lambda \vert ^{n}q^{\frac{n(n-1)\alpha}{2}},\quad c:=\max _{x\in[0,a]}\bigl\vert x^{1-\alpha}y(x)\bigr\vert ,$$

and

$$I_{q}^{1-\alpha}{\mathcal{D}}_{q}^{n\alpha}y(0)= \lambda^{n}q^{\frac {n(n-1)\alpha}{2}}.$$

Therefore,

\begin{aligned} y(x) =&\sum_{k=0}^{\infty} \frac{I_{q}^{1-\alpha}{\mathcal {D}}_{q}^{k\alpha}y(0)}{ \Gamma_{q}((k+1)\alpha)}x^{(k+1)\alpha-1} \\ =&x^{\alpha-1}\sum_{k=0}^{\infty}q^{\frac{n(n-1)\alpha}{2}} \frac{ (\lambda x^{\alpha})^{k}}{\Gamma_{q}((k+1)\alpha)} \\ =&x^{\alpha-1}E_{\alpha,\alpha}\bigl(\lambda x^{\alpha};q\bigr), \end{aligned}

where, in general, $$E_{\alpha,\beta}(z;q)$$ is a second q-analogue of Mittag-Leffler function defined by

$$E_{\alpha,\beta}(z)=\sum_{n=0}^{\infty}q^{\frac{n(n-1)\alpha }{2}} \frac{z^{n}}{\Gamma_{q}(n\alpha+\beta)},\quad z\in\mathbb{R}.$$

Hence, a can be taken to be any positive value in this example. For some derived properties for these q-analogues of Mittag-Leffler functions, see [9] and the references therein.

## References

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11. Agarwal, RP: Certain fractional q-integrals and q-derivatives. Proc. Camb. Philos. Soc. 66, 365-370 (1969)

12. Mansour, ZSI: Linear sequential q-difference equations of fractional calculus. Fract. Calc. Appl. Anal. 12(2), 159-178 (2009)

13. Trench, WF: Introduction to Real Analysis. Pearson Education, Upper Saddle River (2010)

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## Acknowledgements

The author is grateful to the referees for their valuable comments and suggestions, which have improved the manuscript in its present form.

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Correspondence to HA Hassan.

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Hassan, H. Generalized q-Taylor formulas. Adv Differ Equ 2016, 162 (2016). https://doi.org/10.1186/s13662-016-0894-3

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• DOI: https://doi.org/10.1186/s13662-016-0894-3

• 41A58
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• 26A33

### Keywords

• q-difference operator
• generalized q-Taylor formula
• Riemann-Liouville fractional q-derivative
• Caputo fractional q-derivative