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) 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) 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) 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) 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) 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) 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) 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) 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) 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) 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.