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Taylor theory associated with Hahn difference operator
Journal of Inequalities and Applications volume 2020, Article number: 124 (2020)
Abstract
In this paper, we establish Taylor theory based on Hahn’s difference operator \(D_{q,\omega}\) which is defined by \(D_{q,\omega}f(t)=\frac{f(qt+\omega)-f(t)}{t(q-1)+\omega}\), \(t\neq\frac {\omega}{1-q}\), where \(q\in(0,1)\) and ω is a positive number.
1 Introduction and preliminaries
Let \(q\in(0,1)\), \(\omega>0\) and \({\omega_{0}:=\frac{\omega}{1-q}}\). Let f be a function defined on an interval I of \(\mathbb {R}\) which contains \(\omega_{0}\). Hahn [10] introduced his difference operator which is defined by
and \(D_{q,\omega}f(\omega_{0}):=f'(\omega_{0})\), provided that f is differentiable at \(\omega_{0}\) in the usual sense. In this case we call \(D_{q,\omega}f\) the \(q, \omega\)-derivative and that f is \(q, \omega \)-differentiable at t whenever \(D_{q,\omega}f(t)\) exists. Finally, we say that f is \(q, \omega\)-differentiable, i.e., throughout I if \(D_{q,\omega}f(\omega_{0})\) exists.
Hahn difference operator unifies the two most well-known quantum difference operators: the Jackson q-difference operator [11–13], which is defined by
and the forward difference \(\Delta_{\omega}\), which is defined by
see [4, 5, 14, 15]. Hahn operator has attracted the attention of several researchers and a variety of results can be found in papers [1, 2, 6, 16–22]. In [3] Annaby and Mansour proved analytically the q-Taylor series associated with \(D_{q}\), introduced by Jackson [12], of an analytic function in some complex domain. In the present paper, we establish an overarching \(q, \omega\)-Taylor theory associated with Hahn difference operator \(D_{q,\omega}\). In this theory the Hahn difference operator \(D_{q,\omega}\) replaces the differentiation operator in the usual Taylor series.
First, we introduce some preliminary results and some notations. Let f, g be \(q, \omega\)-differentiable at \(t\in I\), then
provided that in (1.6), \(g(t)g(qt+\omega)\neq0\) [1, 2]. Also, for \(n\in\mathbb {N}\), the following relations hold:
where \(\alpha, \beta\in\mathbb {R}\), see [1, 2].
The q-shifted factorial \((b;q)_{n}\) for a complex number b and \(n\in \mathbb {N}_{0}=\mathbb {N}\cup\{0\}\) is defined to be
The limit \(\lim_{n\to\infty}(b;q)_{n}\) is denoted by \((b;q)_{\infty}\). Moreover \((b;q)_{n}\) has the representation [9]
The q-binomial coefficients [9]
satisfy the following property:
For \(n\in\mathbb {N}_{0}\) and \(0< q<1\), the q-analogues of the natural numbers of the factorial function and of the semifactorial function [7, 13] are defined by
and
\([x-a]_{n}\) is defined by
The following formula was obtained by Euler [8]:
The q-gamma function [9] is defined by
where \(z\in\mathbb {C}\setminus\{-n:n\in\mathbb {N}_{0}\}\). Here, we take the principal values of \(q^{z}\) and \((1-q)^{1-z}\). In particular
It is known that, for \(x>0\), \(\varGamma_{q}(x)\) is the unique logarithmically convex function that satisfies the functional equation:
In [1], Aldowah introduced the \(q,\omega\)-integral of f from a to b as follows.
Definition 1.1
Let I be any interval of \(\mathbb {R}\) containing \(\omega_{0}\). Assume that \(f:I\to\mathbb {R}\) is a function, and let \(a, b\in I\) such that \(a< b\). The \(q, \omega\)-integral of f from a to b is defined by
where
provided that the series converges at \(x=a\) and \(x=b\). In this case f is called \(q, \omega\)-integrable over \([a,b]\) for all \(a, b\in I\).
Lemma 1.2
Let\(f, g:I\to\mathbb {R}\)be\(q, \omega\)-integrable on\(I, k\in\mathbb {R}\)and\(a, b, c\in I\), \(a< c< b\). Then
- (i)
\(\int_{a}^{a}f(t)\,d_{q, \omega}t=0\),
- (ii)
\(\int_{a}^{b}kf(t)\,d_{q, \omega}t=k\int_{a}^{b}f(t)\,d_{q, \omega}t\),
- (iii)
\(\int_{a}^{b}f(t)\,d_{q, \omega}t=-\int_{b}^{a}f(t)\,d_{q, \omega}t\),
- (iv)
\(\int_{a}^{b}f(t)\,d_{q, \omega}t=\int_{a}^{c}f(t)\,d_{q, \omega }t+\int_{c}^{b}f(t)\,d_{q, \omega}t\),
- (v)
\(\int_{a}^{b}(f(t)+g(t))\,d_{q, \omega}t=\int_{a}^{b}f(t)\,d_{q, \omega}t+\int_{a}^{b}g(t)\,d_{q, \omega}t\).
Lemma 1.3
If\(f:I\to\mathbb {R}\)is continuous at\(\omega_{0}\), then\(\{f(sq^{k}+\omega [k]_{q})\}_{k\in\mathbb {N}}\)converges uniformly to\(f(\omega_{0})\)onI.
Corollary 1.4
If\(f:I\to\mathbb {R}\)is continuous at\(\omega_{0}\), then\(\sum_{k=0}^{\infty}|f((sq^{k})+\omega[k]_{q})|\)converges uniformly onI, and consequentlyfis\(q, \omega\)-integrable overI.
Lemma 1.5
If\(f, g:I\to\mathbb {R}\)are continuous at\(\omega_{0}\), then
Theorem 1.6
Assume that\(f:I\to\mathbb {R}\)is continuous at\(\omega_{0}\). Define
ThenFis continuous at\(\omega_{0}\). Furthermore, \(D_{q, \omega}F(x)\)exists for every\(x\in I\)and\(D_{q, \omega}F(x)=f(x)\). Conversely,
2 Main results
We define the \(q, \omega\)-derivative of higher order in the usual way. That is, the nth \(q, \omega\)-derivative, \(n\in\mathbb {N}\), of \(f:I\to \mathbb {R}\) is the function \(D^{n}_{q, \omega}f:I\to\mathbb {R}\) given by \(D^{n}_{q, \omega}f:=D_{q, \omega}(D^{n-1}_{q, \omega}f)\), provided \(D^{n-1}_{q, \omega}f\) is \(q, \omega\)-differentiable on I and \(D^{0}_{q, \omega}f=f\). We consider the following linear spaces:
and
Our target is to obtain Taylor expansion of a function f defined on an interval I that contains \(\omega_{0}\) associated with Hahn difference operator. We need the following lemmas in proving our main results.
Lemma 2.1
Letfbe a function defined onI. Then, for\(x\neq\omega_{0}\), thenth\(q, \omega\)derivative\((D^{n}_{q, \omega}f)(x)\)can be expressed as
Proof
For \(n=1\), the formula above yields (1.1). Assume that formula (2.1) is true for \(n=m\). By relations (1.5), (1.8), and (1.10), we have
This implies that
That is,
Therefore relation (2.1) is true at \(n=m+1\) and by induction it is true for every \(n\in\mathbb {N}\). □
In the following result, a formula of the nth derivative of a power series of center zero is given.
Lemma 2.2
Assume that a functionfhas the power series expansion\(f(x)=\sum_{k=0}^{\infty}a_{k}x^{k}\), \(x\in I\). Then
Proof
It is clear that Eq. (2.2) is true for \(n=0\). From Eq. (2.1) and relation (1.9), we have, for \(n\in\mathbb {N}\),
Then
□
The following result includes a useful formula for the nth derivative of a power series of center \(\omega_{0}\).
Lemma 2.3
Assume that a functionfhas the power series expansion\(f(x)=\sum_{k=0}^{\infty}a_{k}(x-\omega_{0})^{k}\), \(x\in I\). Then
Proof
It is clear that Eq. (2.3) is true for \(n=0\). From Eq. (2.1) and relation (1.9), we have, for \(n\in\mathbb {N}\),
From this it follows that
□
One of the important questions: Is there a relation between the nth \(q, \omega\) derivative and the usual nth derivative? The answer is in the following lemma.
Lemma 2.4
If\(f\in C^{n+1}\), then
- (i)
\(D^{m}_{q, \omega}f\)exists onIand is continuous at\(\omega _{0}\)for all\(m=1,2,\ldots,n+1\);
- (ii)
for\(1\le m\le n+1\),
$$ D^{m}_{q, \omega}f(\omega_{0})= \frac{[m]_{q}!}{m!}f^{(m)}(\omega_{0}), $$(2.4)where\(f^{(m)}\)is the usualmth derivative off.
Proof
The proof is by induction. The \(q, \omega\) derivative \(D_{q, \omega}f\) exists and \(D_{q, \omega}f(\omega_{0})=f'(\omega_{0})\). Also \(D_{q, \omega}f\) is continuous at \(\omega_{0}\). Indeed,
Now, we assume that (i) and (ii) hold for all \(m=1,2,\ldots,l\), where \(l\leq n\) and we want to prove that they are true at \(m=l+1\). By Lemma 2.1, we conclude that
Applying L’Hopital rule \(l+1\) times and using relations (1.12), (1.13), and (1.14), we get
On the other hand, we conclude that
Again, applying L’Hopital rule \(l+1\) times and using relations (1.12), (1.13), and (1.14), we get
Therefore,
□
Corollary 2.5
Assume that f has the power series expansion
Then
Proof
By Lemma 2.4, we have
□
Now we define the two variable polynomials \(H_{n}(x,t)\), \(x, t\in I\), to be
where \(h^{j}(t)=tq^{j}+\omega[j]_{q}, t\in I\) is the jth order iteration of \(h(t)=qt+\omega\), which uniformly converges to \(\omega_{0}\) on I.
Lemma 2.6
For\(n\in\mathbb {N}\)and\(x, t\in I\), we have
where\({}_{t}D_{q, \omega}\)is the\(q, \omega\)-derivative with respect tot,
where\(I^{n}_{q, \omega}\)is the\(q, \omega\)-integral
Now, we establish Taylor’s theorem based on Hahn difference operator.
Theorem 2.7
Letfbe a function defined onI. If\(f\in C^{n}_{q,\omega}\)for some\(n\in\mathbb {N}\), then for\(x, a\in I\),
where
Proof
We prove relation (2.9) by induction. The right-hand side (R.H.S) of (2.9) at \(n=1\) is
Assume that relation (2.9) is true for \(n=m\), that is,
where \(R_{m}(x,a)= \int_{a}^{x}\frac{D^{m}_{q, \omega}f(t)}{[m-1]_{q}!} H_{m-1}(x,h(t))\,d_{q, \omega}t\). We integrate by parts in the remainder term \(R_{m}(x,a)\). We obtain
Then
Therefore, relation (2.9) is true for \(n=m+1\), then it is true for every \(n\in\mathbb {N}\). □
As a direct consequence of the previous theorem, we deduce the following theorem.
Theorem 2.8
Let\(f\in C^{\infty}_{q,\omega}\). If for\(x, a\in I\), \(\lim_{n\to\infty } R_{n}(x,a)=0\), then\(f(x)\)has the following expansion:
Furthermore, if\(\lim_{n\to\infty} R_{n}(x,a)=0\)uniformly with respect toxin some subinterval ofI, then the series given by (2.11) is uniformly convergent in this subinterval.
Corollary 2.9
Let\(f\in C^{\infty}_{q, \omega}\). If for\(x\in I\), \(\lim_{n\to\infty} R_{n}(x,\omega_{0})=0\), then\(f(x)\)has the following expansion:
Theorem 2.10
Let\(f\in C^{\infty}_{q, \omega}\). Assume that there is a nonnegative sequence\(\{M_{n}\}\)such that
- (i)
\(|D^{n}_{q, \omega}f(h^{m}(y))|\le C M_{n}\), \(n, m\in\mathbb {N}_{0}\), \(y\in I\), for some\(C>0\);
- (ii)
\({\lim_{n\to\infty} \frac{M_{n+1}}{M_{n}}=M}\)exists.
Thenfhas the\(q, \omega\)-Taylor expansion
for every\({x\in(\omega_{0}-\frac{1}{M(1-q)},\omega_{0}+\frac{1}{M(1-q)})}\)when\(M>0\) (respectively\({x\in I}\)when\(M=0\)).
Proof
We can write \(R_{n}(x,a)\) as follows:
where
and
From (1.16), we have
Consequently,
Then \(\lim_{n\to\infty}R_{1,n}(x,\omega_{0})=0\), \(x\in(\omega_{0}-\frac {1}{M(1-q)},\omega_{0}+\frac{1}{M(1-q)})\), when \(M>0\) (respectively \(x\in I\), when \(M=0\)). On the other hand, for \(a\in I\), we have
Simple calculations show that
Consequently,
This implies that \(\lim_{n\to\infty}R_{2,n}(x;a,\omega_{0})=0\), \(x\in (\omega_{0}-\frac{1}{M(1-q)},\omega_{0}+\frac{1}{M(1-q)})\), when \(M>0\) (respectively \(x\in I\), when \(M=0\)). Therefore
\({x\in(\omega_{0}-\frac{1}{M(1-q)},\omega_{0}+\frac{1}{M(1-q)})}\), when \(M>0\) (respectively \(x\in I\), when \(M=0\)). □
Theorem 2.11
Assume thatfhas the power series expansion\(f(x)=\sum_{n=0}^{\infty}a_{n}(x-\omega_{0})^{n}\)with interval of convergence\(I_{r}=(\omega_{0}-r,\omega_{0}+r)\), \(r>0\). Then, for any\(a\in I_{r}\), fhas the\(q, \omega\)-Taylor expansion
in any closed subinterval\({\overline{I_{\alpha}}, \alpha< r}\), where the series is absolutely and uniformly convergent on\({\overline{I_{\alpha}}, \alpha< r}\).
Proof
For \(n,m\in\mathbb {N}\) and by Lemma 2.3, we get
Consequently, for \(\alpha< r\),
where \({ C=\sum_{k=0}^{\infty}|a_{k}\alpha^{k}| }\). Then, by Theorem 2.10, f has the \(q, \omega\)-Taylor expansion (2.13). □
Now, we establish some properties of the \(q, \omega\)-exponential functions \(e_{q, \omega}(t)\) and \(E_{q, \omega}(t)\) for \(t\in\mathbb {R}\), \(|t-\omega_{0}|<\frac{1}{1-q}\), where
and
Simple calculations show that the following inequalities are true:
and
where \({ A=\sum_{k=1}^{\infty} \frac{q^{k}}{1-q^{k}}}\).
Finally, we can prove the following power series expansions for \(e_{q,\omega}\) and \(E_{q,\omega}\).
Example 2.12
The exponential functions \(e_{q,\omega}\) and \(E_{q,\omega}\) defined in (2.14) and (2.15) have the following power series expansions of center \(a\in I\):
and
and have the following power series expansions of center \(\omega_{0}\):
and
Furthermore, both \(e_{q,\omega}\) and \(E_{q,\omega}\) are continuous.
Proof
For \(n\in\mathbb {N}_{0}\), we have
Inequality (2.16) shows that \(e_{q, \omega}(t)\) is positive and bounded on every compact subinterval of \({(\omega_{0}-\frac{1}{1-q}, \omega_{0}+\frac{1}{1-q})}\). For fixed \(t\in{(\omega_{0}-\frac{1}{1-q}, \omega_{0}+\frac{1}{1-q})}\), there exists \(0<\alpha\le1\) such that \({|t(1-q)-\omega|<\alpha}\), which implies that
By Theorem 2.10, the \(q,\omega\)-Taylor expansion of \(e_{q, \omega}(t)\) at a is given by
Since \(D^{n}_{q,\omega}e_{q, \omega}(\omega_{0})= 1\), the \(q,\omega\)-Taylor expansion of \(e_{q, \omega}(t)\) at \(\omega_{0}\) is given by
The series in (2.23) is uniformly convergent on every compact subinterval of \((\omega_{0}-\frac{1}{1-q}, \omega_{0}+\frac{1}{1-q})\) by Weierstrass M-test, and consequently \(e_{q, \omega}(t)\) is continuous.
Let \(t\in\mathbb {R}\), \(|t-\omega_{0}|<\frac{1}{1-q}\). First, we show that
by induction. For \(n=1\), we have
Assume that formula (2.24) is true for \(n=m\). We have
Inequality (2.17) shows that \(E_{q, \omega}(t)\) is positive and is bounded on every compact subinterval of \({(\omega_{0}-\frac{1}{1-q}, \omega_{0}+\frac{1}{1-q})}\). Also we can see that
Therefore,
By Theorem 2.10, the \(q,\omega\)-Taylor expansion of \(E_{q, \omega}(t)\) at a is given by
Since \(D^{n}_{q,\omega}f(\omega_{0})= q^{\frac{n(n-1)}{2}}\), the \(q,\omega \)-Taylor expansion of \(E_{q,\omega}(t)\) at \(\omega_{0}\) is given by
The series in (2.25) is uniformly convergent on every compact subinterval of \((\omega_{0}-\frac{1}{1-q}, \omega_{0}+\frac{1}{1-q})\) and consequently \(E_{q, \omega}(t)\) is continuous. □
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Oraby, K., Hamza, A. Taylor theory associated with Hahn difference operator. J Inequal Appl 2020, 124 (2020). https://doi.org/10.1186/s13660-020-02392-y
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DOI: https://doi.org/10.1186/s13660-020-02392-y