R E S E A R C H
Open Access
Generalized
q
-Taylor formulas
HA Hassan
**Correspondence:
[email protected] Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt
Department of Mathematics, Faculty of Basic Education, PAAET, Ardiyah, Kuwait
Abstract
In this paper, new generalizedq-Taylor formulas involving both Riemann-Liouville and Caputoq-difference operators are constructed. Some applications with solutions of fractionalq-difference equations are also given.
MSC: 41A58; 39A13; 26A33
Keywords: q-difference operator; generalizedq-Taylor formula; Riemann-Liouville fractionalq-derivative; Caputo fractionalq-derivative
1 Introduction
Aq-analogue of Taylor series was introduced by Jackson []:
f(x) = ∞
n=
( –q)n (q;q)n
Dnqf(a)[x–a]n, (.)
where <q< ,Dqis theq-derivative, and
[x–a]n:= (x–a)(x–qa)· · ·
x–qn–a, n≥, [x–a]:= .
Al-Salam and Verma [] introduced the followingq-interpolation series:
f(x) = ∞
n=
(–)nq–n(n–)/( –q)
n
(q;q)n
Dnqfaq–n[x–a]n. (.)
Al-Salam and Verma gave only formal proofs for (.); see [, ]. Analytic proofs of (.) and (.) were given in [].
Results of generalized Taylor formulas involving the classical fractional derivative may be found in [, ]. In [], a generalized Taylor formula involving the classical Riemann-Liouville fractional derivative of orderαis deduced, whereas the generalized Taylor for-mula in [] contains Caputo fractional derivative of orderα, where <α≤.
In [], aq-Taylor formula in terms of Riemann-Liouville fractionalq-derivativeDα
q,aof
orderαis obtained. This result can be stated as follows.
Theorem A([]) Let f be a function defined on(,b)andα∈(, ).Then f can be
ex-panded in the form
f(x) =
n–
k=
(Dα+k q,af)(c)
q(α+k+ )
(x–c)(α+k)
+
q(α)
c
a
(x–c)(α–)Dqα,af(t)dqt
–K(a)(x–c)(α–)+Iα+n q,c Dαq,+anf
(x), (.)
where <a<c<x<b,and K(a)does not depend on x.
Also, in [], a generalizedq-Taylor formula in fractionalq-calculus is established and used in deriving certainq-generating functions for the basic hyper-geometric functions.
In this paper, we give generalized Taylor formulas involving Riemann-Liouville frac-tionalq-derivatives of orderα and Caputo fractionalq-derivatives of orderα; see (.) and (.). 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 theq-notations and notions that will be used throughout this paper. In Section , we giveq-analogues of mean value theorems on [,a]. In Section , we give generalizedq-Taylor formulas involving both Riemann-Liouville fractionalq-derivative and Caputo fractionalq-derivative. Then conditions for infinite expansion for some functions are given. In the last section, we apply the obtained results in solving certainq-difference equations.
2 Notation and preliminaries
In the following, q is a positive number,q< . We follow [] for the definition of the q-shifted factorial, Jacksonq-integral,q-derivative,q-gamma functionq(z), andq-beta
functionBq(α,β). Also, we follow [] for the definition of theq-derivative at zero and the
q-regular at zero functions.
The followingq-integral is useful and will be used in the sequel:
x
(qt/x;q)β–tα–dqt=xαBq(α,β), α,β,x> ; (.)
it can be proved by settingξ=t/x. ByL
q(,a),a> , we mean the Banach space of all functions defined on (,a] such that
f:= a
f(t)dqt<∞, (.)
where two functions inL
q(,a) are considered to be the same function if they have the
same values at the sequence{aqn}∞ n=.
LetL
q(,a) denote the space of all functionsf defined on (,a] such thatf ∈Lq(,x) for
f isq-regular at zero and
∞
j=
f
tqj–ftqj+<∞, t∈(qa,a]. (.)
A characterization of the spaceACq[,a] is given as follows (see []).
Theorem B Let f be a function defined on[,a].Then f ∈ACq[,a]if and only if there
exist a constant c and a functionφinLq[,a]such that
f ∈ACq[,a] ⇐⇒ f(x) =c+
x
φ(u)dqu, x∈[,a]. (.)
Moreover,c andφare uniquely determined by c=f()andφ(x) =Dqf(x)for all x∈(,a].
The Riemann-Liouville fractionalq-integral operator is introduced in [] by Al-Salam through
Iqαf(x) := x
α– q(α)
x
(qt/x;q)α–f(t)dqt, α∈ {–, –, . . .}./ (.)
In [], the generalized Riemann-Liouville fractionalq-integral operator forα∈R+is given
as
Iqα,af(x) := x
α– q(α)
x
a
(qt/x;q)α–f(t)dqt. (.)
Using the definition of theq-integral, (.) reduces to
Iqαf(x) =xα( –q)α ∞
n=
qn(q
α;q)
n
(q;q)n
fxqn, (.)
which is valid for allα. For example,
Iqαxβ–= q(β)
q(β+α)
xα+β–. (.)
This basic Riemann-Liouville fractionalq-integral was also given later by Agarwal []. In the same paper, he introduced the following semigroup property:
Iα
qIqβf(x) =IqβIqαf(x) =Iqα+βf(x), α,β≥. (.)
The generalized Riemann-Liouville fractionalq-derivative is given in [] by
Dqα,af(x) =DqIq–,aαf(x), a≥, (.)
andDα
q,f(x) =Dαqf(x). The Caputo fractionalq-derivative of orderα, <α≤, is (see[]) cDα
qf(x) :=I–
α
Let AC(qk)[,a], k ∈ N, be the space of all functions f defined on [,a] such that f,Dqf, . . . ,Dkq–f areq-regular at zero andDkq–f ∈ACq[,a].
Forα> , letk=α, where ·is the ceiling function. Then the Riemann-Liouville fractional derivativeDα
qf(x) exists if (see [])
f ∈Lq[;a], Iqk–αDkqf ∈AC(qk)[,a], andcDα
qf(x) exists iff∈AC (k) q [,a].
The following results are proved in [] for anyα> ; the result for the case <α< is introduced in the following theorems without proof.
Theorem C Assume that f ∈Lq[;a]and I–α
q f ∈ACq[,a],where <α< .Then the
Riemann-Liouville fractional derivative of orderα, <α< ,exists,and
Iα
qDαqf(x) =f(x) –
I–α
q f()
q(α)
xα–. (.)
Theorem D If f ∈ACq[,a],then
Iα
qcDαqf(x) =IqDqf(x) =f(x) –f() (.)
for <α< .
It is worth mentioning that the key point in the proofs of Theorems C and D is the q-integration by parts formula:
b
f(t)Dqg(t)dqt= (fg)(b) – lim n→∞(fg)
bqn–
b
Dqf(t)g(qt)dqt.
Hence, iffgisq-regular at zero, then the limit on the right-hand side is nothing but (fg)().
3 Generalizedq-mean value theorems on[0,a]
In this section, we introduce twoq-analogues of the mean value theorems. The first one is forq-integrals on an interval of the form [,a], and the second is a mean value theorem with both of Riemann-Liouville fractionalq-derivative and Caputo fractionalq-derivative on [,a]. The first one can be stated as follows.
Theorem .(Mean value theorem forq-integrals) Let g be a continuous function de-fined on[,a],and h be a nonnegative function defined on[,a]and q-regular at zero. Then
a
g(t)h(t)dqt=g(ξ)
a
h(t)dqt (.)
for someξ∈[,a].
Proof The proof is similar to the classical case (see [], p.) and is omitted.
Remark .
() We cannot replace the lower end point of theq-integrals in (.) by arbitrary nonzero number because the inequality
caf(t)dqt
≤ a
c
f(t)dqt,
holds only forc∈ {,aqn,n∈N}. In this case, (.) is also true.
() There areq-analogues of mean value theorems on[a,b]in [], but all these analogues are valid only for certain values ofq. For example, one of the mean value theorems forq-integrals in [] is the following:
Letf,gbe continuous functions on[a,b]. Then there existsq∈(, )such that
∀q∈(q, ) ∃ξ∈[a,b]: b
a
g(t)f(t)dqt=g(ξ)
b
a
f(t)dqt.
The second theorem is aq-analogue of the mean value theorem for derivative on [,a]. Throughout the rest of this article, we assume that <α< .
Theorem .
() Iff∈Lq[;a],I–α
q f ∈ACq[,a],andx–αDαqf ∈C[,a],then
f(x) =I
–α
q f()
q(α)
xα–+q(α)ξ
–αDα
qf(ξ)
q(α)
xα–. (.)
() Iff∈ACq[,a]andcDαqf ∈C[,a],then
f(x) =f() +
cDα
qf(ξ)
q(α)
xα (.)
for someξ lying in the interval[,x]and all x∈(,a].
Proof We first prove (.). Since (see [], p.)
Bq(α,β) =
q(α)q(β)
q(α+β)
,
from (.), Theorem ., and (.) we get
IqαDαqf(x) = x
α– q(α)
x
(qt/x;q)α–tα–t–αDαqf(t)dqt
= x
α– q(α)
ξ–αDα
q(ξ)
x
(qt/x;q)α–tα–dqt
= q(α)ξ
–αDα
qf(ξ)
q(α)
xα–
for ≤ξ ≤x. Hence, (.) follows from (.). Similarly, using (.), we can prove
4 Generalizedq-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-derivativeDnα
q and Caputo
frac-tionalq-derivativecDnα
q ,n∈N, are
respectively. The following lemma is important to get these formulas.
Lemma .
Proof We give a proof of (.), and the proof of (.) can be obtained similarly. Applying (.) and (.), we obtain
and the lemma follows.
Proof For (.), applying (.), we obtain
Applying theq-integral mean value theorem and (.) yield
Inα
By using (.), (.) can be treated similarly.
A natural question arises: can we expand a functionf in terms ofq-fractional deriva-tives? That is,
Proof Using (.), we obtain
x–αf(x) –
n–
k=
I–α
q Dkqαf()
q((k+ )α)
xkα
≤cq(α)
(aA)nα
q((n+ )α)
=cq(α)(q
(n+)α;q)
∞ (q;q)∞
(aA)nα
( –q)–(n+)α
=cq(α)(q
(n+)α;q)
∞ (q;q)∞( –q)–α
aA( –q)nα−→ asn→ ∞.
Thus, the result follows.
Theorem . Assume thatcDnα
q f∈C[,a]for n∈N.If
cDnα
q f(x)≤cAnα, ∀x∈[,a],n∈N,
where c is a positive constant,and A is a positive number satisfying A<a(–q),then f has the expansion
f(x) = ∞
k= cDkα
q f()
q(kα+ )
xkα, (.)
and the series on the right-hand side of (.)converges uniformly to f(x)on[,a].
Proof The proof is similar to the proof of Theorem . and is omitted.
Remark .
() If a functionf has the expansion
f(x) = ∞
k=
akx(k+)α–,
then we can deduce that
ak=
I–α
q Dkqαf()
q((k+ )α)
.
Also, if a functionf has the expansion
f(x) = ∞
k=
bkxkα,
then we can deduce that
bk= cDkα
q f()
q(kα+ )
() The results of this paper are valid iff is a function defined on intervals of the form [–a,a]or[–a, ], wherea> . In these two cases,L
q[–a,b],b= ora, is the space
of all functions defined on[–a,b]such that
∞
k=
qk( –q)fxqk<∞ for allx∈[–a,b].
The spaceACq[–a,b]is the space of allq-regular at zero functions that satisfy
condition (.) for allt∈[–a,b].
5 Examples
In this section, we apply the generalizedq-Taylor formula to solve fractionalq-difference equations with constant coefficients. A solution to this type of equations is introduced in [] by usingq-Laplace transforms. In the following examples,λis a real number. We assume that the conditions of Theorems . and . are satisfied.
Example . Consider theq-initial value problem
cDα
qy(x) =λy(x), y() =y,x> . (.)
We assume thaty∈C[,a] for somea> to be determined later. By (.),cDnα
q y(x) =
λny(x). Consequently,
cDnα
q y(x)≤c|λ|n, c:= max x∈[,a]
y(x).
Hence, if we assume that|λaα( –q)α|< , theny(x) can be written as
y(x) = ∞
n= cDnα
q y()
xnα
q(nα+ )
=yeα,
λxα;q, x∈[,a], (.)
whereeν,μ(z;q) is one of theq-Mittag-Leffler function defined by
eν,μ(z;q) =
∞
k=
zk
q(νk+μ)
, |z|< ( –q)ν.
Example . Consider theq-initial value problem
cDα
q y(x) = –y(x), y() = , cD
α
qy() = . (.)
We assume thaty,cDα
qy∈C[,a] for somea> to be determined later. From (.), we
conclude that
cD(n+)α
q y(x) = (–)ncD
α
qy(x), cD(n)
α
q y(x) = (–)ny(x), n∈N.
Hence, ifc=max{maxx∈[,a]|y(x)|,maxx∈[,a]|cDαqy(x)|}, then
cDnα
Therefore, by Theorem ., ifais chosen such thata<(–q), then
It is worth mentioning that if we setα= in (.), then we get the Jacksonq-sine function introduced in []. Thus, we may consider the function in (.) as a fractional analogue of the Jacksonq-sine function.
Example . Consider theq-initial value problem
Dαqy(x) =λy(x), x–αy+= y
We can show that
Iq–αDαqy() =q(α)
x–αy(x)+. (.)
Consequently,I–α
q Dkqαy() =λny. Therefore,
Example . Consider theq-initial value problem
Also,
Example . Consider the initial value problem
Dαqy(x) =λqα(–α)yqαx, x–αy+=
where, in general,Eα,β(z;q) is a secondq-analogue of Mittag-Leffler function defined by
Hence,acan be taken to be any positive value in this example. For some derived properties for theseq-analogues of Mittag-Leffler functions, see [] and the references therein.
Competing interests
The author declares that he has no competing interests.
Acknowledgements
The author is grateful to the referees for their valuable comments and suggestions, which have improved the manuscript in its present form.
Received: 27 February 2016 Accepted: 12 June 2016
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