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R E S E A R C H

Open Access

On fractional Hahn calculus

Tanapat Brikshavana and Thanin Sitthiwirattham

*

*Correspondence:

thanin_sit@dusit.ac.th

Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok, Thailand

Abstract

In this paper, the new concepts of Hahn difference operators are introduced. The properties of fractional Hahn calculus in the sense of a forward Hahn difference operator are introduced and developed.

MSC: 39A10; 39A13; 39A70

Keywords: fractional Hahn integral; fractional Hahn difference operator; Jackson-Nörlund integral

1 Introduction

The quantum calculus, known as calculus without the consideration of limits, involves sets of non-differentiable functions. There are many types of quantum difference opera-tors such as the Jacksonq-difference operator, the forward (delta) difference operator, the backward (nabla) difference operator, and so on. These operators are employed in many applications, for example, combinatorics, orthogonal polynomials, basic hypergeometric functions, hypergeometric series, complex analysis, the calculus of variations, the theory of relativity, quantum mechanics, and particle physics [–].

In , Hahn [] introduced the Hahn difference operatorDq,ωas follows:

Dq,ωf(t) :=

f(qt+ω) –f(t) t(q– ) +ω , t=

ω

 –q.

This operator is created with a combination of two well-known operators, the forward difference operator and the Jacksonq-difference operator. We observe that

Dq,ωf(t) =ωf(t) wheneverq= ,

Dq,ωf(t) =Dqf(t) wheneverω= , and

Dq,ωf(t) =f(t) wheneverq= ,ω→.

Particularly, the Hahn difference operator has been employed to construct families of or-thogonal polynomials as well as to examine some approximation problems (see [–] and the references therein).

In  Aldwoah [, ] (PhD thesis supervised by Annaby and Hamza) defined the right inverse ofDq,ωin terms of both the Jacksonq-integral containing the right inverse ofDq[] and the Nörlund sum involving the right inverse ofω[].

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Malinowska and Torres [, ] studied the Hahn quantum variational calculus in . Moreover, in , Malinowska and Martins [] studied the generalized transversality conditions for the Hahn quantum variational calculus. In the same year, Hamzaet al.[– ] studied the theory of linear Hahn difference equations. They established the existence and uniqueness results for the initial value problems for Hahn difference equations by using the method of successive approximations. In addition, they proved Gronwall’s and Bernoulli’s inequalities with respect to the Hahn difference operator and investigated the mean value theorems, Leibniz’s rule and Fubini’s theorem for this calculus. For the bound-ary value problems, in , Sitthiwirattham [] considered a nonlinear Hahn difference equation with nonlocal boundary value conditions

Dq,ωx(t) +ft,x(t),Dp,θx(pt+θ)

= , t∈[ω,T]q,ω,

x(ω) =ϕ(x), (.)

x(T) =λx(η), η∈(ω,T)q,ω,

where  <q< ,  <ω<T, ω := –qω , ≤λ< T–ωη–ω, p=qm, m∈N, θ =ω(–p–q), f : [ω,T]q,ω×R×R→Ris a continuous function, andϕ:C([ω,T]q,ω,R)→Ris a given functional. Moreover, in this year , Sriphanomwan et al. [] studied a nonlocal boundary value problem for second-order nonlinear Hahn integro-difference equation with integral boundary condition

Dq,ωx(t) =ft,x(t),Dp,θx(pt+θ),p,θx(pt+θ)

, t∈[ω,T]q,ω,

x(ω) =x(T), (.)

x(η) =μ

T

ω

g(s)x(s)dq,ωs, η∈(ω,T)q,ω,

where  <q< ,  <ω<T,ω:=–qω ,μ T

ωg(r)dq,ωr= ,μ∈R,p=q

m,mN,θ=ω(–p –q), fC([ω,T]q,ω×R×R×R→R), andgC([ω,T]q,ω×R+) are given functions, and forϕC([ω,T]q,ω×[ω,T]q,ω, [,∞))

Ψp,θx(t) := t

ω

ϕ(t,ps+θ)x(ps+θ)dp,θs. (.)

In particular, the fractional Hahn difference equations have not been studied. We ob-serve that in , Čermák and Nechvátal [] introduced the fractional (q,h)-difference operator and the fractional (q,h)-integral forq> . Čermák et al.[] studied discrete Mittag-Leffler functions in linear fractional difference equations forq>  in . In the same year Rahmat [, ] investigated the (q,h)-Laplace transform and some (q,h )-analogues of integral inequalities on discrete time scales forq> . Recently Duet al.[] studied the monotonicity and convexity for nabla fractional (q,h)-difference forq> ,q=  in . However, these operators are not satisfied with fractional Hahn operators because fractional Hahn operators require the condition  <q< .

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this paper as follows. In Section , some basic formulas of the Hahn difference operator and the associated Jackson-Nörlund integral calculus are briefly reviewed. In Section , we present the fractional Hahn integral and develop some fundamental properties. The fractional Hahn difference operators are presented in Sections  and .

2 Preliminary definitions and properties

The following notations, definitions, and lemmas will be used in proving the main results. Letq∈(, ),ω> ,ω=–qω , and define

[n]q,ω= [n]q:= –q n

 –q =q

n–+· · ·+q+  and [n]q,ω! = [n]q! := n

k=  –qk

 –q.

Theq-analogue of the power function (ab)nqwithn∈N:= [, , , . . .] is

(ab)q:= , (ab)nq:= n–

k=

abqk, a,b∈R.

Theq,ω-analogue of the power function (ab)nq,ωwithn∈N:= [, , , . . .] is

(ab)q,ω:= , (ab)nq,ω:= n–

k=

abqk+ω[k]q, a,b∈R.

More generally, ifα∈R, we have

(ab)αq=

n=  – (b

a)q n

 – (ba)qα+n, a= ,

(ab)α

q,ω= (aω)α

n=

 – (a–ωb–ω)qn

 – (b–ωa–ω)q

α+n=

(aω) – (bω) α

q, a=ω.

Note thataαq =and (aω)αq,ω= (aω)α. We use the notation ()αq= (ω)αq,ω=  for

α> . Forα,β,γ,λ∈R, we have

(A) (λβγ λ)αq=γα(βλ) α q,

(A) (βγ)α+γq,ω = (βγ)αq,ω(βqαωλ) γ q,ω,

(A) (ts)αq,ω= ,ts,α∈/N, fort,sITq,ω:={qkT+ω[k]q:k∈N}. Theq-gamma andq-beta functions are defined by

q(x) :=

( –q)x–q

( –q)x–, x∈R\ {, –, –, . . .}, Bq(x,s) :=

tx–( –qt)s–q dqt

= ( –q)qn

n=

qnα– –qn+qβ–=q(x)q(s)

q(x+s) ,

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Theq,ω-forward jump operator,q,ω-backward jump operator, andq,ω-forward grain-iness function are defined by

σq,ω(t) :=qt+ω, ρq,ω(t) :=tω the following properties:

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=ρq,ω(t) –ωα–

k=( –

β–ωρq,ω(t)–ωq

k)

k=( – β–ω

ρq,ω(t)–ωqk+α–)

 –  –q

= [α]qρq,ω(t) –βα–

q,ω.

So, (a) holds. Proceeding similarly as above, we find that (b) holds.

Lettinga,bI⊆Rwith a<ω<b and [k]q= –q

k

–q,k∈N:=N∪ {}, we define the q,ω-interval by

[a,b]q,ω:=

qka+ω[k]q:k∈N

qkb+ω[k]q:k∈N

∪ {ω}

= [a,ω]q,ω∪[ω,b]q,ω

= (a,b)q,ω∪ {a,b}=

a,b)q,ω∪ {b}= (a,bq,ω∪ {a}.

Observe that, for eachs∈[a,b]q,ω, the sequence{σq,ωk (s)}∞k=={qks+ω[k]

q}∞k=is uniformly convergent toω.

Also we define the forward jump operatorσq,ωk (t) :=qkt+ω[k]

qand the backward jump

operatorρkq,ω(t) :=t–ω[k]qk q fork∈N.

Definition . LetIbe any closed interval ofRcontaininga,b, andω. Assuming that f :I→Ris a given function, we defineq,ω-integral off fromatobby

b

a

f(t)dq,ωt:= b

ω

f(t)dq,ωta

ω

f(t)dq,ωt,

where

x

ω

f(t)dq,ωt:=

x( –q) –ω

k=

qkfxqk+ω[k]q

, xI,

provided that the series converges atx=aandx=b.fis calledq,ω-integrable on [a,b], and the sum to the right-hand side of the above equation will be called the Jackson-Nörlund sum.

We note that:

(i) The actual domain of functionf is defined on[a,b]q,ω⊂I. (ii) For eachx∈[a,b]q,ω, we havelimk→∞σq,ωk (x) =ω. It implies that

x

ω

f(t)dq,ωt= ρq,ω(ω)

s=x

s( –q) –ωf(s).

Lemma .([]) Let q∈(, ),ω> ,a,bIT

q,ω,and f,g be q,ω-integrable on Iq,ωT .Then the following formulas hold:

(a) aaf(t)dq,ωt= ;

(b) abαf(t)dq,ωt=α b

a f(t)dq,ωt,α∈R;

(c) abf(t)dq,ωt= – a

b f(t)dq,ωt;

(d) abf(t)dq,ωt= b

c f(t)dq,ωt+ c

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(e) ab[f(t) +g(t)]dq,ωt= b

a f(t)dq,ωt+ b

a g(t)dq,ωt;

(f ) ab[f(t)Dq,ωg(t)]dq,ωt= [f(t)g(t)]bab

a[g(σq,ω(t))Dq,ωf(t)]dq,ωt.

We next introduce the fundamental theorem and Leibniz formula of Hahn calculus.

Lemma .(Fundamental theorem of Hahn calculus []) Let f :I→Rbe continuous at

ω.Define

F(x) := x

ω

f(t)dq,ωt, xI.

Then F is continuous atω.Furthermore,Dq,ωF(x)exists for every xI and

Dq,ωF(x) =f(x).

Conversely, b

a

Dq,ωf(t)dq,ωt=f(b) –f(a) for all a,bI.

Lemma .(Leibniz formula of Hahn calculus []) Let f :IT

q,ω×Iq,ωT →R.Then Dq,ω

t

ω

f(t,s)dq,ωs

= t

ω

tDq,ωf(t,s)dq,ωs+f

σq,ω(t),t,

wheretDq,ωis Hahn difference with respect to t.

Next, we give some auxiliary lemmas used for simplifying calculations.

Lemma .([]) Let q∈(, ),ω> and f :I→Rbe continuous atω.Then t

ω r

ω

f(s)dq,ωs dq,ωr= t

ω t

qs+ω

f(s)dq,ωr dq,ωs.

Lemma .([]) Let q∈(, )andω> .Then

t

ω

dq,ωs=tωand t

ω

t– (qs+ω)dq,ωs=

(tω)  +q .

3 Fractional Hahn integral

Now, we introduce fractional Hahn integral.

Definition . Forα,ω> ,q∈(, ) and f defined on [ω,T]q,ω, the fractional Hahn integral is defined by

q,ωf(t) := 

q(α) t

ω

tσq,ω(s)α–q,ωf(s)dq,ωs

= [t( –q) –ω]

q(α) ∞

n=

qntσq,ωn+(t)α–q,ωq,ωn (t),

and (I

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Since ( –q)tω= ( –q)(tω), associated with (A), we have

tσq,ωn+(t)α–q,ω =(tω) –σq,ωn+(t) –ωα–

q

=(tω) –qn+(tω) α–

q

= (tω)α– –qn+α–q .

It implies that

q,ωf(t) =

( –q)(tω)α

q(α) ∞

n=

qn –qn+α–q q,ωn (t).

Next, we provide some auxiliary lemmas for simplifying calculations.

Lemma . Letα,β> ,p,q∈(, ),andω> .Then

t

ω

tσq,ω(s)α–q,ω(sω)q,ωβ dq,ωs= (tω)α+βBq(β+ ,α),

t

ωx

ω

tσp,ω(x)α–p,ωxσq,ω(s)β–q,ωdq,ωs dp,ωx=

(tω)α+β [β]q

Bp(β+ ,α).

Proof From the definition ofq,ω-analogue of the power function and Definition ., we obtain

t

ω

tσq,ω(s)α–q,ω(sω) β q,ωdq,ωs

= (tω)α( –q) ∞

n=

qn –qn+α–q qn(tω)β

= (tω)α+β( –q) ∞

n=

qn –qn+α–q qnβ

= (tω)α+βBq(β+ ,α),

and t

ωx

ω

tσp,ω(x)α–p,ωxσq,ω(s)β–q,ωdq,ωs dp,ωx

= t

ω

tσp,ω(x)α–p,ω

x

ω

xσq,ω(s)β–q,ωdq,ωs

dp,ωx

=  [β]q

t

ω

tσp,ω(x)α–p,ω(xω)βdp,ωx

=(tω) α+β

[β]q Bp(β+ ,α).

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Theorem . For f :IT

q,ω→R,α> ,q∈(, ),ω> ,and aIq,ωT ,

q,ωf(t) =Iq,ωα+

Dq,ωf(t)

+ f(ω)

q(α+ )(tω) α

.

Proof Using Lemma .(b) and Lemma ., we obtain

q,ωf(t) = –  [α]qq(α)

t

a

Dq,ω(ts)αq,ωf(s)dq,ωs

= 

q(α+ )

(tω)αq,ωf(ω) + t

a

(tσq,ω)αq,ωDq,ωf(s)dq,ωs

=Iq,ωα+Dq,ωf(t)

+ f(ω)

q(α+ )(tω) α q,ω

=Iq,ωα+Dq,ωf(t)

+ f(ω)

q(α+ )

(tω)α.

Theorem . For f :IT

q,ω→R,α,β> ,q∈(, ),ω> ,and aIq,ωT , a

ω

tσq,ω(s)β–q,ωIq,ωα f(t)dq,ωs= .

Proof Forn∈N,

q,ωf

σq,ωn (a)= 

q(α) σqn,ω(a)

ω

σq,ωn (a) –σq,ω(s)α–q,ωf(s)dq,ωs

=[σ n

q,ω(a)( –q) –ω]

q(α)

k=

qkσq,ωn (a) –σq,ωn+k(a)α–q,ωq,ωn+k(a).

Employing (A) implies that (σq,ωn (a) –σq,ωn+k(a))α–q,ω = . Thus,

q,ωf

σq,ωn (a)= . (.)

Finally, using Definition ., we have

a

ω

tσq,ω(s)β–q,ωIq,ωα f(t)dq,ωs

=( –q)aω

k=

qktσq,ωk+(a)β–q,ωIq,ωα q,ωk (a)= .

Theorem . For f :IT

q,ω→R,α,β> ,q∈(, ),ω> ,and aIq,ωT ,

q,ω

q,ωf(t)

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Proof FortIT

From Theorem ., we have t

Similarly, we have q,ωI

Proof Using Lemma .(c) and Lemma .(b), we have

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=Dq,ω

Proof Substitutingf asDq,ωf into Theorem ., we have

Repeating the same procedure as abovep–  times, we obtain

4 Fractional Hahn difference operator of Riemann-Liouville type

In this section, we introduce a fractional Hahn difference operator of Riemann-Liouville type.

Definition . Forα,ω> ,q∈(, ), andf defined on [ω,T]q,ω, the fractional Hahn difference operator of Riemann-Liouville type of orderαis defined by

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Proof For someN–  <α<N,N∈N, by using Definition . and Lemma ., we have

q,ωf(t) =DNq,ωIq,ωN–αf(t) =DN–q,ωDq,ωIq,ωN–αf(t)

=DN–q,ω

Dq,ω

q(Nα) t

ω

tσq,ω(s)N–α–q,ω f(s)dq,ωs

=DN–q,ω

q(Nα– ) t

ω

tσq,ω(s)N–α–q,ω f(s)dq,ωs

+ 

q(Nα)

σq,ω(t) –σq,ω(t)N–α–q,ω f(t)

=DN– q,ω

q(Nα– ) t

ω

tσq,ω(s)N–α–q,ω f(s)dq,ωs

.

RepeatingN–  times, we obtain

q,ωf(t) =DN–Nq,ω

q(–α) t

ω

tσq,ω(s)–α–q,ω f(s)dq,ωs

+ 

q( –α)(–α– ) –α

q,ωf(t+α+ )

= 

q(–α) t

ω

tσq,ω(s)–α–q,ω f(s)dq,ωs

=[( –q)tω]

q(–α) ∞

k=

qktσq,ωk+(t)–α–q,ω q,ωk (t)

=( –q)(tω) –α

q(–α) ∞

k=

qk –qk+–α–q,ω q,ωk (t).

In the following theorem, we introduce the properties of fractional Hahn difference op-erator of Riemann-Liouville type.

Theorem . Forα> ,q∈(, ),ω> ,and f :IT q,ω→R, q,ωIq,ωα f(t) =f(t).

Proof For someN–  <α<N,N∈N,

q,ωIq,ωα f(t) =DNq,ωIq,ωN–αIq,ωα f(t)

=DNq,ωIq,ωα Iq,ωN–αf(t)=DNq,ωIq,ωN f(t) =f(t).

Theorem . Forα> ,q∈(, ),ω> ,and f :IT q,ω→R,

q,ωDαq,ωf(t) =f(t) – N–

k=

(tω)α–N+k

q(αN+k+ )

Dα–N+kq,ω f(ω)

,

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Proof From Definition ., we have

q,ωD α

q,ωf(t) =I α q,ω

DNq,ωIq,ωN–αf(t)=Iq,ωα DNq,ωIq,ωN–αf(t).

Using Theorem ., we obtain

q,ωD α

q,ωf(t) =DNq,ω

q,ωIq,ωN–αf(t)

N–

k=

(tω)α–N+k

q(αN+k+ )D k q,ω

IN–α

q,ω f(ω)

=DNq,ωIq,ωN f(t)– N–

k=

(tω)α–N+k

q(αN+k+ )

Dk–(N–α)q,ω f(ω)

=f(t) – N–

k=

(tω)α–N+k

q(αN+k+ )

Dα–N+kq,ω f(ω).

Corollary . Letα> ,q∈(, ),ω> ,and f :Iq,ωT →R,

q,ωD α

q,ωf(t) =f(t) +C(tω)α–+· · ·+CN(tω)α–N

for some Ci∈R,i=N,N and N–  <α<N,N∈N.

5 Fractional Hahn difference operator of Caputo type

Now, we introduce a fractional Hahn difference operator of Caputo type.

Definition . Forα,ω> ,q∈(, ), andf defined on [ω,T]q,ω, the fractional Hahn

difference operator of Caputo type of orderαis defined by

CDα q,ωf(t) :=

IN–α

q,ω DNq,ωf

(t)

= 

q(Nα) t

ω

tσq,ω(s)N–α–q,ω DNq,ωf(s)dq,ωs,

andCD

q,ωf(t) =f(t), whereN–  <α<N,N∈N. Theorem . Forα> ,q∈(, ),ω> ,and f :IT

q,ω→R,

CDα q,ωf(t) =

[( –q)tω]

q(Nα) ∞

k=

qktσq,ωk+(t)q,ωN–α–DNq,ωq,ωk (t)

=( –q)(tω) N–α

q(Nα) ∞

k=

qk –qk+Nq,ω–α–DNq,ωq,ωk (t),

(13)

Proof FortIT

q,ω, by Definition ., we have

CDα

q,ωf(t) =Iq,ωN–αDNq,ωf(t) = 

q(Nα) t

ω

tσq,ω(s)Nq,ω–α–DNq,ωf(s)dq,ωs

=[( –q)tω]

q(Nα) ∞

k=

qktσq,ωk+(t)q,ωN–α–DNq,ωq,ωk (t)

=( –q)(tω) N–α

q(Nα) ∞

k=

qk –qk+Nq,ω–α–DNq,ωq,ωk (t).

In the following theorem, we introduce the properties of fractional Hahn difference op-erator of Caputo type.

Theorem . Forα> ,q∈(, ),ω> ,and f :IT q,ω→R, CDα

q,ωI α

q,ωf(t) =f(t).

Proof For someN–  <α<N,N∈N, under Definition . and Theorem ., we have CDα

q,ωI α

q,ωf(t) =Iq,ωN–αDNq,ωI α

q,ωf(t) =Iq,ωN–αDNq,ω–αf(t)

=f(t) – N–

k=

(tω)k–α

q(kα+ )

Dkq,ω q,ωf(ω)

.

From (.), we have

N–

k=

(tω)k–α

q(kα+ )

Dkq,ωIq,ωα f(ω)= .

It implies that

CDα q,ωI

α

q,ωf(t) =f(t).

Theorem . Forα> ,q∈(, ),ω> ,and f :IT q,ω→R,

q,ωCDαq,ωf(t) =f(t) – N–

k=

(tω)k

q(k+ )

Dkq,ωf(ω)

,

where N–  <α<N,N∈N.

Proof From Definition . and Theorem ., we have

q,ωCD α

q,ωf(t) =I α q,ω

IN–α

q,ω DNq,ωf(t)

=Iq,ωN DNq,ωf(t)

=f(t) – N–

k=

(tω)k

q(k+ )

Dkq,ωf(ω)

.

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Corollary . Letα> ,q∈(, ),ω> ,and f :IT q,ω→R,

q,ωCD α

q,ωf(t) =f(t) +C+C(tω) +· · ·+CN–(tω)N–

for some Ci∈R,i=N,N–,and N–  <α<N,N∈N.

6 Conclusions

In this paper, we have introduced a fractional Hahn integral, Riemann-Liouville and Ca-puto fractional Hahn difference operators. Many properties of these fractional Hahn oper-ators have been proved. This work is certainly not complete and should be a starting point of many other works. For example, in future works, one could define the Laplace trans-form for Hahn calculus. Also, another work will be to find the Hahn-convolution product and compute its Hahn-Laplace transform, so we could be able to solve many more Hahn difference equations.

Acknowledgements

This research is supported by Suan Dusit University, Thailand.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

TS and TB worked together in the derivation of the mathematical results. All authors read and approved the final manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 10 August 2017 Accepted: 27 October 2017

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