# Aβ-Convolution Theorem Associated with the General Quantum Difference Operator

Authors: Enas M. Shehata; Rasha M. El Zafarani
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we prove some properties of theβ-partial derivative. We define the β-convolution of two functions associated with the general quantum difference operator, Dβft=fβt−ft/βt−t; β is a strictly increasing continuous function. Moreover, we study the shift, the associative law, and the β-differentiability of the β-convolution. Furthermore, we prove the β-convolution theorem and give some applications.


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## Body

## 1. Introduction and Preliminaries

The classical convolution of two real functionsu and v is defined by
(1)u∗vt=∫0tut−svsds,where ut−s=ft,s is the usual shift of the function u defined on ℝ which is considered as the unique solution of the first-order partial differential equation
(2)∂∂tft,s=−∂∂sft,s,t,s∈ℝ,ft,0=ut,t∈ℝ.The convolution theorem highly extends the potential of Laplace transform in solving ordinary differential equations. Furthermore, the operation of the convolution is needed in understanding many topics such as partial differential equations, Green’s functions, and forming the general solution of some kinds of boundary value problems. See [1–4] for some applications of the convolution in electrical engineering, physics, and theory of distribution. The classical convolution theorem says that the Laplace transform of the convolution u∗v of the two functions u and v is equal to the Laplace transform of u multiplied by the Laplace transform of v. Recently, there are many versions of the convolution theorem such as h-convolution theorem, q-convolution theorem, convolution theorem on time scale, and q,h-convolution theorem on discrete time scale, see, e.g., [5–8]. Bohner and Guseinov [9] studied the convolution theorem on a time scale T, where the convolution of two functions u,v:T⟶ℝ is defined by
(3)u∗vt=∫t0tu^t,σsvsΔs,t,t0∈T.Here,u^t,s is the shift of the given function u defined on T which is considered as the unique solution of the shifting problem
(4)u∧Δtt,σs=−u∧Δst,s,t,s∈T,u^t,t0=ut,t,t0∈T,where Δ is the delta differentiation and σ is the forward jump operator in T. In [7], the convolution theorem of two positive real functions u1t and u2t is defined by
(5)Lqu1t∗u2t=Lqu1tLqu2t,where Lqut=∫0∞eq−stutdt,s∈ℂ,Rs>0, and
(6)u1t∗u2t=∫0tu1τu2t−τdτ.It should be noted that, here in [7], the Laplace transform denoted by Lq and the related convolution are defined by the usual integral; however, the q-exponential eq−st is used in the form 1+q−1st−1/q−1,t≥0,q>1. In [10], the q-convolution is defined by
(7)u1t∗u2t=∫0tu1τu2t−τdqτ.Hamza et al. [11] defined the general quantum β-difference operator Dβ by:
(8)Dβft=fβt−ftβt−t,t≠s0,f′s0,t=s0,where the functionβ is a strictly increasing and continuous defined on an interval I⊆ℝ, β has only one fixed point s0∈I, and the inequality t−s0βt−t≤0 holds for all t∈I, accordingly s0=limk⟶∞βkt, βkt=β∘β∘⋯∘βt (k-times). A function f is said to be β-differentiable on I, if the ordinary derivative f′s0 exists. The β-difference operator Dβ produces the Jackson q-difference operator, the Hahn difference operator, and the power quantum difference operator when βt=qt, βt=qt+ω, and βt=qtn, respectively; t∈I, q∈0,1, ω>0, and n is odd number, see [12–14]. Quantum difference operators consider a good tool in dealing with sets of nondifferentiable functions in the usual concept; furthermore, it has an interesting role due to their applications in several sciences, see, e.g., [15–18]. Recent quantum calculus applications can be found in [19–21].Theβ-integral of f:I⟶ℝ is defined in [11] by
(9)∫acftdβt=∫s0cftdβt−∫s0aftdβt,a,c∈I,where
(10)Fx=∫s0xftdβt=∑k=0∞βkx−βk+1xfβkx,x∈I,provided that the series in the right hand side converges. Furthermore, if f is continuous at s0, then F exists and is continuous at s0, and DβFx exists for all x∈I, DβFx=fx. The β-exponential functions are defined, in [22], by
(11)eλ,βt=1∏k=0∞1−λβktβkt−βk+1t,Eλ,βt=∏k=0∞1+λβktβkt−βk+1t.If the functionf:I⟶ℝ is β-differentiable and Dβft=0 for all t∈I, then f is constant function and
(12)ft=fs0,forallt∈I,see [11]. In [23] (p. 126), the β-partial derivative of ft,y with respect to t is defined by
(13)∂β∂βtft,y=fβt,y−ft,yβt−t,t≠s0.Also, ifFt≔∫s0bft,ydβy,b,t∈I. Then, DβFt at t≠s0 exists and is given by
(14)DβFt=∫s0b∂β∂βtft,ydβy.Moreover, the Leibniz’s rule is proved in [23] such that if Ft≔∫s0tft,ydβy, then DβFt at t≠s0 exists and is given by
(15)DβFt=fβt,t+∫s0t∂β∂βtft,ydβy.The general quantum Laplace transformLβ and its inverse Lβ−1 and some of their properties were deduced in [24]. The β-Laplace transform Lβ is defined by:
(16)Lβft≔∫s0∞fte⊖βz,ββtdβt,such that the improperβ-integral, see [24] (Definition 3.5), in (16) exists, where s0∈I, f∈Vs0,∞,ℂ, which is the set of β-integrable functions over each compact subinterval of s0,∞, and z satisfies 1+zβt−t≠0. The function f is said to be of exponential order λ>0, if there exist a constant M>0 such that ft≤Meλ,βt for all t∈s0,∞. Let p,q:I⟶ℂ be continuous functions at s0 and satisfy the condition 1+βt−tpt≠0 for all t∈I, then the following properties hold [24]:(i1) e⊖βp,βt=1/ep,βt(i2) e⊖βp,ββt=e⊖βp,βt/1+βt−tpt=−⊖βpt/pte⊖βp,βt(i3) Dβe⊖βp,βt=⊖βpte⊖βp,βtwhere⊖βpt=−pt/1+βt−tpt.Recently, Cardoso [25] investigated the β-Lagrange’s identity for the β-Sturm-Liouville eigenvalue problem and proved that it is self-adjoint in Lβ2a,b. For more results in β-calculus, we refer the readers to see [26–29].The current paper is organized as follows. In Section2, we introduce some properties of the β-partial derivative. In Section 3, we define the β-convolution of two functions and study their shift, associativity, and the β-differentiability. Moreover, we prove the β-convolution theorem and give some applications.


## 2.β-Partial Derivative

In this section, we introduce some properties of theβ-partial derivative.Theorem 1.
Letf,g:I×I⟶ℝ. Assume that the β-partial derivatives of ft,τ,gt,τ, t,τ∈I with respect to t at t≠s0 exist. Then:
(i)
Theβ-partial derivative of the product fgt,τ with respect to t at t≠s0 is given by(17)∂β∂βtfgt,τ=∂β∂βtft,τgβt,τ+ft,τ∂β∂βtgt,τ=∂β∂βtft,τgt,τ+fβt,τ∂β∂βtgt,τ,where fgt,τ=ft,τgt,τ.
(ii)
Theβ-partial derivative of f/gt,τ with respect to t at t≠s0 is given by(18)∂β∂βtfgt,τ=∂β/∂βtft,τgt,τ−ft,τ∂β/∂βtgt,τgt,τgβt,τ,provided that gt,τgβt,τ≠0.Proof. (i).
(19)∂β∂βtfgt,τ=fgβt,τ−fgt,τβt−t=fβt,τ−ft,τβt−tgβt,τ+ft,τgβt,τ−gt,τβt−t=∂β∂βtft,τgβt,τ+ft,τ∂β∂βtgt,τ.
Similarly, we can prove that(20)∂β∂βtfgt,τ=∂β∂βtft,τgt,τ+fβt,τ∂β∂βtgt,τ.
In the same way, we can prove (ii).Lemma 2.
Letf:I×I⟶ℝ and assume that the β-partial derivative of ft,τ, t,τ∈I with respect to τ exists and is continuous at t,s0. Then
(21)∫τβτft,ydβy=βτ−τft,τ.Proof.
LetFt,τ=∫s0τft,ydβy. Since Ft,τ is continuous at t,s0 and ∂β/∂βτFt,τ=ft,τ, then
(22)∫τβτft,ydβy=∫s0βτft,ydβy−∫s0τft,ydβy=Ft,βτ−Ft,τ.
SinceFt,βτ−Ft,τ=βτ−τ∂β/∂βτFt,τ, then
(23)∫τβτft,ydβy=βτ−τft,τ.Lemma 3.
Letf:I×I⟶ℝ. If the β-partial derivative of ft,τ, t,τ∈I with respect to t exists and is continuous at s0,τ, then
(24)∫ab∂β∂βtft,τdβt=fb,τ−fa,τforalla,b∈I.Proof.
We have(25)∫s0b∂β∂βtft,τdβt=∑k=0∞βkb−βk+1b∂β∂βtfβkb,τ=∑k=0∞fβkb,τ−fβk+1b,τ=limn⟶∞∑k=0nfβkb,τ−fβk+1b,τ=fb,τ−fs0,τ.
Similarly,(26)∫s0a∂β∂βtft,τdβt=fa,τ−fs0,τ.
Hence,(27)∫ab∂β∂βtft,τdβt=fb,τ−fa,τ.Lemma 4 (integration by parts).
Letf,g:I×I⟶ℝ. Assume that the β-partial derivatives of ft,τ,gt,τ, t,τ∈I with respect to t exist and are both continuous at s0,τ. Then
(28)∫abft,τ∂β∂βtgt,τdβt=fb,τgb,τ−fa,τga,τ−∫ab∂β∂βtft,τgβt,τdβt,a,b∈I.Proof.
Since(29)∂β∂βtfgt,τ=ft,τ∂β∂βtgt,τ+gβt,τ∂β∂βtft,τ.
Then(30)∫abft,τ∂β∂βtgt,τdβt=∫ab∂β∂βtfgt,τdβt−∫ab∂β∂βtft,τgβt,τdβt,a,b∈I.
By Lemma3, we get
(31)∫ab∂β∂βtfgt,τdβt=fb,τgb,τ−fa,τga,τ.
Therefore,(32)∫abft,τ∂β∂βtgt,τdβt=fb,τgb,τ−fa,τga,τ−∫ab∂β∂βtft,τgβt,τdβt,a,b∈I.


## 3. Theβ-Convolution Theorem

In the following, we define theβ-convolution of two functions and prove some of its properties. Furthermore, we prove the β-convolution theorem and give some applications.Definition 5.
Consider theβ-shifting problem
(33)∂β∂βtzt,βτ=−∂β∂βτzt,τ,t,τ∈I,t≥τ≥s0,(34)zt,s0=ft,t∈I,t≥s0.
We denote the shift of a functionf∈Vs0,∞,ℂ by f^t,τ, t,τ∈I which is the solution of (33).We will assume in this paper that the problem (33) has a unique solution z=f^.Definition 6 (β-convolution).
Theβ-convolution f∗g of two functions f,g∈Vs0,∞,ℂ is defined by
(35)f∗gt=∫s0tf^t,βτgτdβτ,t∈I,
wheref^ is the shift of f given in Definition 5.
Note that in the case ofβt=qt, s0=0, we get the q-shifting problem
(36)∂q∂qtzt,qτ=−∂q∂qτzt,τ,zt,0=ft,and the q-convolution
(37)f∗gt=∫0tf^t,qτgτdqτ,we refer the reader to see [6]. Furthermore, in the case of βt=t, we get the usual shifting problem
(38)∂∂tzt,τ=−∂∂τzt,τ,zt,s0=ft,with the unique solution zt,τ=ft−τ=f^t,τ which is the usual shift, and we get the classical convolution
(39)f∗gt=∫0tft−τgτdτ.Lemma 7.
Letf^ be the shift of f. Then, f^t,t=fs0 for all t∈I.Proof.
Letf^ be the shift of f. Set f^t,t=Ft. From the β-shifting problem (33), we have
(40)0=∂β∂βtf^t,βτ+∂β∂βτf^t,τ=f^βt,βτ−f^t,βτβt−t+f^t,βτ−f^t,τβτ−τ.
Taket=τ, consequently
(41)0=f^βt,βt−f^t,βtβt−t+f^t,βt−f^t,tβt−t=f^βt,βt−f^t,tβt−t=Fβt−Ftβt−t=DβFt,t≠s0.
By equation (12), f^t,t=Ft=Fs0, for all t∈I. Moreover, from the initial condition (33), Fs0=fs0. Then, f^t,t=fs0 for all t∈I.Lemma 8.
Define the functionF by
(42)Ft,τ≔∫τtf^t,βyg^y,τdβy.
Then,∂β/∂βtFt,τ and ∂β/∂βτFt,τ exist and are given by:
(43)∂β∂βtFt,τ=f^βt,βtg^t,τ+∫τt∂β∂βtf^t,βyg^y,τdβy,∂β∂βτFt,τ=∫τtf^t,βy∂β∂βτg^y,τdβy−f^t,βτg^τ,βτ.Proof.
We have(44)Ft,τ=∫s0tf^t,βyg^y,τdβy−∫s0τf^t,βyg^y,τdβy.
Then,(45)∂β∂βtFt,τ=∂β∂βt∫s0tf^t,βyg^y,τdβy−∂β∂βt∫s0τf^t,βyg^y,τdβy,∂β∂βτFt,τ=∂β∂βτ∫s0tf^t,βyg^y,τdβy−∂β∂βτ∫s0τf^t,βyg^y,τdβy.
Now,(46)∂β∂βt∫s0tf^t,βyg^y,τdβy=1βt−t∑k=0∞βk+1t−βk+2tf^βt,βk+2tg^βk+1t,τ−∑k=0∞βkt−βk+1tf^t,βk+1tg^βkt,τ=1βt−t∑k=1∞βkt−βk+1tf^βt,βk+1tg^βkt,τ−f^t,βk+1tg^βkt,τ−t−βtf^t,βtg^t,τβt−t=1βt−t∑k=0∞βkt−βk+1tf^βt,βk+1tg^βkt,τ−f^t,βk+1tg^βkt,τ−t−βtβt−tf^βt,βtg^t,τ−f^t,βtg^t,τ+f^t,βtg^t,τ=f^βt,βtg^t,τ+∫s0t∂β∂βtf^t,βyg^y,τdβy.
Similarly,(47)∂β∂βt∫s0τf^t,βyg^y,τdβy=∫s0τ∂β∂βtf^t,βyg^y,τdβy.
Therefore,(48)∂β∂βtFt,τ=f^βt,βtg^t,τ+∫s0t∂β∂βtf^t,βyg^y,τdβy−∫s0τ∂β∂βtf^t,βyg^y,τdβy=f^βt,βtg^t,τ+∫τt∂β∂βtf^t,βyg^y,τdβy.
Using the same technique, we obtain(49)∂β∂βτFt,τ=∫s0τf^t,βy∂β∂βτg^y,τdβy−f^t,βτg^τ,βτ−∫s0tf^t,βy∂β∂βτg^y,τdβy=∫τtf^t,βy∂β∂βτg^y,τdβy−f^t,βτg^τ,βτ.Theorem 9.
The shift of theβ-convolution of two functions f,g is given by
(50)f∗g^t,τ=∫τtf^t,βyg^y,τdβy.Proof.
We aim to prove that the integral in equation (50) is a solution of the β-shifting problem (33). Set
(51)Ft,τ=∫τtf^t,βyg^y,τdβy.
Takeτ=s0, then
(52)Ft,s0=∫s0tf^t,βyg^y,s0dβy.
From the initial condition of the shift problem (33), we get g^y,s0=gy. Consequently,
(53)Ft,s0=∫s0tf^t,βygydβy=f∗gt.
Using Lemmas7 and 8, we get
(54)∂β∂βtFt,βτ+∂β∂βτFt,τ=−∫βτt∂β∂βyf^t,yg^y,βτdβy+fs0g^t,βτ+∫τtf^t,βy∂β∂βτg^y,τdβy−f^t,βτg^τ,βτ.
We have, using the integration by parts,(55)∫βτt∂β∂βyf^t,yg^y,βτdβy=f^t,tg^t,βτ−f^t,βτg^βτ,βτ−∫βτtf^t,βy∂β∂βyg^y,βτdβy=fs0g^t,βτ−f^t,βτgs0+∫βτtf^t,βy∂β∂βτg^y,τdβy.
Consequently,(56)∂β∂βtFt,βτ+∂β∂βτFt,τ=−fs0g^t,βτ+f^t,βτgs0−∫βτtf^t,βy∂β∂βτg^y,τdβy+fs0g^t,βτ−f^t,βτg^τ,βτ+∫τtf^t,βy∂β∂βτg^y,τdβy=f^t,βτgs0+∫τβτf^t,βy∂β∂βτg^y,τdβy−f^t,βτg^τ,βτ.
By using Lemma2, we get
(57)∂β∂βtFt,βτ+∂β∂βτFt,τ=f^t,βτgs0+βτ−τf^t,βτ∂β∂βτg^τ,τ−f^t,βτg^τ,βτ=f^t,βτgs0+f^t,βτg^τ,βτ−g^τ,τ−f^t,βτg^τ,βτ=0.
Therefore,(58)∂β∂βtFt,βτ=−∂β∂βτFt,τ.
Hence,Ft,τ=f∗g^t,τ is the solution of equation (58).The proof of the following Lemma10 will be omitted since it is a direct consequence of Definitions 5 and 6 and the linearity of the β-integral.Lemma 10.
Theβ-convolution f∗g defined in (35) satisfies the following properties:
(i)
cf∗g=cf∗g=f∗cg, c∈ℝ(ii)
f∗g+h=f∗g+f∗h(iii)
f∗ag+bh=af∗g+bf∗h, a,b∈ℝLemma 11.
Letw:J×J⟶ℝ be continuous function at s0,s0, s0∈J, and J is a compact subinterval of I, where wy,τ=f^t,βyg^y,βτhτ. Then,
(59)∫s0t∫βτtf^t,βyg^y,βτhτdβydβτ=∫s0t∫s0yf^t,βyg^y,βτhτdβτdβy.Proof.
First, we refer the reader to see Fubini’s Theorem in [23] (p. 132). Now, let
(60)Wt,τ=∫βτtwy,τdβy,ϕ1t=∫s0tWt,τdβτ,Gy=∫s0ywy,τdβτ,ϕ2t=∫s0tGydβy.
Then,(61)Dβϕ1t=∫s0t∂β∂βtWt,τdβτ,∂β∂βtWt,τ=1βt−t∑k=0∞βk+1t−βk+2twβk+1t,τ−∑k=0∞βkt−βk+1twβkt,τ=wt,τ.
Therefore,Dβϕ1t=∫s0twt,τdβτ. Also,
(62)Dβϕ2t=Dβ∫s0tGydβy=Gt=∫s0twt,τdβτ=Dβϕ1t.
Hence,ϕ1t=ϕ2t, t∈I.Theorem 12 (Associativity of theβ-convolution).
Theβ-convolution is associative, that is,
(63)f∗g∗h=f∗g∗h.Proof.
Using Theorem9 and Lemma 11, we have
(64)f∗g∗ht=∫s0tf∗g^t,βτhτdβτ=∫s0t∫βτtf^t,βyg^y,βτhτdβydβτ=∫s0t∫s0yf^t,βyg^y,βτhτdβτdβy=∫s0tf^t,βyg∗hydβy=f∗g∗ht.Theorem 13.
Letf and g be β-differentiable functions, then we have
(65)Dβf∗gt=ftgs0+f∗Dβgt.Proof.
We have(66)f∗gt=∫s0tf^t,βτgτdβτ,t∈I.
Using Lemma8, then,
(67)Dβf∗gt=∫s0t∂β∂βtf^t,βτgτdβτ+f^βt,βtgt.
By Definition5 and Lemma 7(68)∂β∂βtf^t,βτ=−∂β∂βτf^t,τ,f^βt,βt=fs0,therefore,
(69)Dβf∗gt=−∫s0t∂β∂βτf^t,τgτdβτ+fs0gt.
We get, by the integration by parts,(70)Dβf∗gt=−f^t,tgt+f^t,s0gs0+∫s0tf^t,βτ∂β∂βτgτdβτ+fs0gt=−fs0gt+ftgs0+∫s0tf^t,βτ∂β∂βτgτdβτ+fs0gt=ftgs0+∫s0tf^t,βτ∂β∂βτgτdβτ.
Since∂β/∂βτgτ=Dβgτ, then,
(71)Dβf∗gt=ftgs0+∫s0tf^t,βτDβgτdβτ=ftgs0+f∗Dβgt.
This completes the proof of equation of (65).Corollary 14.
For allk∈ℕ0, we have
(72)Dβkf∗g=f∗Dβkg+∑r=0k−1DβrfDβk−1−rgs0.Proof.
Using the induction, fork=1 equation (72) is equation (65). Suppose that for k=m∈ℕ0 equation (72) is true, i.e.
(73)Dβmf∗gt=f∗Dβmgt+∑r=0m−1DβrftDβm−1−rgs0.
Then, by using equation (65) and from the linearity of Dβ, we get
(74)Dβm+1f∗gt=Dβf∗Dβmgt+∑r=0m−1DβrftDβm−1−rgs0=ftDβmgs0+f∗Dβm+1gt+∑r=0m−1Dβr+1ftDβm−1−rgs0=f∗Dβm+1gt+∑r=0mDβrftDβm−rgs0.
Therefore, equation (72) is true for k=m+1, and thus, it holds for any k∈ℕ.Lemma 15.
LetΨ be a function defined by
(75)Ψτ=∫τ∞f^t,τez,ββt,τdβt.
Then,Ψ is constant, where ez,ββt,τ=ez,ββt/ez,βτ.Proof.
We will show thatDβΨτ≡0 for all τ, then Ψ is constant.
By using Theorem1 and Lemma 8, we find
(76)DβΨτ=∫τ∞∂β/∂βτf^t,τez,ββt,τ−⊖βzτez,ββt,τf^t,τez,ββt,τez,ββt,βτdβt−f^τ,βτez,ββτ,βτ.
Since∂β/∂βτf^t,τ=−∂β/∂βtf^t,βτ, ez,ββτ,βτ=1, we get
(77)DβΨτ=∫τ∞−∂β/∂βtf^t,βτez,ββt,βτ+z/1+zβτ−τf^t,τez,ββt,βτdβt−f^τ,βτ=−∫τ∞∂β/∂βtf^t,βτez,ββt,βτdβt+z∫τ∞f^t,τez,ββt,τdβt−f^τ,βτ=zΨτ−∫τ∞∂β/∂βtf^t,βτez,ββt,βτdβt−f^τ,βτ=zΨτ−∫τβτ∂β/∂βtf^t,βτez,ββt,βτdβt−∫βτ∞∂β/∂βtf^t,βτez,ββt,βτdβt−f^τ,βτ.
Also, sincef^βτ,βτ=fs0 and
(78)∂β∂βtf^t,βτe⊖βz,ββt,βτ=∂β∂βtf^t,βτe⊖βz,βt,βτ−⊖βzte⊖βz,βt,βτf^t,βτ.
Hence, by Lemma3, we get
(79)∫τβτ∂β/∂βtf^t,βτez,ββt,βτdβt−f^τ,βτ=fs0,∫βτ∞∂β∂βtf^t,βτe⊖βz,ββt,βτdβt=∫βτ∞∂β∂βtf^t,βτe⊖βz,βt,βτdβt−∫βτ∞⊖βzte⊖βz,βt,βτf^t,βτdβt=−fs0+z∫βτ∞f^t,βτe⊖βz,ββt,βτdβt=−fs0+zΨβτ.
Therefore,(80)DβΨτ=zΨτ−fs0+fs0−zΨβτ=−zΨβτ−Ψτ=−zβτ−τDβΨτ,
and then1+βτ−τzDβΨτ=0, i.e., DβΨτ=0.Now, we prove theβ-convolution theorem.Theorem 16 (β-Convolution theorem).
Letf,g∈Vs0,∞,ℂ be two functions of exponential order and their β-convolution f∗g is defined by (35). Then,
(81)Lβf∗g=Lβf·Lβg.Proof.
We have(82)Lβf∗g=∫s0∞f∗gtez,ββtdβt=∫s0∞1ez,ββt∫s0tf^t,βτgτdβτdβt.
Then,(83)Lβf∗g=∫s0∞gτez,ββτ∫βτ∞f^t,βτez,ββτez,ββtdβtdβτ=∫s0∞gτez,ββτΨβτdβτ=∫s0∞gτe⊖βz,ββτΨβτdβτ.
Using Lemma15, equation (12), we get Ψτ=Ψβτ=Ψs0. Moreover,
(84)Ψs0=∫s0∞f^t,s0ez,βs0ez,ββtdβt=∫s0∞fte⊖βz,ββtdβt=Lβf.
Hence,Lβf∗g=Lβf·Lβg.Corollary 17.
Letf^ be the shift of f, τ∈I with τ≥s0. Define the function χτ by
(85)χτt=0,t<τ,1,t≥τ.
Then(86)Lβχτf^·,τ=e⊖βz,βτLβf.Proof.
(87)Lβχτf^·,τ=∫s0∞χτtf^t,τe⊖βz,ββtdβt=∫τ∞f^t,τe⊖βz,ββtdβt=e⊖βz,βτ∫τ∞f^t,τe⊖βz,ββt,τdβt=e⊖βz,βτΨτ,
whereΨ is defined in equation (75). Then, by Lemma 15, Ψτ≡Ψs0=Lβf.


### 3.1. Applications on theβ-Convolution Theorem

Here, we apply theβ-convolution theorem to find the following:
(1)
eλ,βt∗eμ,βt,(2)
t∗eλ,βt,(3)
sinλ,βt∗sinλ,βt.(1)
We have by theβ-convolution Theorem 16 and [24] (Theorem 3.10) that(88)Lβeλ,βt∗eμ,βt=1z−λz−μ=1λ−μz−λ−1λ−μz−μ.Then, by [24] (Corollary 3.22), we get
(89)eλ,βt∗eμ,βt=1λ−μLβ−11z−λ−1λ−μLβ−11z−μ=1λ−μeλ,βt−eμ,βt.(2)
We haveLβt=1/z2 by [24] (Example 3.23), then(90)Lβt∗eλ,βt=Lβt·Lβeλ,βt=1z2z−λ.Therefore,(91)t∗eλ,βt=Lβ−11z2z−λ=−1λ2Lβ−11z−1λLβ−11z2+1λ2Lβ−11z−λ=−1λ2−1λt+1λ2eλ,βt.(3)
Since(92)Lβsinλ,βt∗sinλ,βt=Lβsinλ,βt·Lβsinλ,βt.From [24] (Example 3.13), we have
(93)Lβsinλ,βt∗sinλ,βt=λ2z2+λ22.By [24] (Theorem 3.21),
(94)Lβ−11z2+λ22=sinλ,βt2λ3−cosλ,βt2λ2∫s0t11+λ2βτ−τ2dβτ−sinλ,βt2λ∫s0tβτ−τ1+λ2βτ−τ2dβτ.Hence,(95)sinλ,βt∗sinλ,βt=Lβ−1λ2z2+λ22=λ2Lβ−11z2+λ22=sinλ,βt2λ−cosλ,βt2∫s0t11+λ2βτ−τ2dβτ−λsinλ,βt2∫s0tβτ−τ1+λ2βτ−τ2dβτ.


## 3.1. Applications on theβ-Convolution Theorem

Here, we apply theβ-convolution theorem to find the following:
(1)
eλ,βt∗eμ,βt,(2)
t∗eλ,βt,(3)
sinλ,βt∗sinλ,βt.(1)
We have by theβ-convolution Theorem 16 and [24] (Theorem 3.10) that(88)Lβeλ,βt∗eμ,βt=1z−λz−μ=1λ−μz−λ−1λ−μz−μ.Then, by [24] (Corollary 3.22), we get
(89)eλ,βt∗eμ,βt=1λ−μLβ−11z−λ−1λ−μLβ−11z−μ=1λ−μeλ,βt−eμ,βt.(2)
We haveLβt=1/z2 by [24] (Example 3.23), then(90)Lβt∗eλ,βt=Lβt·Lβeλ,βt=1z2z−λ.Therefore,(91)t∗eλ,βt=Lβ−11z2z−λ=−1λ2Lβ−11z−1λLβ−11z2+1λ2Lβ−11z−λ=−1λ2−1λt+1λ2eλ,βt.(3)
Since(92)Lβsinλ,βt∗sinλ,βt=Lβsinλ,βt·Lβsinλ,βt.From [24] (Example 3.13), we have
(93)Lβsinλ,βt∗sinλ,βt=λ2z2+λ22.By [24] (Theorem 3.21),
(94)Lβ−11z2+λ22=sinλ,βt2λ3−cosλ,βt2λ2∫s0t11+λ2βτ−τ2dβτ−sinλ,βt2λ∫s0tβτ−τ1+λ2βτ−τ2dβτ.Hence,(95)sinλ,βt∗sinλ,βt=Lβ−1λ2z2+λ22=λ2Lβ−11z2+λ22=sinλ,βt2λ−cosλ,βt2∫s0t11+λ2βτ−τ2dβτ−λsinλ,βt2∫s0tβτ−τ1+λ2βτ−τ2dβτ.


## 4. Conclusion

In this paper, some properties of theβ-partial derivative corresponding to Dβft=fβt−ft/βt−t, βt≠t, were introduced. The associated β-convolution of two functions was presented, and also the shift, associativity, and β-differentiability of the β-convolution were studied. Moreover, the β-convolution theorem and some applications were introduced.



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