Estimates and properties of certain q-Mellin transform on generalized q-calculus theory

Abstract

This paper deals with the generalized q-theory of the q-Mellin transform and its certain properties in a set of q-generalized functions. Some related q-equivalence relations, q-quotients of sequences, q-convergence definitions, and q-delta sequences are represented. Along with that, a new q-convolution theorem of the estimated operator is obtained on the generalized context of q-Boehmians. On top of that, several results and q-Mellin spaces of q-Boehmians are discussed. Furthermore, certain continuous q-embeddings and an inversion formula are also discussed.

1 Introduction and preliminaries

The quantum calculus or the q-calculus theory has been given a noticeable importance and popularity due to its wide application in various fields of mathematics, statistics, and physics [1]. The q-calculus theory has appeared as a connection between mathematics and physics. Recently, this topic has attracted the attention of several researchers, and a variety of results have been derived in various areas of research including number theory, hypergeometric functions, orthogonal polynomials, quantum theory, combinatorics, and electronics as well. The q-calculus begins with the definition of the q-analogue $d_{q}g$ of the differential

$$ d_{q}g ( t ) =g ( qt ) -g ( t ) $$

of the function g, where q is a fixed real number such that $0< q<1$ (see [1–3]). Having said this, we immediately get the q-analogue of the derivative of g as

$$ D_{q}g ( t ):=\frac{d_{q}g ( t ) }{d_{q}t}:= \frac{g ( t ) -g ( qt ) }{ ( 1-q ) t} \quad \text{for }t\neq 0 $$

and $D_{q}g ( 0 )=\lim_{t\longrightarrow 0}D_{q}g (t )=g^{{\prime }} ( 0 )$ provided $g^{{\prime }} ( 0 )$ exists. Also, when g is differentiable, the q-derivative $D_{q}g$ tends to $g^{{\prime }} ( 0 )$ as q tends to 1. It also satisfies the q-analogue of the Leibniz rule

$$ D_{q} \bigl( g_{1} ( t ) g ( t ) \bigr) =g ( t ) D_{q}g_{1} ( t ) +g_{1} ( qt ) D_{q}g ( t ). $$

The Jackson q-integrals from 0 to x and respectively from 0 to ∞ are defined by [1, 4]

$$\begin{aligned} & \int _{0}^{x}g ( t ) \,d_{q}t= ( 1-q ) t \sum_{0}^{\infty }g \bigl( tq^{k} \bigr) q^{k}, \end{aligned}$$

(1)

$$\begin{aligned} & \int _{0}^{\infty }g ( t ) \,d_{q}t= ( 1-q ) t \sum_{-\infty }^{\infty }g \bigl( q^{k} \bigr) q^{k}, \end{aligned}$$

(2)

when the sums converge absolutely. The Jackson q-integral on the generic interval $[ a,b ]$ is, therefore, given by [1, 5]

$$ \int _{a}^{b}g ( t ) \,d_{q}t= \int _{0}^{b}g ( t ) \,d_{q}t- \int _{0}^{a}g ( t ) \,d_{q}t. $$

The q-integration by parts for two functions f and g is defined by

$$ \int _{0}^{b}g_{2} ( t ) D_{q}g_{1} ( t ) \,d_{q}t=g_{1} ( b ) g_{2} ( b ) -g_{1} ( a ) g_{2} ( a ) - \int _{a}^{b}g_{1} ( qt ) D_{q}g_{2} ( t ) \,d_{q}t. $$

Arising from the notion of regular operators [6], the notion of a Boehmian was firstly introduced by Mikusinski and Mikusinski [7] to generalize distributions and regular operators [8]. Boehmians are equivalence classes of quotients of sequences constructed from an integral domain when the operations are interpreted as addition and convolution, see, e.g., [9–20]. In terms of the q-calculus concept, we introduce the concept of q-Boehmians to popularize the concept of q-calculus theory as follows:

For a complex linear space V and a subspace $( W,\ast ^{q} )$ of V, let $\overset{q}{\bullet }:V\times W\rightarrow V$ be a binary operation such that the undermentioned axioms (1)–(5) hold:

• (1) $( g_{1}+g_{2} ) \overset{q}{\bullet } \psi =g_{1}\overset{q}{\bullet }\psi +g_{2}\overset{q}{\bullet }\psi,\forall g_{1},g_{2} \in V\text{ and }\psi \in W$.

• (2) $( \alpha g ) \overset{q}{\bullet }\psi = \alpha ( g\overset{q}{\bullet }\psi ),\forall \alpha \in \mathbb{C},\forall g\in V\text{ and }\psi \in W$.

• (3) $g\overset{q}{\bullet } ( \psi _{1} \overset{q}{\bullet }\psi _{2} ) = ( g\overset{q}{\bullet }\psi _{1} ) \overset{q}{\bullet }\psi _{2},\forall g\in V\text{ and }\psi _{1},\psi _{2}\in W$.

• (4)
  $$\begin{aligned} & \text{If }g_{n}\rightarrow g\text{ in }V\text{ as }n\rightarrow \infty \text{ and }\psi \in W, \\ &\quad\text{then }g_{n} \overset{q}{\bullet }\psi \rightarrow g\overset{q}{ \bullet }\psi \text{ as }n \rightarrow \infty \text{ in }V. \end{aligned}$$

  (3)

• (5) A collection $\Delta _{q}$ of sequences from W such that, for all $( \varepsilon _{n} ), ( \phi _{n} ) \in \Delta _{q}$ and $( g_{n} ) \in W$, we have $\varepsilon _{n}\overset{q}{\bullet }\phi _{n}\in \Delta _{q}$ and

  $$ \text{if }g_{n}\rightarrow g\text{ in }V\text{ as }n\rightarrow \infty,\text{ then }g_{n}\overset{q}{\bullet }\varepsilon _{n}\rightarrow g \text{ as }n\rightarrow \infty. $$

Once the preceding axioms are applied, the name of a q-Boehmian is set to mean the equivalence class $\frac{g_{n}}{\delta _{n}}$ that arises from the equivalence relation

$$ g_{n}\overset{q}{\bullet }\varepsilon _{m}=g_{m} \overset{q}{\bullet }\varepsilon _{n},\quad\forall m,n\in \mathbb{N}, $$

(4)

where $( g_{n} ) \in V$ and $( \varepsilon _{n} ) \in \Delta _{q}$. The collection of all q-Boehmians is denoted by $\mathbb{B}_{q}$ which is the so-called Boehmian space. The classical linear space V is identified as a subset of the space $\mathbb{B}_{q}$ which can be recognized from the relation

$$ g\longrightarrow \frac{g\overset{q}{\bullet }\varepsilon _{n}}{\varepsilon _{n}}, $$

(5)

where $( \varepsilon _{n} ) \in \Delta _{q}$ is arbitrary. Two q-Boehmians $\frac{g_{n}}{\varepsilon _{n}}$ and $\frac{\varphi _{n}}{\epsilon _{n}}$ are said to be equal in $\mathbb{B}_{q}$ if $g_{n}\overset{q}{\bullet }\epsilon _{m}=\varphi _{m}\overset{q}{\bullet }\varepsilon _{n}, \forall m,n\in \mathbb{N}$. Addition in the space $\mathbb{B}_{q}$ is defined as

$$ \frac{g_{n}}{\varepsilon _{n}}+\frac{\varphi _{n}}{\epsilon _{n}}= \frac{g_{n}\overset{q}{\bullet }\epsilon _{n}+\varphi _{n}\overset{q}{\bullet }\varepsilon _{n}}{\varepsilon _{n}\overset{q}{\bullet }\epsilon _{n}}. $$

(6)

The scalar multiplication in the space $\mathbb{B}_{q}$ is defined as

$$ \alpha \frac{g_{n}}{\varepsilon _{n}}= \frac{\alpha g_{n}}{\varepsilon _{n}},\quad \alpha \in \mathbb{C}. $$

The q-convergence of type δ, $\beta _{n}\overset{\delta }{\rightarrow }\beta$, is defined in the space $\mathbb{B}_{q}$ when for $( \psi _{n} ) \in \Delta _{q}$ and each $k\in \mathbb{N}$ such that

$$ \beta _{n}\overset{q}{\bullet }\varepsilon _{k}\in V,\quad \forall k,n \in \mathbb{N}, \beta \overset{q}{\bullet }\varepsilon _{k} \in V, $$

(7)

we have $\beta _{n}\overset{q}{\bullet }\varepsilon _{k}\rightarrow \beta \overset{q}{\bullet }\varepsilon _{k}$ as $n\rightarrow \infty$ in V. The q-convergence $\beta _{n}\overset{\Delta _{q}}{\rightarrow }\beta$ of type $\Delta _{q}$ is defined when for some $( \varepsilon _{n} ) \in \Delta _{q}$ we have

$$ ( \beta _{n}-\beta ) \overset{q}{\bullet }\varepsilon _{n} \in V,\quad \forall n\in \mathbb{N} \quad\text{and}\quad ( \beta _{n}- \beta ) \overset{q}{\bullet }\varepsilon _{n}\rightarrow 0\quad\text{as }n \rightarrow \infty \text{ in }V. $$

(8)

The space of q-Boehmians emerging from the q-convergence assigns a complete quasi-normed space.

In recent work, several remarkable integral transforms were given different q-analogues in a q-calculus context [4, 21–24]. In the sequence of such q-integral transforms, we recall the q-Laplace integral transform [25–29], the q-Sumudu integral transform [2, 30–32], the q-Weyl fractional integral transform [33], the q-wavelet integral transform [34], the q-Mellin type integral transform [35], the Mangontarum integral transform [36, 37], the $E_{2;1}$ integral transform [38, 39], the natural integral transform [3], and many others, to mention but a few. In this paper, we discuss the generalized q-theory of the q-Mellin transform and obtain several results.

Let g be a function defined on $\mathbb{R}_{q,+},\mathbb{R}_{q,+}=\{q^{n}:n\in \mathbb{Z}\}$, then the q-Mellin transform was defined by [40], p. 521 as

$$ M_{q} \bigl( g ( t ) \bigr) ( \zeta ) = \int _{0}^{\infty }t^{\zeta -1}g ( t ) \,d_{q}t, $$

(9)

provided the q-integral converges. The integral (9) is analytic on the fundamental strip $\langle \alpha _{q,g};\beta _{q,g} \rangle$ and periodic with period $2i\pi \log ( q )$. The inversion formula for the q-analogue (9) is given by

$$ g(t)=\frac{\log (q)}{2i\pi (1-q)} \int _{c- \frac{i\pi }{\log (q)}}^{c+\frac{i\pi }{\log (q)}}M_{q}(g) (\zeta )t^{-\zeta }\,d\zeta, \quad t\in \mathbb{R}_{q,+}, $$

where $\alpha _{q,g}< c<\beta _{q,g}$. The asymptotic properties as well as the asymptotic singularities of the q-Mellin transform into asymptotic expansions of the original function for $x\rightarrow 0$ and $x\rightarrow \infty$ are given in [40]. Additionally, the asymptotic behavior at 0 or ∞ is studied using the q-Mellin transform.

Definition 1

The function g is said to be q-integrable on an interval $[0,\infty[$ provided the infinite series

$$ \sum_{n\in \mathbb{Z}}q^{n}g \bigl(q^{n} \bigr) $$

converges absolutely. The space of all q-integrable functions on $[0,\infty[$ is denoted by $L_{q}^{1} ( \mathbb{R}_{q,+} )$. In a better recognition, the space $L_{q}^{1} ( \mathbb{R}_{q,+} )$ is defined to be the space of all q-integrable functions g on $\mathbb{R}_{q,+}$ such that

$$ L_{q}^{1}g ( t ) =\frac{1}{1-q} \int _{0}^{ \infty } \bigl\vert g ( t ) \bigr\vert \,d_{q}t< \infty. $$

(10)

We denote by $\mathbb{D}_{q}$ the q-space of test functions of compact supports on $\mathbb{R}_{q,+}$, i.e., $\mathbb{D}_{q}$ is the q-space of all smooth functions $\kappa \in C^{\infty } ( \mathbb{R}_{q,+} )$ such that

$$ \mathbb{D}_{q}= \Bigl\{ \kappa \in C^{\infty } ( \mathbb{R}_{q,+} ):\sup_{0< t< \infty } \bigl\vert D_{q}\kappa ( t ) \bigr\vert < \infty \Bigr\} . $$

(11)

However, this theory is new and might be developing a new area of research. It investigates a generalization to the q-theory of calculus [40] and hence all results can be popularized. Every element in the space $L_{q}^{1} ( \mathbb{R}_{q,+} )$ is identified as a member in the generalized theory. To this aim, we spread our results into five sections. In Sect. 1, we recall some definitions and preliminaries from the q-calculus theory. In Sect. 2, we derive q-delta sequences, q-convolution theorems and establish a space of q-Boehmians. In Sect. 3, we establish a space of q-ultraBoehmians. In Sect. 4, we generalize definitions and obtain several properties of the q-Mellin transform. In Sect. 5 we include several results.

2 The space $\mathbb{B}$

In this section, we strive to establish the space $\mathbb{B}$ of q-Boehmians. Henceforth, we denote by $\Delta _{q}$ the set of all sequences from $\mathbb{D}_{q}$ such that the undermentioned identities $\Delta _{q}^{1}-\Delta _{q}^{3}$ hold, where

$$ \begin{aligned}& \Delta _{q}^{1}: \int _{0}^{\infty } \bigl\vert \varepsilon _{n} ( t ) \bigr\vert \,d_{q}t=1,\quad \forall n\in \mathbb{N}, \\ &\Delta _{q}^{2}: \bigl\vert \varepsilon _{n} ( t ) \bigr\vert < M,\quad M>0,M\in \mathbb{R} _{+}, \\ &\Delta _{q}^{3}:\operatorname{supp}(\varepsilon _{n})\subseteq ( 0,b_{n} ),\quad b_{n}\rightarrow 0 \text{ as }n\rightarrow \infty,0< b_{n}, \forall n\in \mathbb{N}.\end{aligned} $$

(12)

On the other hand, we denote by $\overset{q}{\bullet }$ the Mellin type q-convolution product defined on $L_{q}^{1} ( \mathbb{R}_{q,+} )$ by

$$ ( g_{1}\overset{q}{\bullet }g_{2} ) ( x ) = \int _{0}^{\infty }t^{-1}g_{1} \bigl( t^{-1}x \bigr) g_{2} ( t ) \,d_{q}t, $$

(13)

provided the integral part exists for every $x>0$. It is clear from the context that $g_{1}\overset{q}{\bullet }g_{2}\in L_{q}^{1} ( \mathbb{R}_{q,+} )$ for all $g_{1}$ and $g_{2}$ in $L_{q}^{1} ( \mathbb{R}_{q,+} )$. On that account, the q-convolution theorem of the q-Mellin transform of the product $g_{1}\overset{q}{\bullet }g_{2}$ can be easily established as follows.

Theorem 2

Let $L_{q}^{1} ( \mathbb{R}_{q,+} )$ be the space of all q-integrable functions on $\mathbb{R}_{q,+}$. Then the q-convolution theorem of the transform $M_{q}$ is given by

$$ M_{q} ( g_{1}\overset{q}{\bullet }g_{2} ) ( \zeta ) =M_{q}g_{1} ( \zeta ) M_{q}g_{2} ( \zeta ) \quad\textit{for }g_{1}\textit{ and }g_{2} \textit{ in }L_{q}^{1} ( \mathbb{R}_{q,+} ). $$

Proof

By applying the definition of the $M_{q}$ transform to the product $g_{1}\overset{q}{\bullet }g_{2}$, we get

$$\begin{aligned} M_{q} ( g_{1}\overset{q}{\bullet }g_{2} ) ( \zeta ) &= \int _{0}^{\infty } ( g_{1} \overset{q}{ \bullet }g_{2} ) ( x ) x^{\zeta -1}\,d_{q}x \\ &= \int _{0}^{\infty } \biggl( \int _{0}^{\infty }g_{1} ( t ) g_{2} \bigl( t^{-1}x \bigr) x^{-1}\,d_{q}t \biggr) x^{ \zeta -1}\,d_{q}x. \end{aligned}$$

Therefore, employing the substitution $z=t^{-1}x$ and, hence, $d_{q}z=t^{-1}\,d_{q}x$, in collaboration with simple computations, reveals

$$ M_{q} ( g_{1}\overset{q}{\bullet }g_{2} ) ( \zeta ) =M_{q} ( g_{1} ) ( \zeta ) M_{q} ( g_{2} ) ( \zeta ). $$

Hence, the proof of this theorem is completed. □

The following is an imperative result for initiating the q-delta sequence concept.

Lemma 3

Let $( \varepsilon _{n} )$ and $( \epsilon _{n} )$ be sequences in $\Delta _{q}$. Then $( \varepsilon _{n}\overset{q}{\bullet }\epsilon _{n} )$ is a sequence in $\Delta _{q}$.

Proof

To establish this lemma, we examine the performance of the sequence $( \varepsilon _{n}\overset{q}{\bullet }\epsilon _{n} )$. To inspect the correctness of the property $\Delta _{q}^{1}$, we use the integral equation (3) to get

$$ \int _{0}^{\infty } ( \varepsilon _{n} \overset{q}{\bullet }\epsilon _{n} ) ( x ) \,d_{q}x= \int _{0}^{ \infty }t^{-1}\epsilon _{n} ( t ) \biggl( \int _{0}^{ \infty }\varepsilon _{n} \bigl( t^{-1}x \bigr) \,d_{q}x \biggr) \,d_{q}t. $$

(14)

Therefore, by using the change of variables $t^{-1}x=y$ and, hence, $d_{q}x=t\,d_{q}y$, (14) we indicate

$$ \int _{0}^{\infty } ( \varepsilon _{n} \overset{q}{\bullet }\epsilon _{n} ) ( x ) \,d_{q}x= \biggl( \int _{0}^{\infty }\epsilon _{n} ( t ) \,d_{q}t \biggr) \biggl( \int _{0}^{\infty }\varepsilon _{n} ( y ) \,d_{q}y \biggr) =1. $$

This proves the $\Delta _{q}^{1}$ part. The proof of the $\Delta _{q}^{2}$ part follows from similar techniques, whereas the $\Delta _{q}^{3}$ part is clearly valid, by conducting the fact

$$ \operatorname{supp} ( \varepsilon _{n}\overset{q}{\bullet }\epsilon _{n} ) \subset \operatorname{supp} ( \varepsilon _{n} ) + \operatorname{supp} ( \epsilon _{n} ) \quad\text{for } ( \varepsilon _{n} ), ( \epsilon _{n} ) \in \Delta _{q}. $$

This ends the proof of the lemma. □

Lemma 3, hence, displays that every sequence in $\Delta _{q}$ forms, to a great extent, the q-delta sequence concept.

Lemma 4

Let $g_{1},g_{2}\in L_{q}^{1} ( \mathbb{R}_{q,+} )$, $\kappa _{1},\kappa _{2}\in \mathbb{D}_{q}$, and $\alpha \in \mathbb{C}$. Then the following assertions are valid:

$$\begin{aligned} &\mathrm{(i)}\quad \kappa _{1}\overset{q}{\bullet }\kappa _{2}=\kappa _{2}\overset{q}{\bullet }\kappa _{1},\qquad \mathrm{(ii)}\quad ( g_{1}+g_{2} ) \overset{q}{ \bullet }\kappa _{1}=g_{1}\overset{q}{\bullet } \kappa _{1}+g_{2}\overset{q}{\bullet }\kappa _{1}, \\ &\mathrm{(iii)}\quad ( \alpha g_{1} ) \overset{q}{\bullet }\kappa _{1}= \alpha ( g_{1} \overset{q}{\bullet }\kappa _{1} ),\qquad \mathrm{(iv)}\quad g_{1} \overset{q}{\bullet } ( \kappa _{1}\overset{q}{ \bullet }\kappa _{2} ) = ( g_{1}\overset{q}{ \bullet }\kappa _{1} ) \overset{q}{\bullet }\kappa _{2}. \end{aligned}$$

Proof

(i) As the convolution product of the functions $\kappa _{1}$ and $\kappa _{2}$ in $\mathbb{D}_{q}$ is exceptionally given by

$$ ( \kappa \overset{q}{\bullet }\kappa _{2} ) ( x ) = \int _{0}^{\infty }t^{-1}\kappa _{1} \bigl( t^{-1}x \bigr) \kappa _{2} ( t ) \,d_{q}t, $$

(15)

the change of variables $t^{-1}x=y$ reveals us to write (15) into the form

$$ ( \kappa _{1}\overset{q}{\bullet }\kappa _{2} ) ( x ) = \int _{0}^{\infty }y^{-1}\kappa _{2} \bigl( x^{-1}y \bigr) \kappa _{1} ( y ) \,d_{q}y. $$

Hence (i) follows. To prove (ii) and (iii), we merely follow simple integral calculus. To prove (iv), we employ the definition of the product $\overset{q}{\bullet }$ to get

$$\begin{aligned} \bigl( g_{1}\overset{q}{\bullet } ( \kappa _{1} \overset{q}{\bullet }\kappa _{2} ) \bigr) ( x ) &= \int _{0}^{ \infty }t^{-1}g_{1} \bigl( t^{-1}x \bigr) ( \kappa _{1} \overset{q}{\bullet }\kappa _{2} ) ( t ) \,d_{q}t \\ &= \int _{0}^{\infty }t^{-1}g_{1} \bigl( t^{-1}x \bigr) \biggl( \int _{0}^{\infty }y^{-1}\kappa _{1} \bigl( y^{-1}t \bigr) \kappa _{2} ( y ) \,d_{q}y \biggr) \,d_{q}t. \end{aligned}$$

That is,

$$ \bigl( g_{1}\overset{q}{\bullet } ( \kappa _{1} \overset{q}{\bullet }\kappa _{2} ) \bigr) ( x ) = \int _{0}^{ \infty }y^{-1} \biggl( \int _{0}^{\infty }t^{-1}g_{1} \bigl( t^{-1}x \bigr) \kappa _{1} \bigl( y^{-1}t \bigr) \,d_{q}t \biggr) \kappa _{2} ( y ) \,d_{q}y. $$

(16)

Now, by employing the change of variables $y^{-1}t=z$, we write down equation $( 16 )$ into the form

$$\begin{aligned} \bigl( g_{1}\overset{q}{\bullet } ( \kappa _{1} \overset{q}{\bullet }\kappa _{2} ) \bigr) ( x ) &= \int _{0}^{ \infty }y^{-1} \biggl( \int _{0}^{\infty }z^{-1}g_{1} \bigl( z^{-1} \bigl( y^{-1}x \bigr) \bigr) \kappa _{1} ( z ) \,d_{q}z \biggr) \kappa _{2} ( y ) \,d_{q}y \\ &= \int _{0}^{\infty }y^{-1} ( g_{1} \overset{q}{\bullet }\kappa _{1} ) \bigl( y^{-1}x \bigr) \kappa _{2} ( y ) \,d_{q}y. \end{aligned}$$

This ends the proof of the lemma. □

To proceed in our construction, we establish the following lemma.

Lemma 5

(i) Let $g_{1}$ and $g_{2}$ be integrable functions in $L_{q}^{1} ( \mathbb{R}_{q,+} )$ and $( \varepsilon _{n} )$ be a delta sequence in the set $\Delta _{q}$ such that $g_{1}\overset{q}{\bullet }\varepsilon _{n}=g_{2} \overset{q}{\bullet }\varepsilon _{n}$. Then $g_{1}=g_{2}$ in $L_{q}^{1} ( \mathbb{R}_{q,+} )$ for every $n\in \mathbb{N}$.

(ii) Let g and $( g_{n} )$ be integrable functions in $L_{q}^{1} ( \mathbb{R}_{q,+} )$ such that $g_{n}\rightarrow g$ as $n\rightarrow \infty$ in $L_{q}^{1} ( \mathbb{R}_{q,+} )$. Then

$$ g_{n}\overset{q}{\bullet }\psi \rightarrow g\overset{q}{\bullet } \psi \quad\textit{for every }\psi \in \mathbb{D}_{q}\textit{ as }n \rightarrow \infty. $$

Proof

To prove (i), we merely need to show that $g_{1}\overset{q}{\bullet }\varepsilon _{n}=g_{1}\in L_{q}^{1} ( \mathbb{R}_{q,+} )$. By using $\Delta _{q}^{1}$ and $\Delta _{q}^{3}$, we obtain

$$\begin{aligned} \int _{0}^{\infty } \bigl\vert ( g_{1} \overset{q}{\bullet }\varepsilon _{n} ) ( x ) -g_{1} ( x ) \bigr\vert \,d_{q}x &\leq \int _{0}^{\infty } \int _{0}^{\infty } \bigl\vert t^{-1}g_{1} \bigl( t^{-1}x \bigr) -g_{1} ( x ) \bigr\vert \bigl\vert \varepsilon _{n} ( t ) \bigr\vert \,d_{q}t \,d_{q}x \\ &= \int _{0}^{\infty } \int _{a_{n}}^{b_{n}} \bigl\vert t^{-1}g_{1} \bigl( t^{-1}x \bigr) -g_{1} ( x ) \bigr\vert \bigl\vert \varepsilon _{n} ( t ) \bigr\vert \,d_{q}t \,d_{q}x. \end{aligned}$$

Therefore,

$$\begin{aligned} & \int _{0}^{\infty } \bigl\vert ( g_{1} \overset{q}{\bullet }\varepsilon _{n} ) ( x ) -g_{1} ( x ) \bigr\vert \,d_{q}x \\ &\quad \leq \int _{0}^{\infty } \int _{a_{n}}^{b_{n}} \bigl\vert t^{-1}g_{1} \bigl( t^{-1}x \bigr) \bigr\vert \bigl\vert \varepsilon _{n} ( t ) \bigr\vert \,d_{q}t\,d_{q}x \\ &\qquad{}+ \int _{0}^{\infty } \int _{a_{n}}^{b_{n}} \bigl\vert g_{1} ( x ) \bigr\vert \bigl\vert \varepsilon _{n} ( t ) \bigr\vert \,d_{q}t\,d_{q}x. \end{aligned}$$

(17)

Hence, for $g_{1}\in L_{q}^{1} ( \mathbb{R}_{q,+} )$, by using (17) we turn to write

$$ \int _{0}^{\infty } \bigl\vert ( g_{1} \overset{q}{\bullet }\varepsilon _{n} ) ( x ) -g_{1} ( x ) \bigr\vert \,d_{q}x\leq A \int _{0}^{b_{n}} \bigl\vert t^{-1} \bigr\vert \bigl\vert \varepsilon _{n} ( t ) \bigr\vert \,d_{q}t+A \int _{0}^{b_{n}} \bigl\vert \varepsilon _{n} ( t ) \bigr\vert \,d_{q}t. $$

Therefore, by the properties of the delta sequences $\Delta _{q}^{2}$ and $\Delta _{q}^{3}$, we conclude that

$$ \int _{0}^{\infty } \bigl\vert ( g_{1} \overset{q}{\bullet }\varepsilon _{n} ) ( x ) -g_{1} ( x ) \bigr\vert \,d_{q}x\leq AM\ln ( b_{n} ) +AM ( b_{n} ) \rightarrow 0 $$

as $n\rightarrow \infty$.

Proof of (ii) follows from simple integration. We therefore omit the details. Hence the proof of this lemma is ended. □

Lemma 6

Let $g_{1}$ be an integrable function in the space $L_{q}^{1} ( \mathbb{R}_{q,+} )$. Then $g_{1}\overset{q}{\bullet }\varepsilon _{n}\rightarrow g_{1}$ as $n\rightarrow \infty$ for every $( \varepsilon _{n} ) \in \Delta _{q}$.

The proof of this lemma is a straightforward conclusion from the proof of Lemma 4. Hence, we delete the details.

Thus, the space $\mathbb{B}$ with $( L_{q}^{1} ( \mathbb{R}_{q,+} ),\overset{q}{\bullet } ), ( \mathbb{D}_{q}, \overset{q}{\bullet } )$, and $\Delta _{q}$ is defined. The canonical embedding of $L_{q}^{1} ( \mathbb{R}_{q,+} )$ in $\mathbb{B}$ is given by

$$ g\rightarrow \frac{g\overset{q}{\bullet }\varepsilon _{n}}{\varepsilon _{n}}. $$

(18)

That is, every element in the space $L_{q}^{1} ( \mathbb{R}_{q,+} )$ can be identified as a member of the space $\mathbb{B}$. Addition, scalar multiplication, differentiation, $\Delta _{q}$ and $\delta _{q}$ convergence are defined in a natural way as follows:

If $( \varphi _{n} ) \in L_{q}^{1} ( \mathbb{R}_{q,+} )$ and $( \varepsilon _{n} ) \in \Delta _{q}$, then the pair $( \varphi _{n},\varepsilon _{n} )$ $( \text{or }\frac{\varphi _{n}}{\varepsilon _{n}} )$ is said to be a q-quotient of sequences if $\varphi _{n}\overset{q}{\bullet }\varepsilon _{m}=\varphi _{m} \overset{q}{\bullet }\varepsilon _{n},\forall n,m\in \mathbb{N}$. Therefore, if $\frac{\varphi _{n}}{\epsilon _{n}}$ and $\frac{g_{n}}{\varepsilon _{n}}$ are q-quotients of sequences and $g\in L_{q}^{1} ( \mathbb{R}_{q,+} )$, then it is easy to see that

$$ \frac{g\overset{q}{\bullet }\epsilon _{n}}{\epsilon _{n}}\quad\text{and}\quad \frac{\varphi _{n}\overset{q}{\bullet }\epsilon _{n}+g_{n}\overset{q}{\bullet }\epsilon _{n}}{\epsilon _{n}\overset{q}{\bullet }\varepsilon _{n}} $$

are q-quotients of sequences. Two q-quotients of sequences $\frac{\varphi _{n}}{\epsilon _{n}}$ and $\frac{g_{n}}{\varepsilon _{n}}$ are said to be equivalent if

$$ \varphi _{n}\overset{q}{\bullet }\varepsilon _{m}=g_{m} \overset{q}{\bullet }\epsilon _{n},\quad \forall n,m\in \mathbb{N}. $$

We can easily check the following equivalence relations:

$$ \frac{\varphi _{n}}{\epsilon _{n}\overset{q}{\bullet }g}\sim \frac{\varphi _{n}\overset{q}{\bullet }g}{\epsilon _{n}}\quad\text{and}\quad \frac{\varphi _{n}}{\epsilon _{n}\overset{q}{\bullet }g_{n}} \sim \frac{\varphi _{n}\overset{q}{\bullet }g_{n}}{\epsilon _{n}}. $$

The equivalent class $\breve{w}= ( \frac{\varphi _{n}}{\epsilon _{n}} )$ of q-quotients of sequences containing $\frac{\varphi _{n}}{\epsilon _{n}}$ is said to be a q-Boehmian. The space of such q-Boehmians is denoted by $\mathbb{B}$.

Remark 7

For two q-Boehmians $\breve{w}= ( \frac{\varphi _{n}}{\epsilon _{n}} )$ and $\breve{z}= ( \frac{g_{n}}{\varepsilon _{n}} )$ in $\mathbb{B}$, we have the following identities:

$$\begin{aligned} &\mathrm{(i)}\quad \breve{w}+\breve{z}= \biggl( \frac{\varphi _{n}\overset{q}{\bullet }\epsilon _{n}+g_{n}\overset{q}{\bullet }\epsilon _{n}}{\epsilon _{n}\overset{q}{\bullet }\varepsilon _{n}} \biggr), \\ &\mathrm{(ii)}\quad \beta \breve{w}= \biggl( \frac{\beta \varphi _{n}}{\epsilon _{n}} \biggr), \\ &\mathrm{(iii)}\quad \breve{w}\overset{q}{\bullet }\breve{z}= \biggl( \frac{\varphi _{n}\overset{q}{\bullet }g_{n}}{\epsilon _{n}\overset{q}{\bullet }\varepsilon _{n}} \biggr), \\ &\mathrm{(iv)}\quad D^{k}\breve{w}= \biggl( \frac{D^{k}\varphi _{n}}{\epsilon _{n}} \biggr), \\ &\mathrm{(v)}\quad \breve{w}\overset{q}{\bullet }g= \biggl( \frac{\varphi _{n}\overset{q}{\bullet }g}{\epsilon _{n}} \biggr), \end{aligned}$$

where $k\in \mathbb{R}$, $\beta \in \mathbb{C}$ and $D^{k}\breve{w}$ is the kth derivative of w̆, and $\psi \in L_{q}^{1} ( \mathbb{R}_{q,+} )$.

Definition 8

(i) For $n=1,2,3,\ldots$ and $\breve{w}_{n},\breve{w}\in \mathbb{B}$, the sequence $( \breve{w}_{n} )$ is $\delta _{q}$-convergent to w̆, denoted by $\delta _{q}-\lim_{n\rightarrow \infty }\breve{w}_{n}=\breve{w}$, provided there can be found a q-delta sequence $( \epsilon _{n} )$ such that

$$ ( \breve{w}_{n}\overset{q}{\bullet }\epsilon _{k} ), ( \breve{w}\overset{q}{\bullet }\epsilon _{k} ) \quad\text{in }L_{q}^{1} ( \mathbb{R}_{q,+} ) \quad \text{and}\quad\lim_{n \rightarrow \infty }\breve{w}_{n}\overset{q}{ \bullet } \epsilon _{k}= \breve{w}\overset{q}{\bullet } \epsilon _{k}\quad\text{in }L_{q}^{1} ( \mathbb{R}_{q,+} )\ ( \forall \text{ }k\in \mathbb{N} ). $$

(ii) For $n=1,2,3,\ldots$ and $\breve{w}_{n},\breve{w}\in \mathbb{B}$, the sequence $( \breve{w}_{n} )$ is said to be $\Delta _{q}$-convergent to w̆, denoted by $\Delta _{q}$-$\lim_{n\rightarrow \infty }\breve{w}_{n}=\breve{w}$, provided there can be found a q-delta sequence $( \epsilon _{n} )$ such that

$$ ( \breve{w}_{n}-\breve{w} ) \overset{q}{\bullet } \epsilon _{n}\in L_{q}^{1} ( \mathbb{R}_{q,+} )\quad ( \forall n\in \mathbb{N} ) \quad\text{and}\quad\lim _{n\rightarrow \infty } ( \breve{w}_{n}- \breve{w} ) \overset{q}{\bullet }\epsilon _{n}=0 \quad\text{in }L_{q}^{1} ( \mathbb{R}_{q,+} ). $$

Now we have the following few corollaries.

Corollary 9

(i) Let $g\in L_{q}^{1} ( \mathbb{R}_{q,+} )$ and $( \epsilon _{n} ) \in \Delta _{q}$ be fixed. Then the mapping

$$ g\rightarrow \breve{w}, $$

where $\breve{w}=\frac{g\overset{q}{\bullet }\epsilon _{n}}{\epsilon _{n}}$ is an injective mapping from $L_{q}^{1} ( \mathbb{R}_{q,+} )$ into $\mathbb{B}$.

(ii) Let $( \epsilon _{n} ) \in \Delta _{q}$. Then, if $g_{n}\rightarrow g$ in $L_{q}^{1} ( \mathbb{R}_{q,+} )$ as $n\rightarrow \infty$, then for all $k\in \mathbb{N}$,

$$ g_{n}\overset{q}{\bullet }\epsilon _{k}\rightarrow g \overset{q}{\bullet }\epsilon _{k}\quad\textit{and}\quad \breve{w}_{n}\rightarrow \breve{w} \quad\textit{in }\mathbb{B} \textit{ as }n\rightarrow \infty. $$

Therefore, it can be easily checked that $L_{q}^{1} ( \mathbb{R}_{q,+} )$ can be mathematically identified as a subspace of $\mathbb{B}$.

The above corollary leads to the following corollary.

Corollary 10

The q-embedding, $g\rightarrow \breve{w}$, $\breve{w}=\frac{g\overset{q}{\bullet }\epsilon _{n}}{\epsilon _{n}}$, of the space $L_{q}^{1} ( \mathbb{R}_{q,+} )$ into the space $\mathbb{B}$ is continuous.

3 The q-ultraBoehmian space $\mathbb{B}_{\mathbb{M}}$

In this section, we provide sufficient axioms to define the q-ultraBoehmian space $\mathbb{B}_{\mathbb{M}}$ with the set $( L_{\mathbb{M}},\circ )$, the subset $( \mathbb{D}_{\mathbb{M}},\circ )$, the set $( \Delta _{q,\mathbb{M}},\circ )$ of q-delta sequences, and the product ∘, where $L_{\mathbb{M}},\mathbb{D}_{\mathbb{M}}$, and $\Delta _{q,\mathbb{M}}$ are the q-Mellin transforms of the sets $L_{q}^{1} ( \mathbb{R} ),\mathbb{D}_{q}$, and $\Delta _{q}$ respectively. To this end, we introduce the following convolution operation.

Definition 11

Let $\omega _{1}$ and $\omega _{2}$ be in $\mathbb{B}_{\mathbb{M}}$. For $\omega _{1}$ and $\omega _{2}$, we define a product ∘ as

$$ ( \omega _{1}\circ \omega _{2} ) ( t ) = \omega _{1} ( t ) \omega _{2} ( t ). $$

(19)

The following assertion holds in the space $L_{\mathbb{M}}$.

Theorem 12

Let $\omega _{1}$ be in $L_{\mathbb{M}}$. Then $\omega _{1}\circ \eta \in L_{\mathbb{M}}$ for all $\eta \in \mathbb{D}_{\mathbb{M}}$.

Proof

Let $\omega _{1}\in L_{\mathbb{M}}$. Then, by the definition of the space $L_{\mathbb{M}}$ and the definition of the product ∘, we write

$$ ( \omega _{1}\circ \omega _{2} ) ( t ) = \omega _{1} ( t ) \omega _{2} ( t ) =M_{q} ( g_{1} ) M_{q} ( g_{2} ) $$

(20)

for some $g_{1},g_{2}\in L_{q}^{1} ( \mathbb{R}_{q,+} )$. Hence, by virtue of Def. 11, (20) can be written in the form

$$ ( \omega _{1}\circ \omega _{2} ) ( t ) =M_{q} ( g_{1}\overset{q}{\bullet }g_{2} ). $$

(21)

Therefore, as $g_{1}\circ g_{2}\in L_{q}^{1} ( \mathbb{R}_{q,+} )$, it follows from (21) that $\omega _{1}\circ \eta \in L_{\mathbb{M}}$. This ends the proof of the theorem. □

Theorem 13

Let ω be an integrable function in $\mathbb{L}_{\mathbb{M}}$. Then $\omega \circ ( \eta _{1}\circ \eta _{2} ) = ( \omega \circ \eta _{1} ) \circ \eta _{2}$ for all $\eta _{1},\eta _{2}\in \mathbb{D}_{q}$.

Proof

By the concept of the convolution ∘, we get

$$ \bigl( \omega \circ ( \eta _{1}\circ \eta _{2} ) \bigr) ( t ) =\omega ( t ) ( \eta _{1}\circ \eta _{2} ) ( t ) = \omega ( t ) \eta _{1} ( t ) \eta _{2} ( t ). $$

By using Def. 11 twice, we write the preceding equation as

$$ \bigl( \omega \circ ( \eta _{1}\circ \eta _{2} ) \bigr) ( t ) = ( \omega \circ \eta _{1} ) ( t ) \eta _{2} ( t ) = \bigl( ( \omega \circ \eta _{1} ) \circ \eta _{2} \bigr) ( t ). $$

This ends the proof of the theorem. □

The following axioms are straightforward.

Theorem 14

(i) Let $\omega _{1}$ and $\omega _{2}$ be in $L_{\mathbb{M}}$. Then $( \omega _{1}+\omega _{2} ) \circ \eta =\omega _{1} \circ \eta +\omega _{2}\circ \eta$ for all $\eta \in \mathbb{D}_{q}$.

(ii) Let $\omega _{1}$ be in $L_{\mathbb{M}}$. Then $( \alpha \omega _{1}\circ \eta ) =\alpha ( \omega _{1}\circ \eta )$ for all $\eta \in \mathbb{D}_{q}$ and $\alpha \in \mathbb{C}$.

Proof

(i) Let $\omega _{1}$ and $\omega _{2}$ be in $L_{\mathbb{M}}$. Then, by Def. 11, we write

$$ \bigl( ( \omega _{1}+\omega _{2} ) \circ \eta \bigr) ( t ) = ( \omega _{1}+\omega _{2} ) ( t ) \eta ( t ) =\omega _{1} ( t ) \eta ( t ) +\omega _{2} ( t ) \eta ( t ) = ( \omega _{1}\circ \eta ) ( t ) + ( \omega _{2}\circ \eta ) ( t ). $$

The proof of the first part is finished. The proof of the second part is trivial. This completes the proof of the theorem. □

Theorem 15

(i) Let $\omega _{1}$ and $( \omega _{n} )$ be members of the space $L_{\mathbb{M}}$ and $\eta \in \mathbb{D}_{\mathbb{M}}$. If $\omega _{n}\rightarrow \omega _{1}$ in $L_{\mathbb{M}}$ as $n\rightarrow \infty$, then $\omega _{n}\circ \eta \rightarrow \omega _{1}\circ \eta$ as $n\rightarrow \infty$.

(ii) Let $\omega _{1}$ and $\omega _{2}$ be in $L_{\mathbb{M}}$ and $( \upsilon _{n} ) \in \Delta _{q,\mathbb{M}}$. If $\omega _{1}\circ \upsilon _{n}=\omega _{2}\circ \upsilon _{n}$, then $\omega _{1}=\omega _{2}$ in $L_{\mathbb{M}}$.

(iii) Let $\omega _{1}$ be an integrable function in $L_{\mathbb{M}}$ and $( \upsilon _{n} ) \in \Delta _{q,\mathbb{M}},\upsilon _{n} ( t ) \neq 0$ for all $t\in \mathbb{R}_{q,+}$. Then $\omega _{1}\circ \upsilon _{n}\rightarrow 0$ in $L_{\mathbb{M}}$ as $n\rightarrow \infty$.

Proof

To prove (i), let $\omega _{1}$ and $( \omega _{n} )$ be members of $L_{\mathbb{M}}$ and $\eta \in \mathbb{D}_{\mathbb{M}}$. If $\omega _{n}\rightarrow \omega _{1}$ in $L_{\mathbb{M}}$ as $n\rightarrow \infty$, then by Def. 11 and Theo. 14, we have

$$ ( \omega _{n}\circ \eta -\omega _{1}\circ \eta ) ( t ) = \bigl( ( \omega _{n}-\omega _{1} ) \circ \eta \bigr) ( t ) = ( \omega _{n}-\omega _{1} ) ( t ) \eta ( t ) =\omega _{n} ( t ) \eta ( t ) -\omega _{1} ( t ) \eta ( t ). $$

Hence, by the hypothesis of the theorem, we obtain

$$ \omega _{n}\circ \eta -\omega _{1}\circ \eta \rightarrow \omega _{1} \circ \eta -\omega _{1}\circ \eta \rightarrow 0 \quad\text{as }n \rightarrow \infty. $$

Hence, the first part of the theorem is completely proved. To prove (ii), let $\omega _{1}$ and $\omega _{2}$ be in $L_{\mathbb{M}}$ and $( \upsilon _{n} ) \in \Delta _{q,\mathbb{M}}$. If $\omega _{1}\circ \upsilon _{n}=\omega _{2}\circ \upsilon _{n}$, then $\omega _{1} ( t ) \upsilon _{n} ( t ) =\omega _{2} ( t ) \upsilon _{n} ( t )$. Hence,

$$ ( \omega _{1}-\omega _{2} ) ( t ) \upsilon _{n} ( t ) =0\quad\text{for all }t\in \mathbb{R}_{q,+}. $$

Therefore, $( \omega _{1}-\omega _{2} ) ( t ) =0$ for all $\mathbb{R}_{q,+}$. Thus, $\omega _{1}=\omega _{2}$ in $L_{\mathbb{M}}$. The proof of (iii) is similar. Hence, the theorem is completely proved.

If $( \omega _{n} ) \in L_{\mathbb{M}}$ and $( \upsilon _{n} ) \in \Delta _{q,\mathbb{M}}$, then the pair $( \omega _{n},\upsilon _{n} )$ $( \text{or }\frac{\omega _{n}}{\upsilon _{n}} )$ is said to be a q-quotient of sequences if

$$ \omega _{n}\circ \upsilon _{m}=\omega _{m}\circ \upsilon _{n}, \quad\forall n,m\in \mathbb{N}. $$

Therefore, if $\frac{\omega _{n}}{\epsilon _{n}}$ and $\frac{g_{n}}{\upsilon _{n}}$ are q-quotients of sequences and $\omega \in L_{\mathbb{M}}$, then it is easy to see that

$$ \frac{\omega \circ \epsilon _{n}}{\epsilon _{n}}\quad\text{and}\quad \frac{\omega _{n}\circ \epsilon _{n}+g_{n}\circ \epsilon _{n}}{\epsilon _{n}\circ \upsilon _{n}} $$

are q-quotients of sequences. Furthermore, it is easy to see the following equivalence relations:

$$ \frac{\omega _{n}}{\epsilon _{n}\circ \omega }\sim \frac{\omega _{n}\circ \omega }{\epsilon _{n}}\quad\text{and}\quad \frac{\omega _{n}}{\epsilon _{n}\circ g_{n}}\sim \frac{\omega _{n}\circ g_{n}}{\epsilon _{n}}. $$

Two q-quotients of sequences $\frac{\omega _{n}}{\epsilon _{n}}$ and $\frac{g_{n}}{\upsilon _{n}}$ are said to be equivalent if $\omega _{n}\circ \upsilon _{m}=g_{m}\circ \epsilon _{n},\forall n,m \in \mathbb{N}$. The equivalent class $\breve{w}= ( \frac{\omega _{n}}{\epsilon _{n}} )$ of q-quotients of sequences containing $\frac{\varphi _{n}}{\epsilon _{n}}$ is said to be a q-Boehmian. The space of such q-Boehmians is denoted by $\mathbb{B}_{\mathbb{M}}$. □

Remark 16

For two q-Boehmians $\breve{w}= ( \frac{\omega _{n}}{\epsilon _{n}} )$ and $\breve{z}= ( \frac{g_{n}}{\upsilon _{n}} )$ in $\mathbb{B}_{\mathbb{M}}$, the following are well defined on $\mathbb{B}_{\mathbb{M}}$:

$$\begin{aligned} &\mathrm{(i)}\quad \breve{w}+\breve{z}= \biggl( \frac{\omega _{n}\circ \epsilon _{n}+g_{n}\circ \epsilon _{n}}{\epsilon _{n}\circ \upsilon _{n}} \biggr), \\ &\mathrm{(ii)}\quad \beta \breve{w}= \biggl( \frac{\beta \omega _{n}}{\epsilon _{n}} \biggr), \\ &\mathrm{(iii)}\quad \breve{w}\circ \breve{z}= \biggl( \frac{\omega _{n}\circ g_{n}}{\epsilon _{n}\circ \upsilon _{n}} \biggr), \\ &\mathrm{(iv)}\quad D^{k}\breve{w}= \biggl( \frac{D^{k}\omega _{n}}{\epsilon _{n}} \biggr), \\ &\mathrm{(v)} \quad\breve{w}\circ \omega = \biggl( \frac{\omega _{n}\circ \omega }{\epsilon _{n}} \biggr), \end{aligned}$$

where $k\in \mathbb{R}$, $\beta \in \mathbb{C}$ and $D^{k}\breve{w}$ is the kth derivative of w̆, and $\psi \in L_{\mathbb{M}}$.

Definition 17

(i) For $n=1,2,3,\ldots$ and $\breve{w}_{n},\breve{w}\in \mathbb{B}_{\mathbb{M}}$, the sequence $( \breve{w}_{n} )$ is said to be $\delta _{q}$-convergent to w̆, denoted by $\delta _{q}-\lim_{n\rightarrow \infty }\breve{w}_{n}=\breve{w}$, provided there can be found a q-delta sequence $( \upsilon _{n} )$ such that

$$ ( \breve{w}_{n}\circ \upsilon _{k} ), ( \breve{w} \circ \upsilon _{k} ) \quad\text{in }L_{\mathbb{M}}\ ( \forall n,k\in \mathbb{N} ) \quad\text{and}\quad\lim_{n\rightarrow \infty } \breve{w}_{n}\circ \upsilon _{k}=\breve{w}\circ \upsilon _{k}\quad\text{in }L_{\mathbb{M}}\ ( \forall \text{ }k\in \mathbb{N} ). $$

(ii) For $n=1,2,3,\ldots$ and $\breve{w}_{n},\breve{w}\in \mathbb{B}_{\mathbb{M}}$, the sequence $( \breve{w}_{n} )$ is said to be $\Delta _{q}$-convergent to w̆, denoted by $\Delta _{q}$-$\lim_{n\rightarrow \infty }\breve{w}_{n}=\breve{w}$, provided there can be found a q-delta q-sequence $( \upsilon _{n} )$ such that

$$ ( \breve{w}_{n}-\breve{w} ) \circ \upsilon _{n} \in L_{\mathbb{M}} \quad( \forall n\in \mathbb{N} ) \quad\text{and}\quad \lim_{n\rightarrow \infty } ( \breve{w}_{n}-\breve{w} ) \circ \upsilon _{n}=0 \quad\text{in }L_{\mathbb{M}}. $$

Corollary 18

(i) Let $\omega \in L_{\mathbb{M}}$ and $( \upsilon _{n} ) \in \Delta _{q}$ be fixed. Then the mapping

$$ \omega \rightarrow \breve{w}, $$

where $\breve{w}=\frac{\omega \circ \upsilon _{n}}{\upsilon _{n}}$ is an injective mapping from $L_{\mathbb{M}}$ into $\mathbb{B}_{\mathbb{M}}$.

(ii) Let $( \upsilon _{n} ) \in \Delta _{q,\mathbb{M}}$. Then, if $\omega _{n}\rightarrow \omega$ in $L_{\mathbb{M}}$ as $n\rightarrow \infty$, then for all $k\in \mathbb{N}$,

$$ \omega _{n}\circ \upsilon _{k}\rightarrow \omega \circ \upsilon _{k} \quad\textit{and}\quad\breve{w}_{n}\rightarrow \breve{w} \quad\textit{in }\mathbb{B}_{ \mathbb{M}}\textit{ as }n \rightarrow \infty. $$

(22)

Therefore, it can be easily checked that $L_{\mathbb{M}}$ may be identified as a subspace of $\mathbb{B}_{\mathbb{M}}$.

The above corollary can be stated as follows.

Corollary 19

The q-embedding $\psi \rightarrow \breve{w}$, $\breve{w}=\frac{\omega \circ \upsilon _{n}}{\upsilon _{n}}$, of the space $L_{\mathbb{M}}$ into the space $\mathbb{B}_{\mathbb{M}}$ is continuous.

4 The q-Mellin transform of the generalized q-theory

This section aims to discuss a definition and some basic properties of the generalized q-Mellin transform in a context of the new q-theory. All results are brief and concise, and may give the reader a general overview of the generalized q-theory of the Mellin operator. However, by virtue of the preceding analysis, we introduce the following definition.

Definition 20

Let $\frac{g_{n}}{\varepsilon _{n}}\in \mathbb{B}$, then we define the q-Mellin transform of the q-Boehmian $\frac{g_{n}}{\varepsilon _{n}}$ as

$$ \mathbb{M}_{q}\frac{g_{n}}{\varepsilon _{n}}=\tilde{\omega }_{n}, $$

(23)

where $\tilde{\omega }_{n}=\frac{\omega _{n}}{\upsilon _{n}},\omega _{n}=M_{q}g$, and $\upsilon _{n}=M_{q}\varepsilon _{n}$. Indeed $\tilde{\omega }_{n}$ belongs to $\mathbb{B}_{\mathbb{M}}$.

Theorem 21

The operator $\mathbb{M}_{q}:\mathbb{B}\rightarrow \mathbb{B}_{\mathbb{M}}$ is sequentially continuous, i.e., if $\Delta _{q}-\lim_{k\rightarrow \infty }\tilde{\omega }_{n,k}= \tilde{\omega }_{n}$ in $\mathbb{B}$, then $\Delta _{q,\mathbb{M}}-\lim_{n\rightarrow \infty }\mathbb{M}_{q} \tilde{\omega }_{n,k}=\mathbb{M}_{q}\tilde{\omega }_{n}$ in $\mathbb{B}_{\mathbb{M}}$.

Proof

Let $\Delta _{q}-\lim_{k\rightarrow \infty }\tilde{\omega }_{n,k}=\tilde{\omega }_{n}$ in $\mathbb{B}$, then there is $(\varepsilon _{n} )\in \Delta _{q}$ such that

$$ \Delta _{q}-\lim_{n\rightarrow \infty } ( \tilde{\omega }_{n,k}- \tilde{\omega }_{n} ) \overset{q}{\bullet }\varepsilon _{n}=0 \quad\text{in }\mathbb{B}. $$

The continuity of the integral operator gives

$$ \Delta _{q,\mathbb{M}}-\lim_{n\rightarrow \infty }\mathbb{M}_{q} \bigl( ( \tilde{\omega }_{n,k}-\tilde{\omega }_{n} ) \overset{q}{\bullet }\varepsilon _{n} \bigr) =\Delta -\lim _{n\rightarrow \infty } \bigl( ( \mathbb{M}_{q}\tilde{\omega }_{n,k}-\mathbb{M}_{q} \tilde{\omega }_{n} ) \circ \upsilon _{n} \bigr) =0, $$

where $\mathbb{M}_{q}\varepsilon _{n}=\upsilon _{n}$. Thus, we have $\Delta _{q,\mathbb{M}}-\lim_{n\rightarrow \infty }\mathbb{M}_{q} \tilde{\omega }_{n,k}=\mathbb{M}_{q}\tilde{\omega }_{n}$ in $\mathbb{B}_{\mathbb{M}}$.

This finishes the proof of the theorem. □

Theorem 22

(i) $\mathbb{M}_{q}$ is a linear isomorphism from the space $\mathbb{B}$ onto the space $\mathbb{B}_{\mathbb{M}}$.

(ii) $\mathbb{M}_{q}$ is continuous with respect to $\delta _{q}$ and $\Delta _{q}$-convergence.

(iii) The operator $\mathbb{M}_{q}$ coincides with the operator $M_{q}$.

Proof

We prove Part (iii) since similar proofs for Part (i)–Part (ii) are available in literature. Let $g\in L_{q}^{1} ( \mathbb{R}_{q,+} )$ and $\frac{g\overset{q}{\bullet }\varepsilon _{n}}{\varepsilon _{n}}$ be its representative in $\mathbb{B}$, where $( \varepsilon _{n} ) \in \Delta _{q}$ $( \forall n\in \mathbb{N} )$. Clearly, for all $n\in \mathbb{N}{\small,}$ $( \varepsilon _{n} )$ is independent from the representative. Let $\mathbb{M}_{q}\varepsilon _{n}=\upsilon _{n}$, then, by the q-convolution theorem, we get

$$ \mathbb{M}_{q} \frac{g\overset{q}{\bullet }\varepsilon _{n}}{\varepsilon _{n}}=\mathbb{M}_{q} \frac{g\overset{q}{\bullet }\varepsilon _{n}}{\varepsilon _{n}}=\frac{M_{q}g\circ M_{q}\varepsilon _{n}}{M_{q}\varepsilon _{n}}=M_{q}g\circ \frac{M_{q}\varepsilon _{n}}{M_{q}\varepsilon _{n}}= \omega \circ \frac{\upsilon _{n}}{\upsilon _{n}}. $$

Hence, the q-Boehmian $\frac{\omega \circ \upsilon _{n}}{\upsilon _{n}}$ is the representative of $\mathbb{M}_{q}$ in the space $L_{\mathbb{M}}$, where $\omega =M_{q}g$.

The proof is, therefore, ended. □

We introduce the inverse transform of $\mathbb{M}_{q}$ as follows.

Definition 23

We define the inverse integral operator of $\mathbb{M}_{q}$ of a q-Boehmian $\frac{\omega _{n}}{\upsilon _{n}}$ in $\mathbb{B}_{\mathbb{M}}$ as follows:

$$ \mathbb{N}_{q}\frac{\omega _{n}}{\upsilon _{n}}= \frac{g_{n}}{\varepsilon _{n}}\in \mathbb{B}, $$

where $\upsilon _{n}=\mathbb{M}_{q}\varepsilon _{n}$ and $\omega _{n}=M_{q}g_{n}$ for some $( \varepsilon _{n} ) \in \Delta _{q}$ and $\{g_{n}\}\in L_{q}^{1} ( \mathbb{R}_{q,+} )$.

Theorem 24

Let $\frac{\omega _{n}}{\upsilon _{n}}\in \mathbb{B}_{\mathbb{M}}$ and $\omega \in L_{\mathbb{M}}$. Then we have

$$ \mathbb{N}_{q} \biggl( \frac{\omega _{n}}{\upsilon _{n}}\circ \omega \biggr) =\frac{g_{n}}{\varepsilon _{n}}\overset{q}{\bullet }g\quad\textit{and}\quad \mathbb{M}_{q} \biggl( \frac{g_{n}}{\varepsilon _{n}}\overset{q}{\bullet }g \biggr) =\frac{\omega _{n}}{\upsilon _{n}}\circ \omega. $$

Proof

Assume $\frac{\omega _{n}}{\upsilon _{n}}\in \mathbb{B}_{\mathbb{M}}$ where $\omega _{n}=M_{q}g_{n}$. Then, for every $\omega =M_{q}g\in L_{\mathbb{M}}$ and $\upsilon _{n}=M_{q}\varepsilon _{n}$, we have

$$ \mathbb{N}_{q} \biggl( \frac{\omega _{n}}{\upsilon _{n}}\circ M_{q}g \biggr) =\mathbb{N}_{q}\frac{\omega _{n}\circ \omega }{\upsilon _{n}}= \mathbb{N}_{q}\frac{M_{q} ( g_{n}\overset{q}{\bullet }g ) }{\upsilon _{n}}=\frac{g_{n}\overset{q}{\bullet }g}{\varepsilon _{n}}= \frac{g_{n}}{\varepsilon _{n}}\overset{q}{\bullet }g. $$

The proof of the first part is finished. The proof of the second part is almost similar. Hence, we omit the details.

This completely ends the proof of the theorem. □

5 Conclusion

This paper has given an extension of the quantum theory of the q-Mellin transform operator [40] to sets of q-generalized functions named q-Boehmians and q-ultraBoehmians. Every element g in the function space $L_{q}^{1} ( \mathbb{R}_{q,+} )$ is identified as a member in the generalized space $\mathbb{B}$ by the identification formula

$$ g\rightarrow \frac{g\overset{q}{\bullet }\varepsilon _{n}}{\varepsilon _{n}}, $$

where $( \varepsilon _{n} )$ is an arbitrary delta sequence. It also shows that the q-embedding

$$ g\rightarrow \breve{w},\text{ }\breve{w}= \frac{g\overset{q}{\bullet }\varepsilon _{n}}{\varepsilon _{n}} $$

of the space $L_{q}^{1} ( \mathbb{R}_{q,+} )$ into the space $\mathbb{B}$ is continuous, $( \varepsilon _{n} )$ being an arbitrary q-delta sequence. The q-Mellin transform operator is extended to the generalized q-calculus theory, and many properties are discussed. Further, the inversion of the q-Mellin transform operator is also discussed.