# 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.

## 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 . 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 ). 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 , the notion of a Boehmian was firstly introduced by Mikusinski and Mikusinski  to generalize distributions and regular operators . 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., . 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. (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. (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. (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. (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. (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, 2124]. In the sequence of such q-integral transforms, we recall the q-Laplace integral transform , the q-Sumudu integral transform [2, 3032], the q-Weyl fractional integral transform , the q-wavelet integral transform , the q-Mellin type integral transform , the Mangontarum integral transform [36, 37], the $$E_{2;1}$$ integral transform [38, 39], the natural integral transform , 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 , 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 . 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  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.

## 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 , 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 , 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 , 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.

## 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 , 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 , 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 , 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.

## 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. □

## Conclusion

This paper has given an extension of the quantum theory of the q-Mellin transform operator  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.

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## Acknowledgements

The author would like to express deepest thanks to the reviewers for their insightful comments on his paper.

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