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

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 [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][2][3]). Having said this, we immediately get the q-analogue of the derivative of g as D q g(t) := d q g(t) d q t := g(t)g(qt) (1q)t for t = 0 and D q g(0) = lim t− →0 D q g(t) = g (0) provided g (0) exists. Also, when g is differentiable, the q-derivative D q g tends to g (0) as q tends to 1. It also satisfies the q-analogue of the Leibniz rule D q g 1 (t)g(t) = 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] x 0 when the sums converge absolutely. The Jackson q-integral on the generic interval [a, b] is, therefore, given by [1,5] b a g(t) The q-integration by parts for two functions f and g is defined by 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][10][11][12][13][14][15][16][17][18][19][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 , * q ) of V , let q • : V × W → V be a binary operation such that the undermentioned axioms (1)-(5) hold: If g n → g in V as n → ∞ and ψ ∈ W , (5) A collection q of sequences from W such that, for all (ε n ), (φ n ) ∈ q and (g n ) ∈ W , we have ε n q • φ n ∈ q and if g n → g in V as n → ∞, then g n q • ε n → g as n → ∞.
Once the preceding axioms are applied, the name of a q-Boehmian is set to mean the equivalence class g n δ n that arises from the equivalence relation where (g n ) ∈ V and (ε n ) ∈ q . The collection of all q-Boehmians is denoted by B q which is the so-called Boehmian space. The classical linear space V is identified as a subset of the space B q which can be recognized from the relation where (ε n ) ∈ q is arbitrary. Two q-Boehmians g n ε n and ϕ n n are said to be equal in B q if g n q • m = ϕ m q • ε n , ∀m, n ∈ N. Addition in the space B q is defined as The scalar multiplication in the space B q is defined as The q-convergence of type δ, β n δ → β, is defined in the space B q when for (ψ n ) ∈ q and each k ∈ N such that we have β n The space of q-Boehmians emerging from the q-convergence assigns a complete quasinormed space. In recent work, several remarkable integral transforms were given different q-analogues in a q-calculus context [4,[21][22][23][24]. In the sequence of such q-integral transforms, we recall the q-Laplace integral transform [25][26][27][28][29], the q-Sumudu integral transform [2,[30][31][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 R q,+ , R q,+ = {q n : n ∈ Z}, then the q-Mellin transform was defined by [40], p. 521 as provided the q-integral converges. The integral (9) is analytic on the fundamental strip α q,g ; β q,g and periodic with period 2iπ log(q). The inversion formula for the q-analogue (9) is given by where α q,g < c < β 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 → 0 and x → ∞ are given in [40]. Additionally, the asymptotic behavior at 0 or ∞ is studied using the q-Mellin transform.
. In a better recognition, the space L 1 q (R q,+ ) is defined to be the space of all qintegrable functions g on R q,+ such that We denote by D q the q-space of test functions of compact supports on R q,+ , i.e., D q is the q-space of all smooth functions κ ∈ C ∞ (R q,+ ) such that 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 1 q (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 B
In this section, we strive to establish the space B of q-Boehmians. Henceforth, we denote by q the set of all sequences from D q such that the undermentioned identities 1 On the other hand, we denote by q • the Mellin type q-convolution product defined on provided the integral part exists for every x > 0. It is clear from the context that g 1 q • g 2 ∈ L 1 q (R q,+ ) for all g 1 and g 2 in L 1 q (R q,+ ). On that account, the q-convolution theorem of the q-Mellin transform of the product g 1 q • g 2 can be easily established as follows.
Then the qconvolution theorem of the transform M q is given by Proof By applying the definition of the M q transform to the product g 1 q • g 2 , we get Therefore, employing the substitution z = t -1 x and, hence, d q z = t -1 d q x, in collaboration with simple computations, reveals Hence, the proof of this theorem is completed.
The following is an imperative result for initiating the q-delta sequence concept.

Lemma 3
Let (ε n ) and ( n ) be sequences in q . Then (ε n q • n ) is a sequence in q .
Proof To establish this lemma, we examine the performance of the sequence (ε n q • n ). To inspect the correctness of the property 1 q , we use the integral equation (3) to get Therefore, by using the change of variables t -1 x = y and, hence, d q x = t d q y, (14) we indicate This proves the 1 q part. The proof of the 2 q part follows from similar techniques, whereas the 3 q part is clearly valid, by conducting the fact This ends the proof of the lemma.
Lemma 3, hence, displays that every sequence in q forms, to a great extent, the q-delta sequence concept. Lemma 4 Let g 1 , g 2 ∈ L 1 q (R q,+ ), κ 1 , κ 2 ∈ D q , and α ∈ C. Then the following assertions are valid: Proof (i) As the convolution product of the functions κ 1 and κ 2 in D q is exceptionally given by the change of variables t -1 x = y reveals us to write (15) into the form Hence (i) follows. To prove (ii) and (iii), we merely follow simple integral calculus. To prove (iv), we employ the definition of the product q • to get That is, Now, by employing the change of variables y -1 t = z, we write down equation (16) into the form 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 1 q (R q,+ ) and (ε n ) be a delta sequence in the set q such that g 1 q • ε n = g 2 q • ε n . Then g 1 = g 2 in L 1 q (R q,+ ) for every n ∈ N.
(ii) Let g and (g n ) be integrable functions in L 1 q (R q,+ ) such that g n → g as n → ∞ in L 1 q (R q,+ ). Then g n q • ψ → g q • ψ for every ψ ∈ D q as n → ∞.
Proof To prove (i), we merely need to show that g 1 q • ε n = g 1 ∈ L 1 q (R q,+ ). By using 1 q and 3 q , we obtain Therefore, Hence, for g 1 ∈ L 1 q (R q,+ ), by using (17) we turn to write Therefore, by the properties of the delta sequences 2 q and 3 q , we conclude that 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
The proof of this lemma is a straightforward conclusion from the proof of Lemma 4. Hence, we delete the details.
Thus, the space B with (L 1 q (R q,+ ), •), and q is defined. The canonical embedding of L 1 q (R q,+ ) in B is given by That is, every element in the space L 1 q (R q,+ ) can be identified as a member of the space B. Addition, scalar multiplication, differentiation, q and δ q convergence are defined in a natural way as follows: If (ϕ n ) ∈ L 1 q (R q,+ ) and (ε n ) ∈ q , then the pair (ϕ n , ε n ) (or ϕ n ε n ) is said to be a q-quotient of sequences if ϕ n q • ε m = ϕ m q • ε n , ∀n, m ∈ N. Therefore, if ϕ n n and g n ε n are q-quotients of sequences and g ∈ L 1 q (R q,+ ), then it is easy to see that • ε n are q-quotients of sequences. Two q-quotients of sequences ϕ n n and g n ε n are said to be equivalent if We can easily check the following equivalence relations: The equivalent classw = ( ϕ n n ) of q-quotients of sequences containing ϕ n n is said to be a q-Boehmian. The space of such q-Boehmians is denoted by B.
Remark 7 For two q-Boehmiansw = ( ϕ n n ) andz = ( g n ε n ) in B, we have the following identities: where k ∈ R, β ∈ C and D kw is the kth derivative ofw, and ψ ∈ L 1 q (R q,+ ). n = 1, 2, 3, . . . andw n ,w ∈ B, the sequence (w n ) is δ q -convergent tow, denoted by δ q -lim n→∞wn =w, provided there can be found a q-delta sequence ( n ) such that

Definition 8 (i) For
(ii) For n = 1, 2, 3, . . . andw n ,w ∈ B, the sequence (w n ) is said to be q -convergent tow, denoted by q -lim n→∞wn =w, provided there can be found a q-delta sequence ( n ) such that Now we have the following few corollaries.
Corollary 9 (i) Let g ∈ L 1 q (R q,+ ) and ( n ) ∈ q be fixed. Then the mapping g →w, q (R q,+ ) as n → ∞, then for all k ∈ N, Therefore, it can be easily checked that L 1 q (R q,+ ) can be mathematically identified as a subspace of B.
The above corollary leads to the following corollary.

Corollary 10
The q-embedding, g →w,w = g q • n n , of the space L 1 q (R q,+ ) into the space B is continuous.

The q-ultraBoehmian space B M
In this section, we provide sufficient axioms to define the q-ultraBoehmian space B M with the set (L M , •), the subset (D M , •), the set ( q,M , •) of q-delta sequences, and the product •, where L M , D M , and q,M are the q-Mellin transforms of the sets L 1 q (R), D q , and q respectively. To this end, we introduce the following convolution operation.

Definition 11
Let ω 1 and ω 2 be in B M . For ω 1 and ω 2 , we define a product • as The following assertion holds in the space L M .
for some g 1 , g 2 ∈ L 1 q (R q,+ ). Hence, by virtue of Def. 11, (20) can be written in the form Therefore, as g 1 • g 2 ∈ L 1 q (R q,+ ), it follows from (21) that ω 1 • η ∈ L M . This ends the proof of the theorem.

Theorem 13 Let ω be an integrable function in
Proof By the concept of the convolution •, we get By using Def. 11 twice, we write the preceding equation as This ends the proof of the theorem.
The following axioms are straightforward.
Proof (i) Let ω 1 and ω 2 be in L M . Then, by Def. 11, we write The proof of the first part is finished. The proof of the second part is trivial. This completes the proof of the theorem. Proof To prove (i), let ω 1 and (ω n ) be members of L M and η ∈ D M . If ω n → ω 1 in L M as n → ∞, then by Def. 11 and Theo. 14, we have Hence, by the hypothesis of the theorem, we obtain Hence, the first part of the theorem is completely proved. To prove (ii), let ω 1 and ω 2 be in L M and (υ n ) ∈ q,M . If ω 1 • υ n = ω 2 • υ n , then ω 1 (t)υ n (t) = ω 2 (t)υ n (t). Hence, (ω 1ω 2 )(t)υ n (t) = 0 for all t ∈ R q,+ . Therefore, (ω 1ω 2 )(t) = 0 for all R q,+ . Thus, ω 1 = ω 2 in L M . The proof of (iii) is similar. Hence, the theorem is completely proved.
If (ω n ) ∈ L M and (υ n ) ∈ q,M , then the pair (ω n , υ n ) (or ω n υ n ) is said to be a q-quotient of sequences if Therefore, if ω n n and g n υ n are q-quotients of sequences and ω ∈ L M , then it is easy to see that ω • n n and ω n • n + g n • n n • υ n are q-quotients of sequences. Furthermore, it is easy to see the following equivalence relations: Two q-quotients of sequences ω n n and g n υ n are said to be equivalent if ω n • υ m = g m • n , ∀n, m ∈ N. The equivalent classw = ( ω n n ) of q-quotients of sequences containing ϕ n n is said to be a q-Boehmian. The space of such q-Boehmians is denoted by B M .
Remark 16 For two q-Boehmiansw = ( ω n n ) andz = ( g n υ n ) in B M , the following are well defined on B M : where k ∈ R, β ∈ C and D kw is the kth derivative ofw, and ψ ∈ L M .
(ii) For n = 1, 2, 3, . . . andw n ,w ∈ B M , the sequence (w n ) is said to be q -convergent tȏ w, denoted by q -lim n→∞wn =w, provided there can be found a q-delta q-sequence (υ n ) such that (w n -w) • υ n ∈ L M (∀n ∈ N) and lim n→∞ (w n -w) • υ n = 0 in L M .
of the space L 1 q (R q,+ ) into the space B is continuous, (ε 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.