On Transformation Involving Basic Analogue to the Aleph-Function of Two Variables

: In our work, we derived the fractional order q -integrals and q -derivatives concerning a basic analogue to the Aleph-function of two variables (AFTV). We discussed a related application and the q -extension of the corresponding Leibniz rule. Finally, we presented two corollaries concerning the basic analogue to the I -function of two variables and the basic analogue to the Aleph-function of one variable.


Introduction
Fractional calculus represents an important part of mathematical analysis. The concept of fractional calculus was born from a famous correspondence between L'Hopital and Leibniz in 1695. In the last four decades, it has gained significant recognition and found many applications in diverse research fields (see [1][2][3][4][5][6]). The fractional basic (or q−) calculus is the extension of the ordinary fractional calculus in the q-theory (see [7][8][9][10]). We recall that basic series and basic polynomials, particularly the basic (or q−) hypergeometric functions and basic (or q−) hypergeometric polynomials, are particularly applicable in several fields, e.g., Finite Vector Spaces, Lie Theory, Combinatorial Analysis, Particle Physics, Mechanical Engineering, Theory of Heat Conduction, Non-Linear Electric Circuit Theory, Cosmology, Quantum Mechanics, and Statistics. In 1952, Al-Salam introduced the q-analogue to Cauchy's formula [11] (see also [12]). Agarwal [13] studied certain fractional q-integral and q-derivative operators. In addition, various researchers reported image formulas of various q-special functions under fractional q-calculus operators, for example, Kumar et al. [14], Sahni et al. [15], Yadav and Purohit [16], Yadav et al. [17,18], and maybe more. The q-extensions of the Saigo's fractional integral operators were defined by Purohit and Yadav [19]. Several authors utilised such operators to evaluate a general class of q-polynomials, the basic analogue to Fox's H-function, basic analogue to the I-function, fractional q-calculus formulas for various special functions, etc. The readers can see more related new details in [16][17][18]20] on fractional q-calculus.
The purpose of the present manuscript is to discuss expansion formulas, involving the basic analogue to AFTV [21]. The q-Leibniz formula is also provided.
We recall that q-shifted factorial (a; q) n has the following form [22,23] (a; q) n = 1, such that a, q ∈ C and it is assumed that a = q −m (m ∈ N 0 ). The expression of the q-shifted factorial for negative subscript is written by Additionally, we have Using (1)-(3), we conclude that its extension to n = α ∈ C as: such that the principal value of q α is considered. We equivalently have a form of (1), given as where the q-gamma function is expressed as [8]: The expression of the q-analogue to the Riemann-Liouville fractional integral operator (RLI) of f (x) has the following expression [12]: here, (µ) > 0, |q| < 1 and The basic integral [8] is given by Equation (8), in conjunction with (10); then, we have the series representation of (RLI), as follows We mention that for f (x) = x λ−1 , the following can be written [16]

Basic Analogue to Aleph-Function of Two Variables
We recall that AFTV [21] is an extension of the I-function possessing two variables [24]. Here, in the present article, we define a basic analogue to AFTV.
We record Next, we have ℵ(z 1 , z 2 ; q) = ℵ 0,n 1 ;m 2 ,n 2 :m 3 ,n 3 where ω = √ −1, and where z 1 , z 2 = 0 and are real or complex. An empty product is elucidated as unity, and · · · , r; i = 1, · · · , r ; i = 1, · · · , r ). All the As, αs, γs, δs, Es, and Fs are presumed to be positive quantities for standardization intention, the as, bs, cs, ds, es, and f s are complex numbers. The definition of a basic analogue to AFTV will, however, make sense, even if some of these quantities are equal to zero. The contour L 1 is in the s 1 -plane and goes from −ω∞ to +ω∞, with loops where necessary, to make sure that the poles of G q d j −δ j s 1 (j = 1, · · · , m 2 ) are to the right-hand side and all the poles of G q 1−a j +α j s 1 +A j s 2 (j = 1, · · · , n 1 ), G q 1−c j +γs 1 (j = 1, · · · , n 2 ) lie to the left-hand side of L 1 . The contour L 2 is in the s 2 -plane and goes from −ω∞ to +ω∞, with loops where necessary, to ensure that the poles of G q f j −F j s 2 (j = 1, · · · , m 3 ) are to the right-hand side and all the poles of G q 1−a j +α j s 1 +A j s 2 (j = 1, · · · , n 1 ), G q 1−e j +E j s 2 (j = 1, · · · , n 2 ) lie to the left-hand side of L 2 . For values of |s 1 | and |s 2 |, the integrals converge, if (s 1 log(z 1 ) − log sin πs 1 ) < 0 and (s 2 log(z 2 ) − log sin πs 2 ) < 0.
Proof. We apply the definitions (8) and (14) in the left-hand side of (19), we have (say I) By using standard calculations, we arrive at Next, we apply formula (12) to the equation above; then, we get Considering the above q-Mellin-Barnes double contour integrals in terms of the basic analogue to AFTV, we obtain (19).
Proof. If we replace µ by −µ in (19), and follow the proof of Theorem 1, then we can easily obtain (24).

Particular Cases
By setting τ i , τ i , τ i → 1, the basic analogue to AFTV reduces to the basic analogue to the I-function of two variables [24]. Let C 1 = c j , γ j 1,n 2 , c ji , γ ji n 2 +1,P i ; e j , E j 1,n 3 , e ji , γ ji n 3 +1,P i .
Proof. By setting τ i , τ i , τ i → 1 and following the proof of Theorem 3, we can easily obtain the desired result (33).

Remark 1.
If the basic analogue to the I-function of two variables reduces to the basic analogue to the H-function of two variables [25], then we can obtain the result due to Yadav et al. [18].
The basic analogue to AFTV reduces to the basic analogue to AFTV, defined by Ahmad et al. [26].
Then, we have following relation: Data Availability Statement: Not applicable.