Certain Results of q-Sheffer – Appell Polynomials

In this paper, the class of q-Sheffer–Appell polynomials is introduced. The generating function, series definition, determinant definition and some other identities of this class are established. Certain members of q-Sheffer–Appell polynomials are investigated and some properties of these members are derived. In addition, the class of 2D q-Sheffer–Appell polynomials is introduced. Further, the graphs of some members of q-Sheffer–Appell polynomials and 2D q-Sheffer–Appell polynomials are plotted for different values of indices by using Matlab.


Introduction and Preliminaries
The subject of q-calculus leads to a new method for computations and classifications of q-special functions.It was launched in the 1920s.However, it has gained importance and considerable popularity during the last three decades [1][2][3][4][5][6][7][8][9].In the last decades, q-calculus has been developed into an interdisciplinary subject and served as a bridge between physics and mathematics.The recent interest in the subject is due to the fact that q-series has popped in such various areas as quantum groups, statistical mechanics, transcendental number theory, etc.The definitions and notations of q-calculus reviewed here are taken from [10] (see also [11,12]).
The q-analog of the Pochhammer symbol (δ) κ , also called a q-shifted factorial, are defined by The q-analogs of a complex number δ and of the factorial function are given as follows: [κ] q = κ ∑ ν=1 The q-binomial coefficients [ κ ν ] q are defined by κ ν q = (q; q) κ (q; q) ν (q; q) κ−ν = , ν = 0, 1, 2, ..., κ. ( The q-analog of the classical derivative D u of a function u at a point 0 = τ ∈ C is given as In addition, we note that , where d dτ denotes the classical ordinary derivative, (ii) D q (a 1 u(τ) + a 2 v(τ)) = a 1 D q u(τ) + a 2 D q v(τ), (iii) D q (uv)(τ) = u(qτ)D q v(τ) + v(τ)D q u(τ) = u(τ)D q v(τ) + D q u(τ)v(qτ), (vi The q-exponential functions e q (τ) and E q (τ) are defined as: which satisfy the following properties: D q e q (τ) = e q (τ), D q E q (τ) = E q (qτ), e q (τ)E q (−τ) = E q (τ)e q (−τ) = 1.(13) The class of Appell polynomials was introduced and characterized completely by Appell [13].Further, Throne [14], Sheffer [15] and Varma [16] studied this class of polynomials from different point of views.Sharma and Chak [17] introduced a q-analog for the class of Appell polynomials and called this sequence of polynomials as q-Harmonic.Later, Al-Salam [1] established the class of q-Appell polynomials {A κ,q (z)} ∞ κ=0 and investigated some of its properties.These polynomials appear in several problems of theoretical physics, applied mathematics, approximation theory and many other branches of mathematics.The polynomials A κ,q (z) (of degree κ) are called q-Appell polynomials provided that they satisfy the following q-differential equation The generating function for the q-Appell polynomials A κ,q (z) is given as: where is an analytic function at τ = 0 and A κ,q := A κ,q (0) denotes the q-Appell numbers.
We note that the function A q (τ) is called the determining function for the set A κ,q (z).Based on suitable selection for the function A q (τ), different members belonging to the family of q-Appell polynomial A κ,q (z) can be obtained.These members along with their notations, names and generating functions are listed in Table 1.
Table 1.Certain members of q-Appell family.
In addition, the q-Sheffer polynomials may be alternatively defined as: where In view of Equations ( 17) and (18), we have and The q-Sheffer polynomials for the pair (φ(τ), τ) q is called the q-Appell polynomials A κ,q (z) and for the pair (1, H(τ)) q becomes the q-associated Sheffer polynomials s κ,q (z).
Recently, Duran et al. [25] introduced the q-Hermite polynomials (qHP) H κ,q (z) by means of the following generating function: In [25], (p, q)-number is defined by [x] p,q = p x −q x p−q .It is worth noting that [x] p,q = c[x] q for some constant c in p.Thus, there is no need to deal with the family of (p, q)-Sheffer-Appell polynomials.
In the present article, a new family of q-Sheffer-Appell polynomials (qSAP) is introduced by means of generating functions, series and determinant definitions.Further, some results are obtained for some members of this family.In the next section, the q-Sheffer-Appell polynomials are introduced by means of the generating functions and series definition.In addition, the determinant definition and many interesting properties of these q-hybrid special polynomials are derived.In Section 3, we consider some members of q-Sheffer-Appell polynomials and obtain the determinant definitions and some other properties of these members.In Section 4, the class of 2D q-Sheffer-Appell polynomials (2DqSAP) is also introduced.In Section 5, the graphs of some members of q-Sheffer-Appell polynomials and 2D q-Sheffer-Appell polynomials are plotted for different values of indices by using Matlab.

q-Sheffer-Appell Polynomials
In this section, the generating function, series definition and determinant definition for the q-Sheffer-Appell polynomials s A κ,q (z) are introduced.
To establish the generating function for the qSAP by making use of replacement technique, the following result is proved: Theorem 1.The following generating function for the q-Sheffer-Appell polynomials s A κ,q (z) holds true: Proof.By expanding the q-exponential function e q (zτ) in the left hand side of Equation ( 15) and then replacing the powers of z, i.e., z 0 , z, z 2 , ..., z κ , by the corresponding polynomials s 0,q (z), s 1,q (z), s 2,q (z), ..., s κ,q (z) in the left hand side and z by s 1,q (z) in the right hand side of the resultant equation, we have Further, summing up the series in left hand side and then using Equation ( 18) in the resultant equation, we get Finally, indicating resultant qSAP by s A κ,q (z), that is the assertion in Equation ( 22) is proved.
Next, we introduce the series definition for the qSAP s A κ,q (z) by proving the following result: Theorem 2. The q-Sheffer-Appell polynomials s A κ,q (z) are defined by the following series definition: Proof.In view of Equations ( 16) and ( 18), Equation ( 22) can be written as: which on using the Cauchy product rule [26] gives Now, comparing the coefficients of identical powers of τ in above equation, we arrive at our assertion in Equation ( 26).
Theorem 3. The q-Sheffer-Appell polynomials s A κ,q (z) satisfy the following linear homogeneous recurrence relation: where Proof.Consider the generating function Taking the q-derivative of Equation ( 31) partially with respect to τ, we get Now, factorizing F q (z, τ) from its left hand side and after that multiplying both sides by τ, it follows that In view of the assumption in Equations ( 30) and (31), Equation (33) can be expressed as which on using the Cauchy product rule, gives Finally, equating the coefficients of identical powers of τ in above equation and after that dividing both sides of the resultant equation by [κ] q , we get the assertion in Equation (29).
Due to the importance of determinant form for the computational and applied purposes, we derive the determinant definition for the qSAP s A κ,q (z).
Proof.Consider s A κ,q (z) to be a sequence of the qSAP defined by Equation ( 22) and A κ,q , B κ,q be two numerical sequences (the coefficients of q-Taylor's series expansions of functions) such that On using Cauchy product rule for the two series production A q (τ) Âq (τ), we get Next, multiplying both sides of Equation ( 22) by Âq (t), we get Further, in view of Equations ( 18), ( 39) and (40), the above equation can be expressed as Now, on using Cauchy product rule for the two series in the right hand side of Equation (44), we obtain the following infinite system for the unknowns s A κ,q (z): Obviously, the first equation of the system in Equation (45) leads to our first assertion in Equation (36).The coefficient matrix of the system in Equation ( 45) is lower triangular, thus this assist us to obtain the unknowns s A κ,q (z) by applying Cramer rule to the first κ + 1 equations of the system in Equation (45).According to this, we can obtain where κ = 1, 2, 3, ..., which on expanding the determinant in the denominator and taking the transpose of the determinant in the numerator, yields to Finally, after κ circular row exchanges, i.e., after moving the jth row to the (j + 1)th position for j = 1, 2, 3, ..., κ − 1, we arrive at our assertion in Equation (37).
Theorem 5.The following identity for the qSAP s A κ,q (z) holds true: Proof.Expanding the determinant in Equation (37) with respect to the (κ + 1)th row and using a similar approach used in ([27], Theorem 3.1), the assertion in Equation ( 48) is proved.

Examples
Several members belonging to the q-Sheffer-Appell family s A κ,q (z) can be derived by making suitable selections for the functions A q (τ), φ q (τ) and H(τ).The q-Hermite polynomials (qHP) H κ,q (z) [25] are one of the important members of q-Sheffer family.In addition, the q-Bernoulli polynomials B κ,q (z), q-Euler polynomials E κ,q (z) and q-Genocchi polynomials G κ,q (z) are considerable members of the q-Appell family.In this section, we introduce the q-Hermite-Bernoulli polynomials H B κ,q (z), q-Hermite-Euler polynomials H E κ,q (z) and q-Hermite-Genocchi polynomials H G κ,q (z) by means of the generating functions, series definitions and also explore other properties of these members.
Theorem 6.The q-Hermite-Bernoulli polynomials H B κ,q (z) satisfy the following q-recurrence relations: Proof.Applying the q-derivative with respect to z to both sides of Equation (49), we get Now, equating the coefficient of like powers of τ in both sides of the above equation, we get the assertion in Equation (54).Similarly, on applying the q-derivative with respect to z to both sides of Equation ( 49) k times, we get the assertion in Equation (55).
Theorem 7. The q-Hermite-Euler polynomials H E κ,q (z) satisfy the following q-recurrence relations: Proof.Using a similar approach used in the proof of Theorem 6, we are led to the assertions in Equations ( 62) and (63).
Theorem 8.The q-Hermite-Genocchi polynomials H G κ,q (z) satisfy the following q-recurrence relations: Proof.Using a similar approach used in the proof of Theorem 6, we are led to the assertions in Equations ( 69) and (70).
In the next section, we introduce a new class of the 2D q-Sheffer-Appell polynomials by means of generating function and series representation.
The approach used in the previous section is further exploited to introduce the 2D q-Sheffer-Appell polynomials (2DqSAP) and the focus is on deriving its generating functions and series definitions.
Theorem 10.The 2D q-Sheffer-Appell polynomials s A κ,q (z 1 , z 2 ) are defined by the following series definitions: Proof.Using Equations ( 11) and (1) in Equation ( 73), we get Now, using the Cauchy product rule in the left hand side of the above equation and then equating the coefficients of like powers of τ in both sides of the resultant equation, we get the assertion in Equation (77).
Since for φ q (τ) = e q (−τ 2 ), H(τ) = [2] q τ the qSP s κ,q (z) reduce to qHP H κ,q (z), by making same choices for the functions φ q (τ) and H(τ) in Equations ( 73) and (77), we get A q (τ)e q ([2] q z 1 τ)e q (−τ Certain members belonging to the 2D q-Appell family are given in Table 2.By making suitable choices for the functions A q (t) in Equations ( 79) and (80), the generating functions and series definitions for the corresponding member belonging to the 2D q-Hermite-Appell family can be obtained.The resultant 2D q-Hermite-Appell polynomials (2DqHAP) along with their generating functions and series definitions are given in Table 3.

S. No.
A q (τ) Generating Functions Series Definition Polynomials II.

Graphical Representation
In this section, the shapes of some members of the q-Sheffer-Appell polynomials and 2D q-Sheffer-Appell polynomials are displayed with the help of Matlab.
To draw the graphs of qHBP H B κ,q (z), qHEP H E κ,q (z) and qHGP H G κ,q (z), we considered the first four values of q-Hermite polynomials H κ,q (z) [25]; the expressions of these polynomials are listed in Table 4.
Table 4. Expressions of the first four H κ,q (z).

Further Remarks
It is worth noting that the results derived in the previous sections can be exploited to establish further new relations.
Now, taking summation on both sides of the above equation and then multiplying both sides of the resultant equation by τ e q (τ)−1 and using Equation (49), we get where x = −z.
Similarly, we can obtain the following results: where x = −z.

Conclusions
We would like to underline that the q-series and q-polynomials have many applications in different fields of mathematics, physics and engineering.In the present article, we demonstrate how a new replacement technique has been adopted to introduce mixed type q-special polynomials and different method to establish their q-recurrence relation.
To extend this new and significant approach, the hybrid class of the q-Sheffer-Appell polynomials and 2D q-Sheffer-Appell polynomials are introduced by means of series expansion and generating functions.The determinant form related to q-Sheffer-Appell polynomials are derived, which are important for the computational and applied purposes.This process can be used to establish further a wide variety of formulas and new relations for several other q-special polynomials.
The q-difference equation for the two iterated q-Appell and mixed type q-Appell polynomials are established in [29,30].This aspect may be considered in future investigation.