1. Introduction
In last few decades, a progressive instantaneous development has been found in the field of
q-calculus because of its fundamental importance in numerous areas such as applied mathematics, mechanics, mathematical physics, Lie theory and quantum algebra, see for example [
1,
2,
3,
4]. The subject of
q-calculus began to appear in the nineteenth century because of its applications in several branches of science and engineering. Throughout the present paper,
indicates the set of complex numbers,
designates set of natural numbers and
designates set of non-negative integers. Further, the variable
such that
.
We review certain definitions and concepts related to the
q-calculus from [
1], which will be used throughout this work.
q-analogue of shifted factorial
is specified as
q-analogues of
is specified as
and
q-factorial function is specified as
q-binomial coefficient
is specified as
q-exponential function is specified as
The
q-analogue of derivative of a function
f at a point
, denoted by
is specified by
For any two arbitrary functions
and
, the following relation for
q-derivative operator
holds true:
Next, we review the concepts and results related to the q-Appell, q-Bernoulli, q-Euler, q-Genochhi and q-Hermite polynomials.
In 1880, Appell [
5] introduced an indispensable class of polynomial sequences, known as the Appell polynomials. After that, an important generalization of these Appell polynomials is given by Al-Salam [
6] and some of its properties are studied. The
q-Appell polynomials
satisfy the following
q-difference equation:
and have the following generating equation [
6]:
where
is an analytic function at
and
The
q-Bernoulli,
q-Euler and
q-Genocchi polynomials perform an essential role in various expansions and approximation formulae, which are useful both in analytic theory of numbers and in classical and numerical analysis. Different members of
q-Appell family can be obtained by selecting appropriate generating function
. These members along with their notations, names, generating functions and series definitions are listed in
Table 1.
The
q-Bernoulli ,
q-Euler and
q-Genocchi numbers have deep connections with number theory and occur in combinatorics. The first few values of
[
8],
[
8] and
[
10] are given in
Table 2.
The
q-Hermite polynomials are special or limiting cases of the orthogonal polynomials as they contains no parameter other than
q and appears to be at the bottom of a hierarchy of the classical
q-orthogonal polynomials [
2]. These polynomials are very important as many more complicated families of the orthogonal polynomials can be represented in terms of the
q-Hermite polynomials
. The
possess the following generating relation [
11]:
In the present article, a new family of q-Hermite-Appell polynomials is introduced and many interesting properties of these q-special polynomials are studied. The generating function, series representation and determinant forms for this new family of polynomials are established. Further, the q-recurrence relations and q-difference equations for this family and certain specific q-members related to this family are derived. In addition, shapes of this newly introduced q-special polynomials are shown graphically and their zeros are also explored using numerical simulations.
2. q-Hermite-Appell Polynomials
This section aims to introduce q-Hermite-Appell polynomials (q-HAP) by means of generating function. Further, the series expansion and determinant definition of these q-special polynomials are established.
Utilizing expansion (
5) in Relation (
9) and then replacing powers of
of
x by the correlating polynomials
and after summing up the terms of the resultant equation and indicating resultant
q-HAP in r.h.s. by
, the following generating relation for
q-Hermite-Appell polynomials
is obtained:
Using Equations (
10) and (
11) in generating Equation (
15), simplifying and comparing the coefficients of identical powers of
t in obtained relation, the following series definition of
q-Hermite-Appell polynomials
is obtained:
Certain members of
q-Appell family are given in
Table 1. For suitable selections of function
in generating Relation (
15), different members related to
q-Hermite-Appell family are obtained. These members along with their notations, names, generating functions and series definitions are mentioned in
Table 3.
On taking
in Relation (
16), the following series definition for
q-Hermite-Appell numbers (
q-HAN)
is obtained:
where
denotes
q-Hermite numbers.
Next, on taking
in series definitions of
q-HBP
,
q-HEP
and
q-HGP
given in
Table 3 and in view of notations given in
Table 1, the
q-Hermite-Bernoulli,
q-Hermite-Euler and
q-Hermite-Genocchi numbers are obtained. These numbers are listed in
Table 4.
In 1982, several characterization and clarification of
q-Appell family are provided by Srivastava [
12]. During the past few decades, the
q-Appell polynomials have been studied from various aspects, see for example [
3,
4] and using numerous techniques in [
13]. The determinant form of
q-Appell polynomials is obtained by Keleshteri and Mahmudov in [
14]. Due to the importance of determinant forms for computational and applied purposes, the determinant definitions of
q-HAP
together with certain members belonging to this
q-Hermite-Appell family are obtained. By using a similar approach ([
14], p. 359 (Theorem 7)) and in view of Equations (
11) and (
15), the following determinant form for
q-HAP
is obtained:
Definition 1. The q-Hermite-Appell polynomials of degree n are defined bywhere and are q-Hermite polynomials; and Remark 1. Since the q-HBP -HEP and q-HGP mentioned in Table 3 are particular members of q-Hermite-Appell family . Thus, by making suitable choices for the coefficients and in determinant definition of q-HAP the determinant definitions of q-HBP -HEP and q-HGP can be obtained. For instance, taking in Equation (18), the following determinant form of q-HBP is obtained: Definition 2. The q-Hermite-Bernoulli polynomials of degree n are defined bywhere are q-Hermite polynomials. Next, taking
and
in Equation (
18), the following determinant definition of
q-HEP
is obtained:
Definition 3. The q-Hermite-Euler polynomials of degree n are defined bywhere are q-Hermite polynomials. Further, taking
and
in Equation (
18), the following determinant definition of
q-HGP
is obtained:
Definition 4. The q-Hermite-Genocchi polynomials of degree n are defined bywhere are q-Hermite polynomials. Remark 2. The determinant definitions of q-HBN -HEN and q-HGN can be obtained by taking in Equations (20)–(22) respectively and then using suitable notations from Table 4 (I–III). In the next section, q-recurrence relations and q-difference equations of q-HAP and also for certain members belonging to this family are derived.
3. q-Recurrence Relations and q-Difference Equations
The differential or difference equations, linear second order
q-difference equations are of particular interest in the theory and applications of
q-difference equations. In 2002, differential equation involving Appell polynomials is studied by He and Ricci [
15] by making use of factorization method. Later in 2014, Mahmudov [
11] applied an innovative technique to obtain
q-recurrence relations and
q-difference equations for
q-Appell polynomials
together with certain members using only lowering operators that are
q-derivatives.
The q-recurrence relation for q-HAP is established in the form of the following result:
Theorem 1. The following linear homogeneous q-recurrence relations for q-Hermite-Appell polynomials holds true: Proof. Differentiation of generating function (
15) with respect to
x and in view of fact that
we have
Comparing the coefficients of identical powers of
t, we find
Since, the lowering operator
satisfies operational rule:
it follows that
which is derivative operator for
q-HAP
.
Exchanging
x by
in Equation (
15) and differentiating the obtained relation with respect to
t by making using of Formula (
7), we get
Again, using Formula (
7) in Equation (
29) and after that multiplying by
t, it follows that
which after simplifying gives
Using the facts that
,
and Equations (
13) and (
15) (with
t replaced by
) the r.h.s. of above equation can be expressed as:
which on simplification gives
Comparing the coefficients of identical powers of
t in above equation, we find
Replacing
k by
in the third term of the r.h.s. of above equation, assertion (
23) is proved. □
Remark 3. Using Relation (28) in linear homogeneous recurrence Relation (34), the following consequence is deduced: Corollary 1. The q-Hermite-Appell polynomials satisfy the following q-difference equation: Further, the q-recurrence relations and q-difference equations for several members of q-Hermite-Appell family are obtained by considering the following examples:
Example 1. Taking in Equation (15), we find Using generating function of q-Bernoulli numbers (Table 1 (I)) and result ([16], p. 6 (24)–(25)) in q-recurrence Relation (23), the following linear homogeneous q-recurrence relation for q-HBP is obtained: Similarly, we get the following q-difference equation for q-HBP : Example 2. Taking in Equation (15), we get Using generating function of q-Euler numbers (Table 1 (II)) and result ([16], p. 8 (32)–(33)) in q-recurrence Relation (23), the following linear homogeneous q-recurrence relation for q-HEP is obtained: Also, the following q-difference equation for q-HEP is obtained: Example 3. Taking in Equation (15), we find Using generating function of q-Genocchi numbers (Table 1 (III)) and result ([16], p. 9 (37) and (38)) in q-recurrence relation (23), the following linear homogeneous q-recurrence relation for q-HGP is obtained: Also, the following q-difference equation for q-HGP is obtained: In the next section, the graphs of q-HBP , q-HEP and q-HGP are displayed by using Matlab (R2010a, Mathworks, New Delhi, India). The zeros of these polynomials are also investigated using numerical computations.
4. Concluding Remarks
Nowadays, there has been progressively instantaneous development in the computing environment. The numerical investigation is very much essential to understand the basic concepts of
q-numbers and
q-polynomials. By using computer software, we can examine concepts much more easily. In 2006, Ryoo et al. [
17] provide the numerical investigation of the zeros of certain
q-polynomials. Moreover, the graphs of a new class of
q-Bernoulli polynomials are explored in [
18]. Research in this direction will lead to a captivating approach that employs numerical methods in the fields of these
q-special polynomials to turn out in many areas of mathematics and mathematical physics.
First, we plot the graphs of q-HBP -HEP and q-HGP for and The zeros of these q-special polynomials are also investigated and their behaviour is observed. This will show the four plots combined into one for each of these polynomials.
To show graphical representation of
q-HBP
-HEP
and
q-HGP
, the values of first five
and
are required. Using the values listed in
Table 2, the expressions of first five
and
are obtained and are mentioned in
Table 5.
Further, by making appropriate substitutions from
Table 2 and
Table 5 in series definitions mentioned in
Table 3 (I–III), the expressions of the first five
q-HBP
-HEP
and
q-HGP
for
are obtained and are mentioned in
Table 6.
Using Matlab and by using the expressions of
and
from
Table 6 for
,
Figure 1 is drawn.
Next, the real and complex zeros of
q-HBP
,
q-HEP
and
q-HGP
are computed using Matlab for same values of
n,
q and
. These are given in
Table 7 and
Table 8.
Remark 4. In view of the values given in Table 7 and Table 8, the following important relation between the real and complex zeros of q-HAP is observed:where n is degree of the polynomial. Further, we plot the distribution of zeros of
,
and
for
in
Figure 2.
In order to make the above discussion more clear, we draw the combined graphs of shape and zeros of
,
and
for
in
Figure 3 and
Figure 4.
It is to be noted that in
Figure 3 out of total two complex zeros only one, with positive imaginary part is visible, due to the absence of negative imaginary axis in this graphs.
These figures gives mathematicians an unrestricted capability to create visual mathematical investigations of the behaviour of q-HAP . The approach presented in this paper is general and opens new possibilities to deal with other hybrid families of q-special polynomials. The results established in this article may find applications in certain branches of mathematics, physics and engineering.
The following
Appendix A aims to demonstrate the shape and zeros of
q-HBP
for
.