New approach to generalized Mittag-Leffler function via quantum calculus

TL;DR

This paper introduces a novel q-analogue extension of the Mittag-Leffler function, exploring its properties like integral representation, q-differentiation, and q-Laplace transform, with applications demonstrated through particular cases.

Contribution

It presents the first q-analogue extension of the Mittag-Leffler function along with its fundamental properties and potential applications.

Findings

Derived integral representation of the q-Mittag-Leffler function

Established q-differentiation and q-Laplace transform formulas

Provided specific cases demonstrating applications of the new function

Abstract

We aim to introduce a new extension of Mittag-Leffler function via q-analogue and obtained their significant properties including integral representation, q-differentiation, q-Laplace transform, image formula under q-derivative operators. We also consider some particular cases to give the applications of our main results.

Equations

$$E_{\eta }(u)=\sum _{m=0}^{\infty }\frac{u^m}{\Gamma (\eta m+1)m!},(\eta ,u\in \mathbb{C};\Re (\eta )>0),$$

$$E_{\eta ,\kappa }(u)=\sum _{m=0}^{\infty }\frac{u^m}{\Gamma (\eta m+\kappa )m!},(\eta ,\kappa \in \mathbb{C};\Re (\eta )>0,\Re (\kappa )>0).$$

$$E_{\eta ,\kappa }^{\sigma }(u)=\sum _{m=0}^{\infty }\frac{u^m(\sigma {)}_m}{\Gamma (\eta m+\kappa )m!},(\eta ,\kappa ,\sigma \in \mathbb{C};\Re (\eta )>0,\Re (\kappa )>0,\Re (\sigma )>0),$$

$$e_{\eta ,\kappa }(u;q)=\sum _{m=0}^{\infty }\frac{u^m}{{\Gamma }_q(\eta m+\kappa )}(|z|<(1-q{)}^{-\alpha }),$$

$$E_{\eta ,\kappa }^{\sigma }(u;q)=\sum _{m=0}^{\infty }\frac{(q^{\sigma };q{)}_m}{(q;q{)}_m}\frac{u^m}{{\Gamma }_q(\eta m+\kappa )},$$

$$(\eta ,\kappa ,\sigma \in \mathbb{C};\Re (\eta )>0,\Re (\kappa )>0,\Re (\sigma )>0,|q|<1).$$

$$(\lambda ;q{)}_m=\{\begin{array}{ll}1& ;m=0,\\ (1-\lambda )(1-\lambda q)\dots (1-\lambda q^{m-1})& ;m\in \mathbb{N}\end{array}$$

$$(\lambda ;q{)}_m=\frac{(\lambda ;q{)}_{\infty }}{(\lambda q^m;q{)}_{\infty }}$$

$$(\lambda ;q{)}_{\eta }=\frac{(\lambda ;q{)}_{\infty }}{(\lambda q^{\eta };q{)}_{\infty }},\eta \in \mathbb{C},$$

$$(s-t{)}^{(m)}=\{\begin{array}{ll}1& ;m=0\\ \prod _{i=0}^{m-1}(s-t{q}^i)& ;m\ne 0\end{array}$$

$$(s-t{)}^{(m)}=s^m(t/s;q{)}_m,(s\ne 0).$$

$$(s-t{)}^{(m)}=s^m\frac{(t/s;q{)}_{\infty }}{(q^{\tau }t/s;q{)}_{\infty }},(s;q{)}_{\tau }=\frac{(s;q{)}_{\infty }}{(s{q}^{\tau };q{)}_{\infty }}.$$

$$(s-t{)}^{(\tau )}=s^{\tau }(t/s;q{)}_{\tau }.$$

$${\left(\genfrac{}{}{0ex}{}{s}{t}\right)}_q=\frac{[s{]}_q}{[t{]}_q[s-t{]}_q}=\frac{(q;q{)}_s}{(q;q{)}_t(q;q{)}_{s-t}}={\left(\genfrac{}{}{0ex}{}{s}{s-t}\right)}_q.$$

$${\left(\genfrac{}{}{0ex}{}{\tau }{m}\right)}_q=\frac{(q^{-\tau };q{)}_m}{(q;q{)}_m}(-1{)}^m{q}^{\tau m-\left(\genfrac{}{}{0ex}{}{m}{2}\right)}=\frac{{\Gamma }_q(\tau +1)}{{\Gamma }_q(m+1){\Gamma }_q(\tau -m+1)}$$

$${\Gamma }_q(u)=\frac{(q;q{)}_{\infty }}{(q^u;q{)}_{\infty }}(1-q{)}^{1-u}=(1-q{)}^{(u-1)}(1-q{)}^{1-u},$$

$${\Gamma }_q(u+1)=[u{]}_q{\Gamma }_q(u)$$

$$B_q(\eta ,\kappa )=\frac{{\Gamma }_q(\eta ){\Gamma }_q(\kappa )}{{\Gamma }_q(\eta +\kappa )}={\int }_0^1{u}^{\eta -1}\frac{(qu;q{)}_{\infty }}{(q^{\kappa }u;q{)}_{\infty }}{d}_{q}u={\int }_0^1{u}^{\eta -1}(uq;q{)}_{\kappa -1}{d}_{q}u,$$

$$(\Re (\eta ),\Re (\kappa )>0).$$

$${\Gamma }_q(u+1)=\frac{1-q^u}{1-q}{\Gamma }_q(u)$$

$$D_{q}f(u)=\frac{f(u)-f(uq)}{u(1-q)}(u\ne 0,q\ne 1),(D_{q}f)(0)=\underset{u\to 0}{\lim }(D_{q}f)(u)$$

$${\int }_0^{u}f(t)d(t;q)=u(1-q)\sum _{m=0}^{\infty }{q}^{m}f(u{q}^k).$$

$$\frac{(q^c;q{)}_m}{(q^{\sigma };q{)}_m}=\frac{B_q(\sigma +m,c-\sigma )}{B_q(\sigma ,c-\sigma )}.$$

$$E_{\eta ,\kappa }^{(\sigma ;c)}(u;q)=\sum _{m=0}^{\infty }\frac{B_q(\sigma +m,c-\sigma )}{B_q(\sigma ,c-\sigma )}\frac{(q^c;q{)}_m}{(q;q{)}_m}\frac{u^m}{{\Gamma }_q(\eta m+\kappa )}$$

$$(\Re (c)>\Re (\sigma )>0,|q|<1),$$

$$E_{\eta ,\kappa }^{(\sigma ;1)}(u;q)=\sum _{m=0}^{\infty }\frac{(q^{\sigma };q{)}_m}{(q;q{)}_m}\frac{u^m}{{\Gamma }_q(\eta m+\kappa )}=E_{\eta ,\kappa }^{\sigma }(u;q),$$

$$E_{\eta ,\kappa }^{(1;c)}(u;q)=\sum _{m=0}^{\infty }\frac{u^m}{{\Gamma }_q(\eta m+\kappa )}=e_{\eta ,\kappa }(u;q),$$

$$E_{1,1}^{(1;c)}(u;q)=\sum _{m=0}^{\infty }\frac{(q^c;q{)}_m}{(q;q{)}_m}{u}^m=\frac{(q^{c}u;q{)}_{\infty }}{(q;q{)}_{\infty }}={}_1{\varphi }_0(q^c;-;q,u),$$

$$\lim\limits_{m\rightarrow\infty}\bigg{|}\frac{s_{m+1}}{s_{m}}\bigg{|}=\bigg{|}\frac{B_{q}(\sigma+m+1,c-\sigma)}{B_{q}(\sigma+m,c-\sigma)}\bigg{|}\bigg{|}\frac{(q^{c},q)_{m+1}}{(q^{c},q)_{m}}\bigg{|}\bigg{|}\frac{(q,q)_{m}}{(q,q)_{m+1}}\bigg{|}\bigg{|}\frac{\Gamma(\eta m+\kappa)}{\Gamma(\eta m+\eta+\kappa)}~{}u\bigg{|}$$

$$=\lim\limits_{m\rightarrow\infty}\bigg{|}\frac{(q^{c+m},q)_{\infty}}{(q^{c+m+1},q)_{\infty}}~{}\frac{(q^{\sigma+m},q)_{\infty}}{(q^{\sigma+m+1},q)_{\infty}}~{}\frac{(q^{\eta m+\kappa},q)_{\infty}}{(q^{\eta m+\kappa+\eta},q)_{\infty}}~{}\frac{(q^{m+1},q)_{\infty}}{(q^{m},q)_{\infty}}~{}~{}(1-q)^{-\eta}u\bigg{|}$$

Full text

1. Overture

The Swedish mathematician Gösta Mittag-Leffler discovered a special function in 1903 (see, [3, 5]) defined as

[TABLE]

where $\Gamma(.)$ is a classical gamma function [9]. The special function defined in (1.1) is called Mittag Leffler function (MLf).

For the very first time, in 1905, A. Wiman [12] firstly proposed the generalization of the MLf $E_{\eta}(u)$ as

[TABLE]

Subsequently, the generalized form of series (1.1) and (1.2) studied by Prabhakar [8] in 1971 as:

[TABLE]

where $(\sigma)_{m}=\frac{\Gamma(\sigma+m)}{\Gamma(\sigma)}$ denotes the Pochhammer symbol [9].

The Mittag-Leffler function plays a vital role in the solution of fractional order differential equations and fractional order integral equations. It has recently become a subject of rich interest in the field of fractional calculus and its applications and nowadays some mathematicians consider to refer the classical Mittag-Leffler function as the Queen Function in the Fractional Calculus. An enormous amount of research in the theory of Mittag-Leffler functions has been published in the literature. For detailed account of the various generalizations, properties and applications of the MLf readers may refer to the literatures [1, 13, 14, 17, 18, 19, 20, 21].

The $q$-calculus is the $q$-extension of the ordinary calculus. The theory of $q$-calculus operators in recent past have been applied in the areas of ordinary fractional calculus, optimal control problem, in finding solutions of the $q$-difference and $q$-integral equations and $q$-transform analysis.

In 2009, Mansoor [6] has proposed a new form of $q$-analogue of the Mittag-Leffler function is given as:

[TABLE]

where $\eta>0,\kappa\in\mathbb{C}$.

Recently, Sharma and Jain [10] introduced the $q$-analogue of generalized MLf as given underneath:

[TABLE]

[TABLE]

2. Prelude

In the theory of $q$-series (see[4]), for complex $\lambda$ and $0<q<1$, the $q$-shifted factorial is defined as follows:

[TABLE]

which is equivalent to

[TABLE]

and its extension naturally as:

[TABLE]

where the principal value of $q^{\eta}$ is taken.

For $s,t\in\mathbb{R}$ the $q$-analogue of the exponent $(s-t)^{m}$ is

[TABLE]

and connected by the following relationship

[TABLE]

Obviously, its expansion for $\tau\in\mathbb{R}$ as

[TABLE]

Note that

[TABLE]

The $q$-analogue of binomial coefficient is defined for $s,t>0$ as

[TABLE]

The definition can be generalized in the following way. For arbitrary complex $\tau$ we have

[TABLE]

where $\Gamma_{q}(u)$ is the q-gamma function.

The $q$-gamma and $q$-beta functions([4]) are defined by

[TABLE]

for $u\in\mathbb{R}\setminus\{0,-1,-2,-3,\ldots\};|q|<1.$

Clearly,

[TABLE]

and

[TABLE]

[TABLE]

Further, the $\Gamma_{q}(u)$ satisfies the functional equation

[TABLE]

Also, the $q$-difference operator and $q$-integration of a function $f(u)$ defined on a subset of $\mathbb{C}$ are given by [4] respectively.

[TABLE]

and

[TABLE]

3. Extended $q$-Mittag-Leffler function and their properties

In this section, we extend the definition (1.5) by introducing the following relation for $(q^{c},q)_{m}$

[TABLE]

Now, we define the extension of generalized Mittag-Leffler function (1.5) using above relation as:

[TABLE]

[TABLE]

where $B_{q}(.)$ is the $q$-analog of beta function.

We enumerate the relations and particular cases of $q$-analogue of extended generalized Mittag-Leffler function with other special functions as given below

1. (i)

If we put $c=1$ in (3.2), we obtain

[TABLE]

where the function $E_{\eta,\kappa}^{\sigma}(u;q)$ is the $q$-analogue of Mittag-Leffler function defined in (1.5) . 2. (ii)

Again, if we take $\sigma=1$ in (3.2), we get

[TABLE]

the function $e_{\eta,\kappa}(u;q)$ can be termed as $q$-analogue of Mittag-Leffler function defined in (1.4). 3. (iii)

If we consider $\eta=\kappa=\sigma=1$, in (3.2), we find

[TABLE]

where the function ${}_{2}\phi_{0}(q^{c};-;q,u)=(1-u)^{-c}$ can be termed as $q$- binomial function. 4. (iv)

On setting $c=c+\sigma$, in (3.2), then similarly, we obtain $q$-analogue of Mittag-Leffler function $E_{\eta,\kappa}^{\sigma}(u;q)$ defined in (1.5).

4. Convergence of $E_{\eta,\kappa}^{(\sigma;c)}(u;q)$

Theorem 4.1**.**

The $q$-analogue of the extended generalized Mittag-Leffler function defined by the summation formula (3.2) converges absolutely for $|u|<(1-q)^{-\eta}$ provided that $0<q<1,\eta>0,\Re(c)>\Re(\sigma),c,\sigma\in\mathbb{C}$.

Proof.

Writing the summation formula (3.2) as $E_{\eta,\kappa}^{(\sigma;c)}(u;q)=\sum\limits_{m=0}^{\infty}s_{m}$ and by applying ratio formula, we find

[TABLE]

[TABLE]

[TABLE]

[TABLE]

∎

5. Recurrence Relations

Theorem 5.1**.**

If $\eta,\kappa,\sigma\in\mathbb{C}$, $\Re(\eta)>0,\Re(\kappa)>0,\Re(\sigma)>0$ and $\sigma\neq c$, then

[TABLE]

Proof.

By the definition (3.2), we have

[TABLE]

[TABLE]

Since $(1-q^{c})=(1-q^{c+m})-q^{c}(1-q^{m})$, the above equation reduces to

[TABLE]

[TABLE]

On replacing $m$ with $m+1$ in the second summation, it becomes

[TABLE]

[TABLE]

which leads to the required result (5.1). ∎

6. Some elementary properties of Extended $q$-Mittag-Leffler function

We begin with the underlying theorem, which shows the integral representation of extended $q$-Mittag-Leffler function:

Theorem 6.1**.**

(Integral representation) For the extended $q$-Mittag-Leffler function, we have

[TABLE]

provided that, $\eta,\kappa,\sigma\in\mathbb{C}$, $\Re(\eta)>0,\Re(\kappa)>0,\Re(\sigma)>0$ and $\sigma\neq c$.

Proof.

By the definition of q-analogue of beta function, we can rewrite equation (3.2) as follows:

[TABLE]

[TABLE]

[TABLE]

which leads to the required result (6.1). ∎

Theorem 6.2**.**

For $\eta,\kappa,\sigma\in\mathbb{C},\Re(\eta)>0,\Re(\kappa)>0,\Re(\sigma)>0,c\neq\sigma$, then for any $m\in\mathbb{N}$, we have

[TABLE]

Proof.

By considering the function

[TABLE]

and using the definition (3.2), then, in view of (2.11), we obtain

[TABLE]

[TABLE]

Since, according to the functional equation (2.10), the r.h.s of the above expression can be written as

[TABLE]

Conclusively, we obtain

[TABLE]

Iterating above result $m-1$ times, we obtain the required result (6.2). ∎

Theorem 6.3**.**

Let $\xi,\zeta,\sigma,\kappa\in\mathbb{C};\Re(\xi),\Re(\kappa),\Re(\sigma)>0;\zeta\neq 0,-1,-2,\ldots$ then

[TABLE]

[TABLE]

In particular,

[TABLE]

Proof.

By using the definition (3.2), the l.h.s of equation (6.3) can be written as

[TABLE]

Interchanging the order of summation and integration and in view of equation (2.9), we obtain the required result (6.3).

In equation (6.3) replacing $\rho=\eta$, $\kappa=\zeta$, then in view of equation (3.2), we can clearly obtain (6.4). ∎

Theorem 6.4**.**

(q-Laplace transform) For $q$-analogue of the extended generalized Lapalce transform is defined as follows:

[TABLE]

[TABLE]

provided that $\kappa,\sigma,s\in\mathbb{C};\Re(\beta),\Re(\kappa),\Re(s)>0.$

Proof.

The $q$-Laplace transform of a suitable function is given by means of following $q$-integral [11]

[TABLE]

The $q$-extension of the exponential function [4] is given by

[TABLE]

and

[TABLE]

By using the above $q$-exponential series and the $q$-integral equation (2.12), we can write equation (6.6) as

[TABLE]

Using the definition (3.2) and the definition of $q$-Laplace transform, we obtain

[TABLE]

[TABLE]

On interchanging the order of summation and writing the $j$ series as ${}_{1}\phi_{0}$, which can be summed up as $\frac{1}{(q^{1+\rho m};q)_{\infty}}$ and after some simplifications, we obtain the required result (6.5). ∎

7. Kober type fractional $q$- calculus operators

Agarwal [2] established Kober type fractional $q$-integral operator in the following manner

[TABLE]

where $\Re(\mu)>0$.

Also, Garg et al. [7] introduced Kober fractional $q$-derivative operator given by

[TABLE]

where $m=[\Re(\mu)]+1,\,m\in\mathbb{N}$.

The image formula of the power function $u^{m}$ under the above operators [7] are given as:

[TABLE]

[TABLE]

Theorem 7.1**.**

The underlying assumption holds true:

[TABLE]

particularly,

[TABLE]

provided that if $\eta,c>0,\kappa,\sigma,u\in\mathbb{C};\Re(\kappa),\,\Re(\sigma)>0$.

Proof.

The proof of (7.5) can easily be obtained by making use of the definition (3.2) and the result (7.3).

Now, on setting $\sigma=\nu+\mu$ in the definition (3.2), we obtain the result (7.6). ∎

Theorem 7.2**.**

The underlying assumption holds true:

[TABLE]

particularly,

[TABLE]

provided that if $\eta,c>0,\kappa,\sigma,u\in\mathbb{C};\Re(\kappa),\Re(\sigma)>0$.

Proof.

The proof of (7.7) can easily be obtained by making use of the definition (3.2) and the result (7.4). Similarly, on setting $\sigma=\nu+1$ in the definition (3.2), we obtain the result (7.8). ∎