Generalizations of fractional q-Leibniz formulae and applications

Abstract

In this paper we generalize the fractional q-Leibniz formula introduced by Agarwal in (Ganita 27(1-2):25-32, 1976) for the Riemann-Liouville fractional q-derivative. This extension is a q-version of a fractional Leibniz formula introduced by Osler in (SIAM J. Appl. Math. 18(3):658-674, 1970). We also introduce a generalization of the fractional q-Leibniz formula introduced by Purohit for the Weyl fractional q-difference operator in (Kyungpook Math. J. 50(4):473-482, 2010). Applications are included.

1 q-notions and notations

Let q be a positive number, $0<q<1$. In the following, we follow the notations and notions of q-hypergeometric functions, the q-gamma function ${\mathrm{\Gamma }}_q(x)$, Jackson q-exponential functions $E_q(x)$, and the q-shifted factorial as in [1, 2]. By a q-geometric set A, we mean a set that satisfies if $x\in A$, then $qx\in A$. Let f be a function defined on a q-geometric set A. The q-difference operator is defined by

$$D_{q}f(x):=\frac{f(x)-f(qx)}{x-qx},x\ne 0.$$

(1)

The n th q-derivative, $D_q^{n}f$, can be represented by its values at the points $\{q^{j}x,j=0,1,\dots ,n\}$ through the identity

$$D_q^{n}f(x)={(-1)}^n{(1-q)}^{-n}{x}^{-n}{q}^{-n(n-1)/2}\sum _{r=0}^n{(-1)}^r{\left[\begin{array}{c}n\\ r\end{array}\right]}_q{q}^{r(r-1)/2}f\left(x{q}^{n-r}\right)$$

(2)

for every x in $A\setminus \{0\}$. After some straightforward manipulations, formula (2) can be written as

$$D_q^{n}f(x)={(1-q)}^{-n}{x}^{-n}\sum _{r=0}^n{q}^r\frac{{(q^{-n};q)}_r}{{(q;q)}_r}f\left(x{q}^r\right)\text{for }x\in A\setminus \{0\}.$$

(3)

Moreover, formula (2) can be inverted through the relation

$$f\left(x{q}^n\right)=\sum _{k=0}^n{(-1)}^k{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{(1-q)}^k{x}^k{q}^{\left(\begin{array}{c}k\\ 2\end{array}\right)}{D}_q^{k}f(x).$$

(4)

Formulas (2) and (4) are well known and follow easily by induction. Jackson [3] introduced an integral denoted by

$${\int }_a^{b}f(x)d_{q}x$$

as a right inverse of the q-derivative. It is defined by

$${\int }_a^{b}f(t)d_{q}t:={\int }_0^{b}f(t)d_{q}t-{\int }_0^{a}f(t)d_{q}t,a,b\in \mathbb{C},$$

(5)

where

$${\int }_0^{x}f(t)d_{q}t:=(1-q)\sum _{n=0}^{\mathrm{\infty }}x{q}^{n}f\left(x{q}^n\right),x\in \mathbb{C},$$

(6)

provided that the series at the right-hand side of (6) converges at $x=a$ and b. In [4], Hahn defined the q-integration for a function f over $[0,\mathrm{\infty })$ and $[x,\mathrm{\infty })$, $x>0$, by

$$\begin{array}{r}{\int }_0^{\mathrm{\infty }}f(t)d_{q}t=(1-q)\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{q}^{n}f\left(q^n\right),\\ {\int }_x^{\mathrm{\infty }}f(t)d_{q}t=(1-q)\sum _{n=1}^{\mathrm{\infty }}x{q}^{-n}(1-q)f\left(x{q}^{-n}\right),\end{array}$$

(7)

respectively, provided that the series converges absolutely. Al-Salam [5] defined a fractional q-integral operator $K_q^{-\alpha }$ by

$$\begin{array}{r}{K}_q^{-\alpha }\varphi (x):=\frac{q^{-\frac{1}{2}\alpha (\alpha -1)}}{{\mathrm{\Gamma }}_q(\alpha )}{\int }_x^{\mathrm{\infty }}{t}^{\alpha -1}{(x/t;q)}_{\alpha -1}\varphi \left(t{q}^{1-\alpha }\right)d_{q}t,\\ K_q^0\varphi (x):=\varphi (x),\end{array}$$

(8)

where $\alpha \ne -1,-2,\dots$ , as a generalization of the q-Cauchy formula

$$\begin{array}{rcl}{K}_q^{-n}\varphi (x)& =& {\int }_x^{\mathrm{\infty }}{\int }_{x_{n-1}}^{\mathrm{\infty }}\cdots {\int }_{x_1}^{\mathrm{\infty }}\varphi (t)d_{q}t{d}_q{x}_1\cdots d_q{x}_{n-1}\\ =& \frac{q^{-\frac{1}{2}n(n-1)}}{{\mathrm{\Gamma }}_q(n)}{\int }_x^{\mathrm{\infty }}{t}^{n-1}{(x/t;q)}_{n-1}\varphi \left(t{q}^{1-n}\right)d_{q}t,\end{array}$$

which he introduced in [6] for a positive integer n. Using (7), we can write (8) explicitly as

$$K_q^{-\alpha }\varphi (x)=q^{-\frac{\alpha (\alpha +1)}{2}}{x}^{\alpha }{(1-q)}^{\alpha }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k{q}^{\left(\begin{array}{c}k\\ 2\end{array}\right)}{\left[\begin{array}{c}-\alpha \\ k\end{array}\right]}_q\varphi \left(x{q}^{-\alpha -k}\right),$$

(9)

or in a more simple form

$$K_q^{-\alpha }\varphi (x)=q^{-\frac{\alpha (\alpha +1)}{2}}{x}^{\alpha }{(1-q)}^{\alpha }\sum _{k=0}^{\mathrm{\infty }}{q}^{-k\alpha }\frac{{(q^{\alpha };q)}_k}{{(q;q)}_k}\varphi \left(x{q}^{-\alpha -k}\right).$$

(10)

Using (2), we can prove

$$K_q^n\varphi (x)={(-1)}^n{D}_q^n\varphi (x)(n\in \mathbb{N}).$$

(11)

This paper is organized as follows. In Section 2, we mention in brief some known fractional and q-fractional Leibniz formulae. In Section 3, we generalize the fractional q-Leibniz formula of the Riemann-Liouville fractional q-derivative introduced by Agarwal in [7]. Finally, in Section 4, we extend the fractional q-Leibniz formula introduced by Purohit [8] for the q-Weyl derivatives of nonnegative integral orders to any real order.

2 Fractional and q-fractional Leibniz formulas

The Riemann-Liouville fractional q-integral operator is introduced by Al-Salam in [5] and later by Agarwal in [9] and defined by

$$I_q^{\alpha }f(x):=\frac{x^{\alpha -1}}{{\mathrm{\Gamma }}_q(\alpha )}{\int }_0^x{(qt/x;q)}_{\alpha -1}f(t)d_{q}t,\alpha \notin \{-1,-2,\dots \}.$$

(12)

Using (6), (12) reduces to

$$I_q^{\alpha }f(x)=x^{\alpha }{(1-q)}^{\alpha }\sum _{n=0}^{\mathrm{\infty }}{q}^n\frac{{(q^{\alpha };q)}_n}{{(q;q)}_n}f\left(x{q}^n\right),$$

(13)

which is valid for all α. The Riemann-Liouville fractional q-derivative of order α, $\alpha >0$, is defined by

$$D_q^{\alpha }=D_q^k{I}_q^{k-\alpha },k=\lceil \alpha \rceil .$$

For the definition of Caputo fractional q-derivatives, see [10]. See also [11] for more applications. Liouville [12] introduced the fractional Leibniz rule

$$I^{\alpha }\{f(x)g(x)\}=\sum _{k=0}^{\mathrm{\infty }}\left(\begin{array}{c}-\alpha \\ k\end{array}\right)f^{(k)}(x)I^{\alpha +k}g(x),$$

(14)

where

$$I^{\alpha }\{f(x)\}=\frac{1}{\mathrm{\Gamma }(\alpha )}{\int }_0^x{(x-t)}^{\alpha -1}f(t)dt$$

is the familiar Riemann-Liouville integral operator. While Liouville used Fourier expansions in obtaining (14), Grünwald [13] and Letnikov [14] obtained (14) by a different technique. Other extensions and proofs are in the work of Watanabe [15], Post [16], Bassam [17], and Gaer-Rubel [18]. In a series of papers [19–23], Osler introduced several generalizations of (14). For example, in [19] Osler introduced the fractional Leibniz formula

$$D_{g(z)}^{\alpha }\{u(z)v(z)\}=\sum _{-\mathrm{\infty }}^{\mathrm{\infty }}\left(\begin{array}{c}\alpha \\ \gamma +k\end{array}\right)D_{g(z)}^{\alpha -\gamma -k}u(z)D_{g(z)}^{\gamma +k}v(z),$$

(15)

which coincides with (14) when we set $g(z)=z$, $\gamma =0$ and replace α with −α in (15). For an extensive study of the fractional calculus and its applications in physics and control theory, see [24–28]. There are two q-analogues of the fractional Leibniz rule (14). Al-Salam and Verma [29] introduced the fractional Leibniz formula

$$I_q^{\alpha }(UV)(z)=\sum _{m=0}^{\mathrm{\infty }}{\left[\begin{array}{c}-\alpha \\ m\end{array}\right]}_q{D}_q^{m}U\left(z{q}^{-\alpha -m}\right)I_q^{\alpha +m}V(z),$$

(16)

formally. An analytic proof of (16) is introduced in [10] where the following theorem is introduced.

Theorem 2.1 Let $U(z)$ be an entire function with q-exponential growth of order k, $k<ln{q}^{-1}$, and a finite type δ, $\delta \in \mathbb{R}$. Let V be a function that satisfies

$$\sum _{j=0}^{\mathrm{\infty }}{q}^j\left|V\left(z{q}^j\right)\right|<\mathrm{\infty }(z\in \mathbb{C}).$$

Then (16) holds for $z\in \mathbb{C}\setminus \{0\}$ and $\alpha \in \mathbb{R}$.

For the definition of the q-exponential growth, see [30]. In [7], Agarwal introduced the following fractional q-Leibniz formula.

Theorem 2.2 Let U and V be two analytic functions which have power series representations at $z=0$ with radii of convergence $R_1$ and $R_2$, respectively, and $R=min\{R_1,R_2\}$. Then

$$I_q^{\alpha }(UV)(z)=\sum _{n=0}^{\mathrm{\infty }}{\left[\begin{array}{c}-\alpha \\ n\end{array}\right]}_q{D}_q^{n}U(z)I_q^{\alpha +n}V\left(z{q}^n\right)(|z|<R),$$

(17)

Proof See [7]. □

Recently, Purohit [31] used (17) to derive a number of summation formulae for the generalized basic hypergeometric functions. In the following section, we introduce a generalization of Agarwal’s fractional q-Leibniz formula (17). Let $0<R<\mathrm{\infty }$ and $D_R:=\{z\in \mathbb{C}:|z|<R\}$. In the following, we say that a function $f\in L_q^1(D_R)$ if

$$\sum _{j=0}^{\mathrm{\infty }}{q}^j\left|f\left(z{q}^j\right)\right|<\mathrm{\infty }\text{for all }z\in D_R\setminus \{0\}.$$

In [8], Purohit derived a q-extension of the Leibniz rule for q-derivative via the Weyl q-derivative operator defined in (8). He proved that for a nonnegative integer α,

$$K_q^{\alpha }(UV)(z)=\sum _{r=0}^{\alpha }\frac{{(-1)}^r{q}^{r(r+1)/2}{(q^{-\alpha };q)}_r}{{(q;q)}_r}{K}_q^{\alpha -r}U(z)K_{q,z}^{r}V\left(z{q}^{r-\alpha }\right),$$

(18)

where $U(z)=z^{-p_1}u(z)$, $V(z)=z^{-p_2}v(z)$, u and v are analytic functions having a power series expansion at $z=0$ with radii of convergence ρ, $\rho >0$, and $p_1,p_2\ge 0$. Purohit established some summation formulae as an application of the fractional Leibniz formula (18) which can be represented as

$$K_q^{\alpha }(UV)(z)=\sum _{r=0}^{\alpha }\frac{{(q^{-\alpha };q)}_r}{{(q;q)}_r}{K}_q^{\alpha -r}U(z)D_{q^{-1},z}^r\left\{V\left(z{q}^{\alpha }\right)\right\},$$

(19)

where we used

$$D_{q^{-1},z}^{r}V\left(z{q}^{\alpha }\right)={(-1)}^r{q}^{r(r+1)/2}{K}_{q,z}^{r}V\left(z{q}^{r-\alpha }\right).$$

3 A generalization for Agarwal’s fractional q-Leibniz formula

In this section we introduce a q-analogue of the fractional Leibniz formula (15) when $g(z)=z$. The case $\gamma =0$ of the derived fractional q-Leibniz formula is the fractional q-Leibniz formula (17) introduced by Agarwal [7].

Theorem 3.1 Let G be a branch domain of the logarithmic function. Let a, b be complex numbers and R be a positive number. Let u and v be analytic functions in the disk $D_R$. Let U and V be defined in $G\cap D_R$ through the relations

$$U(z)=z^{a}u(z),V(z)=z^{b}v(z).$$

(20)

If $V(\cdot )$ and $UV(\cdot )$ are in $L_q^1(D_R)$, then

(21)

where $z\in G\cap D_R$, and $\alpha ,\gamma \in \mathbb{R}$.

Remark 3.2 It is worthwhile to notice that if we set $\gamma =0$ in (21), we obtain Agarwal’s fractional Leibniz rule (17) with less restrictive conditions on the functions $U(z)$ and $V(z)$. Actually, the special case $\gamma =0$ of Theorem 3.1 is an extension of the result given by Manocha and Sharma in [32].

Proof Since V, UV are in $L_q^1(D_R)$, then

$$z^{a}V\in L_q^1(D_R),Re(b)>-1\text{and}Re(a+b)>-1.$$

From (13) we obtain

$$I_q^{\alpha }(UV)(z)=z^{\alpha }{(1-q)}^{\alpha }\sum _{n=0}^{\mathrm{\infty }}{q}^n\frac{{(q^{\alpha };q)}_n}{{(q;q)}_n}U\left(z{q}^n\right)V\left(z{q}^n\right).$$

(22)

Substituting with

$$\frac{{(q^{\alpha };q)}_n}{{(q;q)}_n}=\frac{{(q^{\alpha -\gamma };q)}_n}{{(q;q)}_n}{(q^{\alpha };q)}_{-\gamma }{(q^{\alpha -\gamma +n};q)}_{\gamma }$$

into (22), we obtain

$$I_q^{\alpha }(UV)(z)=z^{\alpha }{(1-q)}^{\alpha }{(q^{\alpha };q)}_{-\gamma }\sum _{n=0}^{\mathrm{\infty }}{q}^n\frac{{(q^{\alpha -\gamma };q)}_n}{{(q;q)}_n}{(q^{\alpha -\gamma +n};q)}_{\gamma }U\left(z{q}^n\right)V\left(z{q}^n\right).$$

(23)

The existence of UV in the space $L_q^1(D_R)$ guarantees that the series in (22) or in (23) converges absolutely for all $z\in D_R\setminus \{0\}$. Replace x in (4) with ξ and then let

$$f(\xi )={(q^{\alpha -\gamma }\xi /z;q)}_{\gamma }U(\xi ).$$

Consequently,

(24)

Then substituting (24) into (23), we get

$$\begin{array}{rcl}{I}_q^{\alpha }(UV)(z)& =& z^{\alpha }{(1-q)}^{\alpha }{(q^{\alpha };q)}_{-\gamma }\sum _{n=0}^{\mathrm{\infty }}{q}^n\frac{{(q^{\alpha -\gamma };q)}_n}{{(q;q)}_n}V\left(z{q}^n\right)\\ \times \sum _{k=0}^n{(-1)}^k{(1-q)}^k{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{q}^{k(k-1)/2}{z}^k{D}_q^k({(q^{\alpha -\gamma }\xi /z;q)}_{\gamma }U(\xi )){|}_{\xi =z}.\end{array}$$

(25)

Using (2), we obtain

$$\begin{array}{rcl}{D}_q^k({(q^{\alpha -\gamma }\frac{\xi }{z};q)}_{\gamma }U(\xi )){|}_{\xi =k}& =& {(-1)}^k{(1-q)}^{-k}{z}^{a-k}{q}^{-\frac{k(k-1)}{2}}{(q^{\alpha -\gamma };q)}_{\gamma }\\ \times \sum _{r=0}^k{q}^{\frac{r(r-1)}{2}}{\left[\begin{array}{c}k\\ r\end{array}\right]}_q\frac{{(q^{\alpha };q)}_{k-r}}{{(q^{\alpha -\gamma };q)}_{k-r}}{q}^{(k-r)a}u\left(z{q}^{k-r}\right).\end{array}$$

Therefore, since $u(z)$ is analytic in $D_R$, there exists $M>0$ such that

(26)

Consequently,

(27)

Set $F(\xi ):={(q^{\alpha -\gamma }\frac{\xi }{z};q)}_{\gamma }U(\xi )$. Then substituting (27) into (25), we obtain

(28)

The last series converges for all $z\in D_R\setminus \{0\}$ since $V,z^{a}V\in L_q^1(D_R)$. Consequently, the series in (28) is absolutely convergent, and we can interchange the order of summations in (25). This leads to

(29)

Since

$$\left(I_q^{\alpha -\gamma +k}V\right)\left(z{q}^k\right)={\left(z{q}^k\right)}^{\alpha -\gamma +k}{(1-q)}^{\alpha -\gamma +k}\sum _{j=0}^{\mathrm{\infty }}{q}^j\frac{{(q^{\alpha -\gamma +k};q)}_j}{{(q;q)}_j}V\left(z{q}^{j+k}\right)$$

and

$${(q^{\alpha -\gamma };q)}_{j+k}={(q^{\alpha -\gamma };q)}_k{(q^{\alpha -\gamma +k};q)}_j,$$

the substitution with the last two identities in (29) gives

$$\begin{array}{c}{I}_q^{\alpha }(UV)(z)\hfill \\ =z^{\gamma }{(1-q)}^{\gamma }{(q^{\alpha };q)}_{-\gamma }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k{q}^{-k(k-1)/2+k(-\alpha +\gamma )}\frac{{(q^{\alpha -\gamma };q)}_k}{{(q;q)}_k}\hfill \\ \times \left(I_q^{\alpha -\gamma +k}V\right)\left(z{q}^k\right)D_q^k({(q^{\alpha -\gamma }\frac{\xi }{z};q)}_{\gamma }U(\xi )){|}_{\xi =z}\hfill \\ =z^{\gamma }{(1-q)}^{\gamma }{(q^{\alpha };q)}_{-\gamma }\sum _{k=0}^{\mathrm{\infty }}{\left[\begin{array}{c}-\alpha +\gamma \\ k\end{array}\right]}_q\left(I_q^{\alpha -\gamma +k}V\right)\left(z{q}^k\right)D_q^k({(q^{\alpha -\gamma }\frac{\xi }{z};q)}_{\gamma }U(\xi )){|}_{\xi =z},\hfill \end{array}$$

and the theorem follows. □

Example 3.3 Let γ, λ, μ, and α be complex numbers satisfying

$$Re(\lambda )>0,Re(\lambda +\mu )>0,\mu \notin {\mathbb{N}}_0\text{and}Re(\alpha )>0.$$

Then

(30)

Proof We prove the identity by using Theorem 3.1. Take $U(z)=z^{\mu }$ and $v(z)=z^{\lambda -1}$. Then

$$\begin{array}{c}{D}_q^m\left\{{(q^{\alpha -\gamma }\frac{\xi }{z};q)}_{\gamma }{\xi }^{\mu }\right\}\hfill \\ =D_q^m\sum _{k=0}^{\mathrm{\infty }}\frac{{(q^{-\gamma };q)}_k}{{(q;q)}_k}{\left(\frac{q^{\alpha }}{z}\right)}^k{\xi }^{\mu +k}\hfill \\ =\sum _{k=0}^{\mathrm{\infty }}\frac{{(q^{-\gamma };q)}_k}{{(q;q)}_k}{\left(\frac{q^{\alpha }}{z}\right)}^k\frac{{\mathrm{\Gamma }}_q(\mu +k+1)}{{\mathrm{\Gamma }}_q(\mu +k-m+1)}{\xi }^{\mu +k-m}.\hfill \end{array}$$

Hence,

(31)

and

$$\begin{array}{rcl}\left(I_q^{\alpha -\gamma +m}V\right)\left(z{q}^m\right)& =& \frac{{\mathrm{\Gamma }}_q(\lambda )}{{\mathrm{\Gamma }}_q(\lambda +\alpha -\gamma +m)}{\left(z{q}^m\right)}^{\lambda +\alpha -\gamma +m-1}\\ =& \frac{{\mathrm{\Gamma }}_q(\lambda )}{{\mathrm{\Gamma }}_q(\lambda +\alpha -\gamma )}\frac{{(1-q)}^m}{{(q^{\lambda +\alpha -\gamma };q)}_m}{\left(z{q}^m\right)}^{\lambda +\alpha -\gamma +m-1}.\end{array}$$

(32)

Then applying Theorem 3.1 gives

$$\begin{array}{rcl}{I}_q^{\alpha }(UV)(z)& =& z^{\mu +\lambda +\alpha -1}\frac{{\mathrm{\Gamma }}_q(\alpha ){\mathrm{\Gamma }}_q(\lambda )}{{\mathrm{\Gamma }}_q(\alpha -\gamma ){\mathrm{\Gamma }}_q(\lambda +\alpha -\gamma )}\\ \times \sum _{m=0}^{\mathrm{\infty }}{q}^{m(\lambda +\mu )}{\frac{{(q^{\alpha -\gamma },q^{-\mu };q)}_m}{{(q,q^{\lambda +\alpha -\gamma };q)}_m}}_2{\varphi }_1(q^{-\gamma },q^{\mu +1};q^{\mu -m+1};q,q^{\alpha }).\end{array}$$

(33)

On the other hand,

$$I_q^{\alpha }{z}^{\lambda +\mu -1}=\frac{{\mathrm{\Gamma }}_q(\lambda +\mu )}{{\mathrm{\Gamma }}_q(\lambda +\mu +\alpha )}{z}^{\lambda +\mu +\alpha -1}.$$

(34)

Equating (33) and (34) gives (30). □

Example 3.4 For complex numbers a, b, A, B, d, and D such that $Re(b)>-1$, $Re(B)>0$, and $Re(b+B)>1$,

(35)

for $|z{q}^{-a-A}|<1$.

Proof

The previous identity follows by taking

$$U(z)=z^b{(z;q)}_{-a},V(z)=z^{B-1}{(z{q}^{-a};q)}_{-A},$$

and applying Theorem 3.1 with

$$\alpha =d+D-b-B,\gamma =D-B.$$

Then using (3), we obtain

$$\begin{array}{rcl}{D}_q^m\left({(q^{d-b}\frac{\xi }{z};q)}_{D-B}{\xi }^b{(\xi ;q)}_{-a}\right){|}_{\xi =z}& =& {(1-q)}^{-m}{z}^{-m+b}{(z;q)}_{-a}\frac{{(q^{d-b};q)}_{\mathrm{\infty }}}{{(q^{d-b+D-B};q)}_{\mathrm{\infty }}}\\ {\times }_3{\varphi }_2(q^{-m},q^{d-b+D-B},z{q}^{-a};q^{d-b},z;q,q^{b+1}).\end{array}$$

(36)

In addition,

$$\begin{array}{rcl}\left(I_q^{\alpha -\gamma +m}V\right)\left(z{q}^m\right)& =& z^{d-b+\beta -1+m}{q}^{m^2+(d-b+\beta -1)m}\frac{{\mathrm{\Gamma }}_q(\beta )}{{\mathrm{\Gamma }}_q(B+m+d-b)}\\ {\times }_2{\varphi }_1(q^A,q^{\beta };q^{m-b};q,q^{m-a-A})\end{array}$$

(37)

and

$${\left[\begin{array}{c}\gamma -\alpha \\ m\end{array}\right]}_q={(-1)}^m{q}^{(\gamma -\alpha )m}{q}^{\left(\begin{array}{c}-m\\ 2\end{array}\right)}\frac{{(q^{\alpha -\gamma };q)}_m}{{(q;q)}_m}.$$

(38)

Substituting with (36)-(38) into (21), we obtain

$$\begin{array}{rcl}{I}_q^{\alpha }UV(z)& =& z^{d+D-1}{(1-q)}^{D-B}{(z;q)}_{-a}\frac{{\mathrm{\Gamma }}_q(\beta )}{{\mathrm{\Gamma }}_q(B+d-b)}\\ \times \sum _{m=0}^{\mathrm{\infty }}{(-1)}^m{q}^{\left(\begin{array}{c}m\\ 2\end{array}\right)}{q}^{mB}\frac{{(q^{b-d};q)}_m}{{(q;q)}_m{(q^{d+D-b};q)}_m}\\ {\times }_3{\varphi }_2(q^{-m},q^{d+D-b-B},z{q}^{-a};q^{d-b},z;q,q^{b+1})\\ {\times }_2{\varphi }_1(q^A,q^B;q^{d+B-b+m};q,q^{-a-A+m}).\end{array}$$

(39)

On the other hand,

$$I_q^{\alpha }UV(z)=\frac{{\mathrm{\Gamma }}_q(b+B)}{{\mathrm{\Gamma }}_q(d+D)}z^{d+D-1}{}_2{\varphi }_1(q^{a+A},q^{b+B};q^{d+D};q,q^{-(a+A)}).$$

(40)

Combining (39) and (40), we obtain (35). □

4 A q-extension of the Leibniz rule via Weyl-type of a q-derivative operator

In this section, we prove that the q-expansion in (18) can be derived for any $\alpha \in \mathbb{R}$. The proof we introduce is completely different from the one introduced by Purohit for nonnegative integer values of α. We start with characterizing a sufficient class of functions for which $K_q^{-\alpha }$ exists for some α.

Definition 4.1 Let $\alpha \in \mathbb{C}$ and let f be a function defined on a $q^{-1}$-geometric set A. We say that f is of class $S_{q,\alpha }$ if there exists $\mu \in \mathbb{C}$, $Re\mu >Re\alpha$ such that

$$f\left(x{q}^{-n}\right)=O\left(q^{n\mu }\right)\text{as }n\to \mathrm{\infty },x\in A.$$

Proposition 4.2 If $\alpha \in \mathbb{Z}$, then $K_q^{-\alpha }f$ exists for any function f defined on $(0,\mathrm{\infty })$. If $\alpha \notin \mathbb{Z}$ and $f\in S_{q,\alpha }$, then $K_q^{-\alpha }f$ exists.

Proof If $\alpha \in \mathbb{Z}$, then by (11), $K_q^{-\alpha }f$ exists for any functions f defined on a $(0,\mathrm{\infty })$. If $\alpha \notin \mathbb{Z}$ and $f\in S_{q,\alpha }$, then for each $x>0$, there exists a constant $C>0$, C depends on x and α, such that

$$\left|f\left(x{q}^{-n-\alpha }\right)\right|\le C{q}^{n\mu }.$$

Applying the previous inequality in (10) gives

$$\begin{array}{rcl}|K_q^{-\alpha }f(x)|& \le & C{q}^{-\alpha (\alpha +1)/2}{|x|}^{\alpha }{(1-q)}^{\alpha }\frac{{(-q^{Re\alpha };q)}_{\mathrm{\infty }}}{{(q;q)}_{\mathrm{\infty }}}\sum _{k=0}^{\mathrm{\infty }}{q}^{k(Re\mu -Re\alpha )}\\ \le & C{q}^{-\alpha (\alpha +1)/2}{|x|}^{\alpha }{(1-q)}^{\alpha }\frac{{(-q^{Re\alpha };q)}_{\mathrm{\infty }}}{(1-q^{(Re\mu -Re\alpha )}){(q;q)}_{\mathrm{\infty }}}.\end{array}$$

 □

In the following, we define a sufficient class of functions ${\mathfrak{S}}_{q,\mu }$ for which $K_q^{-\alpha }f$ exists for all α.

Definition 4.3 Let f be a function defined on a $q^{-1}$-geometric set A. We say that f is in the class ${\mathfrak{S}}_{q,\mu }$ if there exist $\mu >0$ and $\nu \in \mathbb{R}$ such that for each $x\in A$,

$$\left|f\left(x{q}^{-n}\right)\right|=O\left(q^{\mu n(n+\nu )}\right)\text{as }n\to \mathrm{\infty }.$$

It is clear that if $f\in {\mathfrak{S}}_{q,\mu }$, then $f\in S_{q,\alpha }$ for all α. The spaces $S_{q,\alpha }$ and ${\mathfrak{S}}_{q,\mu }$ are q-analogues of the spaces of fairly good functions and good functions, respectively, introduced by Lighthill [[33], p.15], see also [[34], Chapter VII].

Example 4.4 An example of a function in a class $S_{q,1/2}$ is any function of the form

$$\frac{P_n(x)}{{(ax;q)}_{\mathrm{\infty }}}\text{for all }n\in {\mathbb{N}}_0,$$

where $P_n(x)$ is a polynomial of degree n and a is a constant such that $ax{q}^k\ne 1$ for all $k\in {\mathbb{N}}_0$.

The keynotes in proving the generalization of Purohit q-fractional Leibniz formula are two identities. The first one is

$$K_q^{\alpha }{z}^{-p}=q^{\alpha (1-\alpha )/2}{q}^{-\alpha p}{z}^{-\alpha -p}\frac{{\mathrm{\Gamma }}_q(\alpha +p)}{{\mathrm{\Gamma }}_q(p)},$$

(41)

which holds for any $p\in \mathbb{R}$ when $\alpha \in \mathbb{N}$ or holds when $\alpha +p>0$. The proof of (41) follows from (10) by replacing α with −α, x with z, and setting $\varphi (z)=z^{-p}$. The second identity follows from the formula (4) with q replaced with $q^{-1}$ and x with z. That is,

$$\begin{array}{rcl}f\left(z{q}^{-n}\right)& =& \sum _{k=0}^n{(q^{-1}-1)}^k{q}^{-\left(\begin{array}{c}k\\ 2\end{array}\right)}{\left[\begin{array}{c}n\\ k\end{array}\right]}_{q^{-1}}{z}^k{D}_{q^{-1}}^{k}f(z)\\ =& \sum _{k=0}^n{q}^{k(k-1)/2}{(1-q)}^k{q}^{-nk}{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{z}^k{D}_{q^{-1}}^{k}f(z),\end{array}$$

(42)

where we use [[1], Eq. (I.47)]

$${\left[\begin{array}{c}n\\ k\end{array}\right]}_{q^{-1}}={\left[\begin{array}{c}n\\ k\end{array}\right]}_q{q}^{k^2-nk}.$$

The identity in (41) leads to the following result.

Lemma 4.5 Let p and α be such that $0<Rep<1$ and $\alpha >Rep$. Let G be the principal branch of the logarithmic function and let $D_R:=\{z\in \mathbb{C}:|z|>R\}\cap G$. Assume that

$$U(z)=z^p\sum _{j=0}^{\mathrm{\infty }}{a}_j{z}^{-j}$$

is analytic on $D_R$. Let

$${\mathrm{\Omega }}_{\alpha }=\{z\in D_R:q^{\alpha }z\in D_R\}.$$

Then $K_q^{\alpha }U(z)$ exists for all $z\in {\mathrm{\Omega }}_{\alpha }$ and is equal to

$$K_q^{\alpha }U(z)=q^{\alpha p}\frac{{\mathrm{\Gamma }}_q(\alpha -p)}{{\mathrm{\Gamma }}_q(-p)}{z}^{p-\alpha }\sum _{j=0}^{\mathrm{\infty }}{a}_j\frac{{(q^{-p};q)}_j}{{(q^{\alpha -p};q)}_j}{q}^{-\alpha j}{z}^{-j}.$$

(43)

Proof From (10) we find that

$$K_q^{\alpha }U(z)=q^{\alpha (1-\alpha )/2}{q}^{\alpha p}{z}^{p-\alpha }{(1-q)}^{-\alpha }\sum _{k=0}^{\mathrm{\infty }}{q}^{k(\alpha -p)}\frac{{(q^{-\alpha };q)}_k}{{(q;q)}_k}\sum _{j=0}^{\mathrm{\infty }}{a}_j{q}^{(-\alpha +k)j}.$$

(44)

From the assumptions of the present lemma, we can easily deduce that the double series in (44) is absolutely convergent for all $z\in {\mathrm{\Omega }}_{\alpha }$. Hence, we can interchange the order of summations in (44). This and the q-binomials theorem [[1], Eq. (1.3.2)] give

$$\begin{array}{rcl}{K}_q^{\alpha }U(z)& =& q^{\alpha (1-\alpha )/2}{q}^{\alpha p}{z}^{p-\alpha }{(1-q)}^{-\alpha }\sum _{j=0}^{\mathrm{\infty }}{a}_j{q}^{-\alpha j}{z}^{-j}\sum _{k=0}^{\mathrm{\infty }}{q}^{k\alpha -kp+kj}\frac{{(q^{-\alpha };q)}_j}{{(q;q)}_j}\\ =& q^{\alpha (1-\alpha )/2}{q}^{\alpha p}{z}^{p-\alpha }{(1-q)}^{-\alpha }\sum _{j=0}^{\mathrm{\infty }}{a}_j{q}^{-\alpha j}{z}^{-j}\frac{{(q^{-p+j};q)}_{\mathrm{\infty }}}{{(q^{\alpha -p+j};q)}_{\mathrm{\infty }}}.\end{array}$$

Simple manipulations give (43). □

Lemma 4.6 Let p, α, G, U, $D_R$, and ${\mathrm{\Omega }}_{\alpha }$ be as in Lemma  4.5. Then

$$\sum _{j=0}^{\mathrm{\infty }}{q}^{\alpha j}\left|U\left(z{q}^{\alpha -j}\right)\right|<\mathrm{\infty }\mathrm{\forall }z\in {\mathrm{\Omega }}_{\alpha }.$$

Proof The proof is easy and is omitted. □

Theorem 4.7 Let U and V be functions defined on a $q^{-1}$-geometric set A and let $\alpha \in \mathbb{R}$. Assume that $UV\in S_{q,\alpha }$ and $U\in S_{q,\mu }$, $\mu >\frac{1}{2}$. Then

$$K_q^{\alpha }UV(z)=\sum _{m=0}^{\mathrm{\infty }}\frac{{(q^{-\alpha };q)}_m}{{(q;q)}_m}{K}_q^{\alpha -m}U(z)D_{q^{-1},z}^m\left\{V\left(z{q}^{\alpha }\right)\right\}$$

(45)

for all $z\in A$ and for all $\alpha \in \mathbb{R}$. If $\mu =1/2$, then (45) may not hold for all α on ℝ but only for α in a subdomain of ℝ.

Proof Let $z\in q^{-\alpha }A$ be arbitrary but fixed. Since $UV\in S_{q,\alpha }$, then

$$\sum _{k=0}^{\mathrm{\infty }}{q}^{k\alpha }\left|U\left(z{q}^{\alpha -k}\right)V\left(z{q}^{\alpha -k}\right)\right|<\mathrm{\infty }.$$

(46)

From (10),

Applying (42) with $f(z)=V(z{q}^{\alpha })$ yields

$$\begin{array}{rcl}\left(K_q^{\alpha }UV\right)(z)& =& q^{\alpha (1-\alpha )/2}{(1-q)}^{-\alpha }{z}^{-\alpha }\sum _{m=0}^{\mathrm{\infty }}{q}^{\alpha m}\frac{{(q^{-\alpha };q)}_m}{{(q;q)}_m}U\left(z{q}^{\alpha -m}\right)\\ \times \sum _{j=0}^m{q}^{j(1-j)/2}{(1-q)}^j{q}^{-mj}{\left[\begin{array}{c}m\\ j\end{array}\right]}_q{z}^j{D}_{q^{-1},z}^j\left\{V\left(z{q}^{\alpha }\right)\right\}.\end{array}$$

(47)

From the assumptions on the function U, there exists a constant $C_1>0$ and $\nu \in \mathbb{R}$ such that

$$\left|U\left(z{q}^{\alpha -m}\right)\right|\le C_1{q}^{\mu m(m+\nu )}.$$

Using (2) with ($q^{-1}$ instead of q), we obtain

$$z^j{D}_{q^{-1},z}^j\left\{V\left(z{q}^{\alpha }\right)\right\}\le C_2.$$

Consequently, the double series on (47) is bounded from above by

(48)

where we applied the identity cf., e.g., [[1], p.11],

$${(a;q)}_n:=\sum _{k=0}^n{(-1)}^k{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{q}^{\frac{k(k-1)}{2}}{a}^k.$$

(49)

Now, it is clear that if $\mu >1/2$, then the series on the most right-hand side of (48) is convergent for all $\alpha \in \mathbb{C}$. On the other hand, it is convergent only for $Re\alpha >-\nu +\frac{1}{2}$ when $\mu =\frac{1}{2}$. Therefore, we can interchange the order of summation in the series on the right-hand side of (47). This gives

$$\begin{array}{rcl}\left(K_q^{\alpha }UV\right)(z)& =& q^{\alpha (1-\alpha )/2}{(1-q)}^{-\alpha }{z}^{-\alpha }\\ \times \sum _{j=0}^{\mathrm{\infty }}\frac{{(q^{-\alpha };q)}_j}{{(q;q)}_j}{z}^j{(1-q)}^j{D}_{q^{-1},z}^{j}V\left(z{q}^{\alpha }\right)\\ \times \sum _{r=0}^{\mathrm{\infty }}{q}^{(\alpha -j)r}\frac{{(q^{-\alpha +j};q)}_r}{{(q;q)}_r}U\left(z{q}^{\alpha -j-r}\right).\end{array}$$

(50)

But

$$\sum _{r=0}^{\mathrm{\infty }}{q}^{(\alpha -j)r}\frac{{(q^{-\alpha +j};q)}_r}{{(q;q)}_r}U\left(z{q}^{\alpha -j-r}\right)=z^{\alpha -j}{(1-q)}^{\alpha -j}{q}^{(\alpha -j)(\alpha -j-1)/2}{K}_q^{\alpha -j}V(z).$$

Combining this latter identity with (50) yields the theorem. □

Example 4.8 Let γ, λ, μ, and α be complex numbers satisfying

$$Re(\lambda )>0,Re(\lambda +\mu )>0,\mu \notin {\mathbb{N}}_0\text{and}Re(\alpha )>0.$$

Then

(51)

Proof We prove the identity by using Theorem 3.1. Take $U(z)=z^{\mu }$ and $v(z)=z^{\lambda -1}$. Then

$$\begin{array}{c}{D}_q^m{(q^{\alpha -\gamma }\frac{\xi }{z};q)}_{\gamma }{\xi }^{\mu }\hfill \\ =D_q^m\sum _{k=0}^{\mathrm{\infty }}\frac{{(q^{-\gamma };q)}_k}{{(q;q)}_k}{\left(\frac{q^{\alpha }}{z}\right)}^k{\xi }^{\mu +k}\hfill \\ =\sum _{k=0}^{\mathrm{\infty }}\frac{{(q^{-\gamma };q)}_k}{{(q;q)}_k}{\left(\frac{q^{\alpha }}{z}\right)}^k\frac{{\mathrm{\Gamma }}_q(\mu +k+1)}{{\mathrm{\Gamma }}_q(\mu +k-m+1)}{\xi }^{\mu +k-m}.\hfill \end{array}$$

Hence,

(52)

and

$$\begin{array}{rcl}\left(I_q^{\alpha -\gamma +m}V\right)\left(z{q}^m\right)& =& \frac{{\mathrm{\Gamma }}_q(\lambda )}{{\mathrm{\Gamma }}_q(\lambda +\alpha -\gamma +m)}{\left(z{q}^m\right)}^{\lambda +\alpha -\gamma +m-1}\\ =& \frac{{\mathrm{\Gamma }}_q(\lambda )}{{\mathrm{\Gamma }}_q(\lambda +\alpha -\gamma )}\frac{{(1-q)}^m}{{(q^{\lambda +\alpha -\gamma };q)}_m}{\left(z{q}^m\right)}^{\lambda +\alpha -\gamma +m-1}.\end{array}$$

(53)

Then applying Theorem 3.1 gives

$$\begin{array}{rcl}{I}_q^{\alpha }(UV)(z)& =& z^{\mu +\lambda +\alpha -1}\frac{{\mathrm{\Gamma }}_q(\alpha ){\mathrm{\Gamma }}_q(\lambda )}{{\mathrm{\Gamma }}_q(\alpha -\gamma ){\mathrm{\Gamma }}_q(\lambda +\alpha -\gamma )}\\ \times \sum _{m=0}^{\mathrm{\infty }}{q}^{m(\lambda +\mu )}{\frac{{(q^{\alpha -\gamma },q^{-\mu };q)}_m}{{(q,q^{\lambda +\alpha -\gamma };q)}_m}}_2{\varphi }_1(q^{-\gamma },q^{\mu +1};q^{\mu -m+1};q,q^{\alpha }).\end{array}$$

(54)

On the other hand,

$$I_q^{\alpha }{z}^{\lambda +\mu -1}=\frac{{\mathrm{\Gamma }}_q(\lambda +\mu )}{{\mathrm{\Gamma }}_q(\lambda +\mu +\alpha )}{z}^{\lambda +\mu +\alpha -1}.$$

(55)

Equating (54) and (55) gives (51). □

Example 4.9 For complex numbers a, b, A, B, d, and D such that $Re(b)>-1$, $Re(B)>0$, and $Re(b+B)>1$,

(56)

for $|z{q}^{-a-A}|<1$.

Proof

The previous identity follows by taking

$$U(z)=z^b{(z;q)}_{-a},V(z)=z^{B-1}{(z{q}^{-a};q)}_{-A}$$

and applying Theorem 3.1 with

$$\alpha =d+D-b-B,\gamma =D-B.$$

Then using (3), we obtain

$$\begin{array}{rcl}{D}_q^m\left({(q^{d-b}\frac{\xi }{z};q)}_{D-B}{\xi }^b{(\xi ;q)}_{-a}\right){|}_{\xi =z}& =& {(1-q)}^{-m}{z}^{-m+b}{(z;q)}_{-a}\frac{{(q^{d-b};q)}_{\mathrm{\infty }}}{{(q^{d-b+D-B};q)}_{\mathrm{\infty }}}\\ {\times }_3{\varphi }_2(q^{-m},q^{d-b+D-B},z{q}^{-a};q^{d-b},z;q,q^{b+1}).\end{array}$$

(57)

In addition,

$$\begin{array}{rcl}\left(I_q^{\alpha -\gamma +m}V\right)\left(z{q}^m\right)& =& z^{d-b+\beta -1+m}{q}^{m^2+(d-b+\beta -1)m}\frac{{\mathrm{\Gamma }}_q(\beta )}{{\mathrm{\Gamma }}_q(B+m+d-b)}\\ {\times }_2{\varphi }_1(q^A,q^{\beta };q^{m-b};q,q^{m-a-A})\end{array}$$

(58)

and

$${\left[\begin{array}{c}\gamma -\alpha \\ m\end{array}\right]}_q={(-1)}^m{q}^{(\gamma -\alpha )m}{q}^{\left(\begin{array}{c}-m\\ 2\end{array}\right)}\frac{{(q^{\alpha -\gamma };q)}_m}{{(q;q)}_m}.$$

(59)

Substituting with (57)-(59) into (21), we obtain

$$\begin{array}{rcl}{I}_q^{\alpha }UV(z)& =& z^{d+D-1}{(1-q)}^{D-B}{(z;q)}_{-a}\frac{{\mathrm{\Gamma }}_q(\beta )}{{\mathrm{\Gamma }}_q(B+d-b)}\\ \times \sum _{m=0}^{\mathrm{\infty }}{(-1)}^m{q}^{\left(\begin{array}{c}m\\ 2\end{array}\right)}{q}^{mB}\frac{{(q^{b-d};q)}_m}{{(q;q)}_m{(q^{d+D-b};q)}_m}\\ {\times }_3{\varphi }_2(q^{-m},q^{d+D-b-B},z{q}^{-a};q^{d-b},z;q,q^{b+1})\\ {\times }_2{\varphi }_1(q^A,q^B;q^{d+B-b+m};q,q^{-a-A+m}).\end{array}$$

(60)

On the other hand,

$$I_q^{\alpha }UV(z)=\frac{{\mathrm{\Gamma }}_q(b+B)}{{\mathrm{\Gamma }}_q(d+D)}z^{d+D-1}{}_2{\varphi }_1(q^{a+A},q^{b+B};q^{d+D};q,q^{-(a+A)}).$$

(61)

Combining (60) and (61), we obtain (56). □