# 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. In the following, we follow the notations and notions of q-hypergeometric functions, the q-gamma function $Γ q (x)$, 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 $x∈A$, then $qx∈A$. Let f be a function defined on a q-geometric set A. The q-difference operator is defined by

$D q f(x):= f ( x ) − f ( q x ) x − q x ,x≠0.$
(1)

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

$D q n f(x)= ( − 1 ) n ( 1 − q ) − n x − n q − n ( n − 1 ) / 2 ∑ r = 0 n ( − 1 ) r [ n r ] q q r ( r − 1 ) / 2 f ( x q n − r )$
(2)

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

(3)

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

$f ( x q n ) = ∑ k = 0 n ( − 1 ) k [ n k ] q ( 1 − q ) k x k q ( k 2 ) D q k f(x).$
(4)

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

$∫ a b f(x) d q x$

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

$∫ a b f(t) d q t:= ∫ 0 b f(t) d q t− ∫ 0 a f(t) d q t,a,b∈C,$
(5)

where

$∫ 0 x f(t) d q t:=(1−q) ∑ n = 0 ∞ x q n f ( x q n ) ,x∈C,$
(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 $[0,∞)$ and $[x,∞)$, $x>0$, by

$∫ 0 ∞ f ( t ) d q t = ( 1 − q ) ∑ n = − ∞ ∞ q n f ( q n ) , ∫ x ∞ f ( t ) d q t = ( 1 − q ) ∑ n = 1 ∞ x q − n ( 1 − q ) f ( x q − n ) ,$
(7)

respectively, provided that the series converges absolutely. Al-Salam [5] defined a fractional q-integral operator $K q − α$ by

$K q − α ϕ ( x ) : = q − 1 2 α ( α − 1 ) Γ q ( α ) ∫ x ∞ t α − 1 ( x / t ; q ) α − 1 ϕ ( t q 1 − α ) d q t , K q 0 ϕ ( x ) : = ϕ ( x ) ,$
(8)

where $α≠−1,−2,…$ , as a generalization of the q-Cauchy formula

$K q − n ϕ ( x ) = ∫ x ∞ ∫ x n − 1 ∞ ⋯ ∫ x 1 ∞ ϕ ( t ) d q t d q x 1 ⋯ d q x n − 1 = q − 1 2 n ( n − 1 ) Γ q ( n ) ∫ x ∞ t n − 1 ( x / t ; q ) n − 1 ϕ ( t q 1 − n ) d q t ,$

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

$K q − α ϕ(x)= q − α ( α + 1 ) 2 x α ( 1 − q ) α ∑ k = 0 ∞ ( − 1 ) k q ( k 2 ) [ − α k ] q ϕ ( x q − α − k ) ,$
(9)

or in a more simple form

$K q − α ϕ(x)= q − α ( α + 1 ) 2 x α ( 1 − q ) α ∑ k = 0 ∞ q − k α ( q α ; q ) k ( q ; q ) k ϕ ( x q − α − k ) .$
(10)

Using (2), we can prove

$K q n ϕ(x)= ( − 1 ) n D q n ϕ(x)(n∈N).$
(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

$I q α f(x):= x α − 1 Γ q ( α ) ∫ 0 x ( q t / x ; q ) α − 1 f(t) d q t,α∉{−1,−2,…}.$
(12)

Using (6), (12) reduces to

$I q α f(x)= x α ( 1 − q ) α ∑ n = 0 ∞ q n ( q α ; q ) n ( q ; q ) n f ( x q n ) ,$
(13)

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

$D q α = D q k I q k − α ,k=⌈α⌉.$

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

$I α { f ( x ) g ( x ) } = ∑ k = 0 ∞ ( − α k ) f ( k ) (x) I α + k g(x),$
(14)

where

$I α { f ( x ) } = 1 Γ ( α ) ∫ 0 x ( x − t ) α − 1 f(t)dt$

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 [1923], Osler introduced several generalizations of (14). For example, in [19] Osler introduced the fractional Leibniz formula

$D g ( z ) α { u ( z ) v ( z ) } = ∑ − ∞ ∞ ( α γ + k ) D g ( z ) α − γ − k u(z) D g ( z ) γ + k v(z),$
(15)

which coincides with (14) when we set $g(z)=z$, $γ=0$ and replace α with −α in (15). For an extensive study of the fractional calculus and its applications in physics and control theory, see [2428]. There are two q-analogues of the fractional Leibniz rule (14). Al-Salam and Verma [29] introduced the fractional Leibniz formula

$I q α (UV)(z)= ∑ m = 0 ∞ [ − α m ] q D q m U ( z q − α − m ) I q α + m V(z),$
(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, and a finite type δ, $δ∈R$. Let V be a function that satisfies

$∑ j = 0 ∞ q j | V ( z q j ) | <∞(z∈C).$

Then (16) holds for $z∈C∖{0}$ and $α∈R$.

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

$I q α (UV)(z)= ∑ n = 0 ∞ [ − α n ] q D q n U(z) I q α + n V ( z q n ) ( | z | < R ) ,$
(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 and $D R :={z∈C:|z|. In the following, we say that a function $f∈ L q 1 ( D R )$ if

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,

$K q α (UV)(z)= ∑ r = 0 α ( − 1 ) r q r ( r + 1 ) / 2 ( q − α ; q ) r ( q ; q ) r K q α − r U(z) K q , z r V ( z q r − α ) ,$
(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, $ρ>0$, and $p 1 , p 2 ≥0$. Purohit established some summation formulae as an application of the fractional Leibniz formula (18) which can be represented as

$K q α (UV)(z)= ∑ r = 0 α ( q − α ; q ) r ( q ; q ) r K q α − r U(z) D q − 1 , z r { V ( z q α ) } ,$
(19)

where we used

$D q − 1 , z r V ( z q α ) = ( − 1 ) r q r ( r + 1 ) / 2 K q , z r V ( z q r − α ) .$

## 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 $γ=0$ 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 $G∩ D R$ through the relations

$U(z)= z a u(z),V(z)= z b v(z).$
(20)

If $V(⋅)$ and $UV(⋅)$ are in $L q 1 ( D R )$, then

(21)

where $z∈G∩ D R$, and $α,γ∈R$.

Remark 3.2 It is worthwhile to notice that if we set $γ=0$ 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 $γ=0$ 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

$z a V∈ L q 1 ( D R ),Re(b)>−1andRe(a+b)>−1.$

From (13) we obtain

$I q α (UV)(z)= z α ( 1 − q ) α ∑ n = 0 ∞ q n ( q α ; q ) n ( q ; q ) n U ( z q n ) V ( z q n ) .$
(22)

Substituting with

$( q α ; q ) n ( q ; q ) n = ( q α − γ ; q ) n ( q ; q ) n ( q α ; q ) − γ ( q α − γ + n ; q ) γ$

into (22), we obtain

$I q α (UV)(z)= z α ( 1 − q ) α ( q α ; q ) − γ ∑ n = 0 ∞ q n ( q α − γ ; q ) n ( q ; q ) n ( q α − γ + n ; q ) γ U ( z q n ) V ( z q n ) .$
(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 $z∈ D R ∖{0}$. Replace x in (4) with ξ and then let

$f(ξ)= ( q α − γ ξ / z ; q ) γ U(ξ).$

Consequently,

(24)

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

$I q α ( U V ) ( z ) = z α ( 1 − q ) α ( q α ; q ) − γ ∑ n = 0 ∞ q n ( q α − γ ; q ) n ( q ; q ) n V ( z q n ) × ∑ k = 0 n ( − 1 ) k ( 1 − q ) k [ n k ] q q k ( k − 1 ) / 2 z k D q k ( ( q α − γ ξ / z ; q ) γ U ( ξ ) ) | ξ = z .$
(25)

Using (2), we obtain

$D q k ( ( q α − γ ξ z ; q ) γ U ( ξ ) ) | ξ = k = ( − 1 ) k ( 1 − q ) − k z a − k q − k ( k − 1 ) 2 ( q α − γ ; q ) γ × ∑ r = 0 k q r ( r − 1 ) 2 [ k r ] q ( q α ; q ) k − r ( q α − γ ; q ) k − r q ( k − r ) a u ( z q k − r ) .$

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

(26)

Consequently,

(27)

Set $F(ξ):= ( q α − γ ξ z ; q ) γ U(ξ)$. Then substituting (27) into (25), we obtain

(28)

The last series converges for all $z∈ D R ∖{0}$ since $V, z a V∈ L q 1 ( D R )$. Consequently, the series in (28) is absolutely convergent, and we can interchange the order of summations in (25). This leads to

(29)

Since

$( I q α − γ + k V ) ( z q k ) = ( z q k ) α − γ + k ( 1 − q ) α − γ + k ∑ j = 0 ∞ q j ( q α − γ + k ; q ) j ( q ; q ) j V ( z q j + k )$

and

$( q α − γ ; q ) j + k = ( q α − γ ; q ) k ( q α − γ + k ; q ) j ,$

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

$I q α ( U V ) ( z ) = z γ ( 1 − q ) γ ( q α ; q ) − γ ∑ k = 0 ∞ ( − 1 ) k q − k ( k − 1 ) / 2 + k ( − α + γ ) ( q α − γ ; q ) k ( q ; q ) k × ( I q α − γ + k V ) ( z q k ) D q k ( ( q α − γ ξ z ; q ) γ U ( ξ ) ) | ξ = z = z γ ( 1 − q ) γ ( q α ; q ) − γ ∑ k = 0 ∞ [ − α + γ k ] q ( I q α − γ + k V ) ( z q k ) D q k ( ( q α − γ ξ z ; q ) γ U ( ξ ) ) | ξ = z ,$

and the theorem follows. □

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

$Re(λ)>0,Re(λ+μ)>0,μ∉ N 0 andRe(α)>0.$

Then

(30)

Proof We prove the identity by using Theorem 3.1. Take $U(z)= z μ$ and $v(z)= z λ − 1$. Then

$D q m { ( q α − γ ξ z ; q ) γ ξ μ } = D q m ∑ k = 0 ∞ ( q − γ ; q ) k ( q ; q ) k ( q α z ) k ξ μ + k = ∑ k = 0 ∞ ( q − γ ; q ) k ( q ; q ) k ( q α z ) k Γ q ( μ + k + 1 ) Γ q ( μ + k − m + 1 ) ξ μ + k − m .$

Hence,

(31)

and

$( I q α − γ + m V ) ( z q m ) = Γ q ( λ ) Γ q ( λ + α − γ + m ) ( z q m ) λ + α − γ + m − 1 = Γ q ( λ ) Γ q ( λ + α − γ ) ( 1 − q ) m ( q λ + α − γ ; q ) m ( z q m ) λ + α − γ + m − 1 .$
(32)

Then applying Theorem 3.1 gives

$I q α ( U V ) ( z ) = z μ + λ + α − 1 Γ q ( α ) Γ q ( λ ) Γ q ( α − γ ) Γ q ( λ + α − γ ) × ∑ m = 0 ∞ q m ( λ + μ ) ( q α − γ , q − μ ; q ) m ( q , q λ + α − γ ; q ) m 2 ϕ 1 ( q − γ , q μ + 1 ; q μ − m + 1 ; q , q α ) .$
(33)

On the other hand,

$I q α z λ + μ − 1 = Γ q ( λ + μ ) Γ q ( λ + μ + α ) z λ + μ + α − 1 .$
(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

$α=d+D−b−B,γ=D−B.$

Then using (3), we obtain

$D q m ( ( q d − b ξ z ; q ) D − B ξ b ( ξ ; q ) − a ) | ξ = z = ( 1 − q ) − m z − m + b ( z ; q ) − a ( q d − b ; q ) ∞ ( q d − b + D − B ; q ) ∞ × 3 ϕ 2 ( q − m , q d − b + D − B , z q − a ; q d − b , z ; q , q b + 1 ) .$
(36)

$( I q α − γ + m V ) ( z q m ) = z d − b + β − 1 + m q m 2 + ( d − b + β − 1 ) m Γ q ( β ) Γ q ( B + m + d − b ) × 2 ϕ 1 ( q A , q β ; q m − b ; q , q m − a − A )$
(37)

and

$[ γ − α m ] q = ( − 1 ) m q ( γ − α ) m q ( − m 2 ) ( q α − γ ; q ) m ( q ; q ) m .$
(38)

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

$I q α U V ( z ) = z d + D − 1 ( 1 − q ) D − B ( z ; q ) − a Γ q ( β ) Γ q ( B + d − b ) × ∑ m = 0 ∞ ( − 1 ) m q ( m 2 ) q m B ( q b − d ; q ) m ( q ; q ) m ( q d + D − b ; q ) m × 3 ϕ 2 ( q − m , q d + D − b − B , z q − a ; q d − b , z ; q , q b + 1 ) × 2 ϕ 1 ( q A , q B ; q d + B − b + m ; q , q − a − A + m ) .$
(39)

On the other hand,

$I q α UV(z)= Γ q ( b + B ) Γ q ( d + D ) z d + D − 1 2 ϕ 1 ( q a + A , q b + B ; q d + D ; q , q − ( a + A ) ) .$
(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 $α∈R$. 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 $K q − α$ exists for some α.

Definition 4.1 Let $α∈C$ and let f be a function defined on a $q − 1$-geometric set A. We say that f is of class $S q , α$ if there exists $μ∈C$, $Reμ>Reα$ such that

Proposition 4.2 If $α∈Z$, then $K q − α f$ exists for any function f defined on $(0,∞)$. If $α∉Z$ and $f∈ S q , α$, then $K q − α f$ exists.

Proof If $α∈Z$, then by (11), $K q − α f$ exists for any functions f defined on a $(0,∞)$. If $α∉Z$ and $f∈ S q , α$, then for each $x>0$, there exists a constant $C>0$, C depends on x and α, such that

$| f ( x q − n − α ) | ≤C q n μ .$

Applying the previous inequality in (10) gives

$| K q − α f ( x ) | ≤ C q − α ( α + 1 ) / 2 | x | α ( 1 − q ) α ( − q Re α ; q ) ∞ ( q ; q ) ∞ ∑ k = 0 ∞ q k ( Re μ − Re α ) ≤ C q − α ( α + 1 ) / 2 | x | α ( 1 − q ) α ( − q Re α ; q ) ∞ ( 1 − q ( Re μ − Re α ) ) ( q ; q ) ∞ .$

□

In the following, we define a sufficient class of functions $S q , μ$ for which $K q − α f$ 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 $S q , μ$ if there exist $μ>0$ and $ν∈R$ such that for each $x∈A$,

It is clear that if $f∈ S q , μ$, then $f∈ S q , α$ for all α. The spaces $S q , α$ and $S q , μ$ 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

where $P n (x)$ is a polynomial of degree n and a is a constant such that $ax q k ≠1$ for all $k∈ N 0$.

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

$K q α z − p = q α ( 1 − α ) / 2 q − α p z − α − p Γ q ( α + p ) Γ q ( p ) ,$
(41)

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

$f ( z q − n ) = ∑ k = 0 n ( q − 1 − 1 ) k q − ( k 2 ) [ n k ] q − 1 z k D q − 1 k f ( z ) = ∑ k = 0 n q k ( k − 1 ) / 2 ( 1 − q ) k q − n k [ n k ] q z k D q − 1 k f ( z ) ,$
(42)

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

$[ n k ] q − 1 = [ n k ] q q k 2 − n k .$

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

Lemma 4.5 Let p and α be such that $0 and $α>Rep$. Let G be the principal branch of the logarithmic function and let $D R :={z∈C:|z|>R}∩G$. Assume that

$U(z)= z p ∑ j = 0 ∞ a j z − j$

is analytic on $D R$. Let

$Ω α = { z ∈ D R : q α z ∈ D R } .$

Then $K q α U(z)$ exists for all $z∈ Ω α$ and is equal to

$K q α U(z)= q α p Γ q ( α − p ) Γ q ( − p ) z p − α ∑ j = 0 ∞ a j ( q − p ; q ) j ( q α − p ; q ) j q − α j z − j .$
(43)

Proof From (10) we find that

$K q α U(z)= q α ( 1 − α ) / 2 q α p z p − α ( 1 − q ) − α ∑ k = 0 ∞ q k ( α − p ) ( q − α ; q ) k ( q ; q ) k ∑ j = 0 ∞ a j q ( − α + k ) j .$
(44)

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

$K q α U ( z ) = q α ( 1 − α ) / 2 q α p z p − α ( 1 − q ) − α ∑ j = 0 ∞ a j q − α j z − j ∑ k = 0 ∞ q k α − k p + k j ( q − α ; q ) j ( q ; q ) j = q α ( 1 − α ) / 2 q α p z p − α ( 1 − q ) − α ∑ j = 0 ∞ a j q − α j z − j ( q − p + j ; q ) ∞ ( q α − p + j ; q ) ∞ .$

Simple manipulations give (43). □

Lemma 4.6 Let p, α, G, U, $D R$, and $Ω α$ be as in Lemma  4.5. Then

$∑ j = 0 ∞ q α j | U ( z q α − j ) | <∞∀z∈ Ω α .$

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 $α∈R$. Assume that $UV∈ S q , α$ and $U∈ S q , μ$, $μ> 1 2$. Then

$K q α UV(z)= ∑ m = 0 ∞ ( q − α ; q ) m ( q ; q ) m K q α − m U(z) D q − 1 , z m { V ( z q α ) }$
(45)

for all $z∈A$ and for all $α∈R$. If $μ=1/2$, then (45) may not hold for all α on but only for α in a subdomain of .

Proof Let $z∈ q − α A$ be arbitrary but fixed. Since $UV∈ S q , α$, then

$∑ k = 0 ∞ q k α |U ( z q α − k ) V ( z q α − k ) |<∞.$
(46)

From (10),

Applying (42) with $f(z)=V(z q α )$ yields

$( K q α U V ) ( z ) = q α ( 1 − α ) / 2 ( 1 − q ) − α z − α ∑ m = 0 ∞ q α m ( q − α ; q ) m ( q ; q ) m U ( z q α − m ) × ∑ j = 0 m q j ( 1 − j ) / 2 ( 1 − q ) j q − m j [ m j ] q z j D q − 1 , z j { V ( z q α ) } .$
(47)

From the assumptions on the function U, there exists a constant $C 1 >0$ and $ν∈R$ such that

$| U ( z q α − m ) | ≤ C 1 q μ m ( m + ν ) .$

Using (2) with ($q − 1$ instead of q), we obtain

$z j D q − 1 , z j { V ( z q α ) } ≤ C 2 .$

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

(48)

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

$( a ; q ) n := ∑ k = 0 n ( − 1 ) k [ n k ] q q k ( k − 1 ) 2 a k .$
(49)

Now, it is clear that if $μ>1/2$, then the series on the most right-hand side of (48) is convergent for all $α∈C$. On the other hand, it is convergent only for $Reα>−ν+ 1 2$ when $μ= 1 2$. Therefore, we can interchange the order of summation in the series on the right-hand side of (47). This gives

$( K q α U V ) ( z ) = q α ( 1 − α ) / 2 ( 1 − q ) − α z − α × ∑ j = 0 ∞ ( q − α ; q ) j ( q ; q ) j z j ( 1 − q ) j D q − 1 , z j V ( z q α ) × ∑ r = 0 ∞ q ( α − j ) r ( q − α + j ; q ) r ( q ; q ) r U ( z q α − j − r ) .$
(50)

But

$∑ r = 0 ∞ q ( α − j ) r ( q − α + j ; q ) r ( q ; q ) r U ( z q α − j − r ) = z α − j ( 1 − q ) α − j q ( α − j ) ( α − j − 1 ) / 2 K q α − j V(z).$

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

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

$Re(λ)>0,Re(λ+μ)>0,μ∉ N 0 andRe(α)>0.$

Then

(51)

Proof We prove the identity by using Theorem 3.1. Take $U(z)= z μ$ and $v(z)= z λ − 1$. Then

$D q m ( q α − γ ξ z ; q ) γ ξ μ = D q m ∑ k = 0 ∞ ( q − γ ; q ) k ( q ; q ) k ( q α z ) k ξ μ + k = ∑ k = 0 ∞ ( q − γ ; q ) k ( q ; q ) k ( q α z ) k Γ q ( μ + k + 1 ) Γ q ( μ + k − m + 1 ) ξ μ + k − m .$

Hence,

(52)

and

$( I q α − γ + m V ) ( z q m ) = Γ q ( λ ) Γ q ( λ + α − γ + m ) ( z q m ) λ + α − γ + m − 1 = Γ q ( λ ) Γ q ( λ + α − γ ) ( 1 − q ) m ( q λ + α − γ ; q ) m ( z q m ) λ + α − γ + m − 1 .$
(53)

Then applying Theorem 3.1 gives

$I q α ( U V ) ( z ) = z μ + λ + α − 1 Γ q ( α ) Γ q ( λ ) Γ q ( α − γ ) Γ q ( λ + α − γ ) × ∑ m = 0 ∞ q m ( λ + μ ) ( q α − γ , q − μ ; q ) m ( q , q λ + α − γ ; q ) m 2 ϕ 1 ( q − γ , q μ + 1 ; q μ − m + 1 ; q , q α ) .$
(54)

On the other hand,

$I q α z λ + μ − 1 = Γ q ( λ + μ ) Γ q ( λ + μ + α ) z λ + μ + α − 1 .$
(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

$α=d+D−b−B,γ=D−B.$

Then using (3), we obtain

$D q m ( ( q d − b ξ z ; q ) D − B ξ b ( ξ ; q ) − a ) | ξ = z = ( 1 − q ) − m z − m + b ( z ; q ) − a ( q d − b ; q ) ∞ ( q d − b + D − B ; q ) ∞ × 3 ϕ 2 ( q − m , q d − b + D − B , z q − a ; q d − b , z ; q , q b + 1 ) .$
(57)

$( I q α − γ + m V ) ( z q m ) = z d − b + β − 1 + m q m 2 + ( d − b + β − 1 ) m Γ q ( β ) Γ q ( B + m + d − b ) × 2 ϕ 1 ( q A , q β ; q m − b ; q , q m − a − A )$
(58)

and

$[ γ − α m ] q = ( − 1 ) m q ( γ − α ) m q ( − m 2 ) ( q α − γ ; q ) m ( q ; q ) m .$
(59)

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

$I q α U V ( z ) = z d + D − 1 ( 1 − q ) D − B ( z ; q ) − a Γ q ( β ) Γ q ( B + d − b ) × ∑ m = 0 ∞ ( − 1 ) m q ( m 2 ) q m B ( q b − d ; q ) m ( q ; q ) m ( q d + D − b ; q ) m × 3 ϕ 2 ( q − m , q d + D − b − B , z q − a ; q d − b , z ; q , q b + 1 ) × 2 ϕ 1 ( q A , q B ; q d + B − b + m ; q , q − a − A + m ) .$
(60)

On the other hand,

$I q α UV(z)= Γ q ( b + B ) Γ q ( d + D ) z d + D − 1 2 ϕ 1 ( q a + A , q b + B ; q d + D ; q , q − ( a + A ) ) .$
(61)

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

## References

1. 1.

Gasper G, Rahman M: Basic Hypergeometric Series. 2nd edition. Cambridge University Press, Cambridge; 2004.

2. 2.

Ismail MEH: Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge University Press, Cambridge; 2005.

3. 3.

Jackson FH: On q -definite integrals. Q. J. Pure Appl. Math. 1910, 41: 193-203.

4. 4.

Hahn W: Beiträge zur Theorie der Heineschen Reihen. Math. Nachr. 1949, 2: 340-379. (German) 10.1002/mana.19490020604

5. 5.

Al-Salam WA: Some fractional q -integrals and q -derivatives. Proc. Edinb. Math. Soc. 1966/1967, 2(15):135-140.

6. 6.

Al-Salam WA: q -analogues of Cauchy’s formulas. Proc. Am. Math. Soc. 1966, 17: 616-621.

7. 7.

Agarwal RP: Fractional q -derivative and q -integrals and certain hypergeometric transformations. Ganita 1976, 27(1-2):25-32.

8. 8.

Purohit SD: On a q -extension of the Leibniz rule via Weyl type of q -derivative operator. Kyungpook Math. J. 2010, 50(4):473-482. 10.5666/KMJ.2010.50.4.473

9. 9.

Agarwal RP: Certain fractional q -integrals and q -derivatives. Proc. Camb. Philos. Soc. 1969, 66: 365-370. 10.1017/S0305004100045060

10. 10.

Annaby MH, Mansour ZS Lecture Notes of Mathematics 2056. In q-Fractional Calculus and Equations. Springer, Berlin; 2012.

11. 11.

Abdeljawad T, Baleanu D: Caputo q -fractional initial value problems and a q -analogue of Mittag-Leffler function. Commun. Nonlinear Sci. Numer. Simul. 2011, 16(12):4682-4688. 10.1016/j.cnsns.2011.01.026

12. 12.

Liouville J: Mèmoire sur le calcul des différentielles à indices quelconques. J. Éc. Polytech. 1832, 13: 71-162.

13. 13.

Grünwald AK: Über begrenzte derivationen und deren anwedung. Z. Angew. Math. Phys. 1867, 12: 441-480.

14. 14.

Letinkov AV: Theory of differentiation of fractional order. Mat. Sb. 1868, 3: 1-68.

15. 15.

Watanabe Y: Notes on the generalized derivative of Riemann-Liouville and its application to Leibniz’s formula. I and II. Tohoku Math. J. 1931, 34: 8-41.

16. 16.

Post EL: Generalized differentiation. Trans. Am. Math. Soc. 1930, 32: 723-781. 10.1090/S0002-9947-1930-1501560-X

17. 17.

Bassam MA: Some properties of Holmgren-Riez transform. Ann. Sc. Norm. Super. Pisa 1961, 15(3):1-24.

18. 18.

Gaer MC, Rubel LA: The fractional derivative and entire functions. Lecture Notes in Math. 457. In Fractional Calculus and Its Applications(Proc. Internat. Conf., Univ. New Haven, West Haven, Conn., 1974). Lecture Notes in Math. Springer, Berlin; 1975:171-206.

19. 19.

Osler TJ: Leibniz rule for fractional derivatives generalized and an application to infinite series. SIAM J. Appl. Math. 1970, 18(3):658-674. 10.1137/0118059

20. 20.

Osler TJ: Fractional derivatives and Leibniz rule. Am. Math. Mon. 1971, 78(6):645-649. 10.2307/2316573

21. 21.

Diaz JB, Osler TJ: Differences of fractional order. Math. Comput. 1974, 28(125):185-202.

22. 22.

Osler TJ: The integral analogue of the Leibniz rule. Math. Comput. 1972, 26(120):903-915.

23. 23.

Osler TJ: A correction to Leibniz rule for fractional derivatives. SIAM J. Math. Anal. 1973, 4: 456-459. 10.1137/0504040

24. 24.

Herrmann R: Fractional Calculus: An Introduction for Physicists. World Scientific, Singapore; 2011.

25. 25.

Baleanu D, Diethelm K, Scalas E, Trujillo JJ Series on Complexity, Nonlinearity, and Chaos 3. In Fractional Calculus Models and Numerical Methods. World Scientific, Singapore; 2012.

26. 26.

Baleanu D, António J, Machado T, Luo ACJ: Fractional Dynamics and Control. Springer, Berlin; 2012.

27. 27.

Golmankhaneh AK: Investigations in Dynamics: With Focus on Fractional Dynamics. LAP Lambert Academic Publishing, Saarbrücken; 2012.

28. 28.

Hilfer R (Ed): Applications of Fractional Calculus in Physics. World Scientific, Singapore; 2000.

29. 29.

Al-Salam WA, Verma A: A fractional Leibniz q -formula. Pac. J. Math. 1975, 60(2):1-9. 10.2140/pjm.1975.60.1

30. 30.

Ramis JP: About the growth of entire functions solutions of linear algebraic q -difference equations. Ann. Fac. Sci. Toulouse 1992, 1(6):53-94.

31. 31.

Purohit SD: Summation formulae for basic hypergeometric functions via fractional q -calculus. Matematiche 2009, 64(1):67-75.

32. 32.

Manocha HL, Sharma BL: Fractional derivatives and summation. J. Indian Math. Soc. 1974, 38(1-4):371-382.

33. 33.

Lighthill MJ: Introduction to Fourier Analysis and Generalized Functions. Cambridge University Press, New York; 1960.

34. 34.

Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York; 1993.

## Acknowledgements

This research is supported by NPST Program of King Saud University; project number 10-MAT1293-02.

## Author information

Authors

### Corresponding author

Correspondence to Zeinab SI Mansour.

### Competing interests

The author declares that they have no competing interests.

## Rights and permissions

Reprints and Permissions

Mansour, Z.S. Generalizations of fractional q-Leibniz formulae and applications. Adv Differ Equ 2013, 29 (2013). https://doi.org/10.1186/1687-1847-2013-29