Open Access

Generalizations of fractional q-Leibniz formulae and applications

Advances in Difference Equations20132013:29

https://doi.org/10.1186/1687-1847-2013-29

Received: 5 September 2012

Accepted: 24 January 2013

Published: 11 February 2013

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 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 q x 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
D q n f ( x ) = ( 1 q ) n x n r = 0 n q r ( q n ; q ) r ( q ; q ) r f ( x q r ) for  x A { 0 } .
(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 ) d t
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 ( U V ) ( 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 < ln q 1 , 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 ( U V ) ( 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 < R < and D R : = { z C : | z | < R } . In the following, we say that a function f L q 1 ( D R ) if
j = 0 q j | f ( z q j ) | < for all  z D R { 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 α ( U V ) ( 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 ( U V ) ( 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 U V ( ) 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 ) > 1 and Re ( a + b ) > 1 .
From (13) we obtain
I q α ( U V ) ( 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 ( U V ) ( 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 and Re ( 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)
In addition,
( 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 U V ( 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 , Re μ > Re α such that
f ( x q n ) = O ( q n μ ) as  n , x A .

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 ,
| f ( x q n ) | = O ( q μ n ( n + ν ) ) as  n .

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
P n ( x ) ( a x ; q ) for all  n N 0 ,

where P n ( x ) is a polynomial of degree n and a is a constant such that a x 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 < Re p < 1 and α > Re p . 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 U V S q , α and U S q , μ , μ > 1 2 . Then
K q α U V ( 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 U V 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 Re α > 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 and Re ( 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)
In addition,
( 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 U V ( 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). □

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Saud University

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