CERTAIN INVERSION AND REPRESENTATION FORMULAS FOR q

CERTAIN INVERSION AND REPRESENTATION FORMULAS FOR q-SUMUDU TRANSFORMS Durmu¸s Albayrak

1

, Sunil Dutt Purohit

2

, Faruk U¸car

∗, 3

Abstract In the present paper, we explore the formal properties of the q-Sumudu transforms to derive a number of inversion and representation formulas. Some interesting applications of the main results are also presented in the concluding section. Key Words q-Laplace transforms, q-Sumudu transforms, q-Bessel functions. 2010 Mathematics Subject Classification. 33D05, 33D15, 05A30. 1. Introduction and Preliminaries In the classical analysis, integral transforms are the most widely used to solve differential equations and integral equations. Thus a lot of work has been done on the theory and applications of integral transforms (for instance, see [15] and [16]). Most popular integral transforms are due to Laplace, Fourier, Mellin and Hankel. In 1993 the Sumudu transform was proposed originally by Watugala [17] as follow: Z 1 ∞ −t/s e f (t) dt, s ∈ (−τ1 , τ2 ) , S{f (t); s} = s 0 over the set of functions o n A = f (t)|∃M, τ1 , τ2 > 0, |f (t)| < M e|t|/τj , t ∈ (−1)j × [0, ∞) and he applied it to the solution of ordinary differential equations in control engineering problems. Subsequently, Weerakoon [18] gave the Sumudu transform of partial derivatives and the complex inversion transform and applied it to the solution of partial differential equations. The Sumudu transform is not a new integral transform, but simply s-multiplied Laplace transform, providing the relation between them (see [12] and [5]). The Sumudu transform which is itself linear, preserves linear functions, and hence in particular does not change units (Belgacem and Karaballı [6]). One may, for instance, refer to such type of works in the recent papers [9] and [10]. The theory of q-analysis, in recent past, have been applied in the many areas of mathematics and physics like ordinary fractional calculus, optimal control problems, qtransform analysis, geometric function theory and in finding solutions of the q-difference and 1Department of Mathematics, University of Marmara, TR-34722 Kadik¨ oy, Istanbul, Turkey.

E-mail: [email protected] 2Department of Basic Science (Mathematics), College of Technology and Engineering, M. P. University of Agriculture and Technology Udaipur-313001 (Rajasthan), India. E-mail: sunil a [email protected] 3Department of Mathematics, University of Marmara, TR-34722 Kadik¨ oy, Istanbul, Turkey. E-mail: [email protected], ∗ Corresponding Author

2

Durmu¸s Albayrak, Sunil Dutt Purohit and Faruk U¸car

q-integral equations (for instance, see [2], [11], [13] and [14]). Albayrak, Purohit and U¸car [3] introduced the q-analogues of the Sumudu transform and established several theorems related to q-Sumudu transforms of some functions. They also introduced the convolution theorem for q-Sumudu transform. Abdi [1] gave certain representation and inversion formulae for the two basic analogues of the Laplace transform. Here, it is worth to note that, the results given in this paper, can be derived from the corresponding results from q-analogue of Laplace transform which is given by Theorem 2 and Theorem 4 in [3]: Theorem 2: Let F1 (s) = Lq {f (t) ; s} and G1 (s) = Sq {f (t) ; s}. Then we have   1 1 G1 (s) = F1 . s s Theorem 4: Let F2 (s) = Lq {f (t) ; s} and G2 (s) = Sq {f (t) ; s}. Then we have   1 1 G2 (s) = F2 . s s The aim of this paper is to give certain inversion and representation formulas for qSumudu transforms. Throughout this paper we will assume that functions are analytic at infinity. This paper is organized in following manner. In Section 2, we will give some theorems, which exhibit the inversion formulae for q-Sumudu transforms. In section 3, we give some examples for theorems. As for prerequisites, the reader is expected to be familiar with notations of q-calculus. We start with basic definitions and facts from the q-calculus necessary for understanding of this study. Throughout this paper, we will assume that q satisfies the condition 0 < |q| < 1. The q-derivative Dq f of an arbitrary function f is given by (Dq f )(x) =

f (x) − f (qx) , (1 − q)x

where x 6= 0. Clearly, if f is differentiable, then lim (Dq f )(x) =

q→1−

df (x) . dx

For any real number α, [α] :=

qα − 1 . q−1

In particular, if n ∈ Z+ , we denote [n] =

qn − 1 = q n−1 + · · · + q + 1, q−1

and the q-binomial coefficients is defined by   n [n]! = , k [k]! [n − k]!

Certain Inversion and Representation formulas for q-Sumudu Transforms

3

where [n]! = [n] [n − 1] · · · [2] [1]. Following usual notations are very useful in the theory of q-calculus: n−1  Y (a; q)n = 1 − aq k , (a; q)∞ =

k=0 ∞  Y

 1 − aq k ,

k=0

(a; q)t =

(a; q)∞ , ( t ∈ R), (aq t ; q)∞

(1.1)

(−q/a)n q n(n−1)/2 . (q/a; q)n The q-analogues of the classical exponential functions are defined by ∞ X tn 1 eq (t) = = , |t| < 1, (q; q)n (t; q)∞ (a; q)−n =

Eq (t) =

n=0 ∞ X

n=0

(−1)n q n(n−1)/2 tn = (t; q)∞ , (q; q)n

(t ∈ C) .

(1.2)

(1.3) (1.4)

q-exponential functions have the following properties lim eq ((1 − q) x) = ex ,

q→1−

lim Eq ((q − 1) x) = ex .

q→1−

The q-integrals are defined as (see [7]) Z x ∞ X f (t)dq t = x(1 − q) q k f (xq k ), 0

Z

∞/A

f (t)dq t = (1 − q) 0

(1.5)

k=0

X qk qk f ( ). A A

(1.6)

k∈Z

The following definition is due to Albayrak, Purohit and U¸car [3]: Definition 1.1. The q-analogues of Sumudu transform are defined as follow: Z s q  1 Eq t f (t) dq t, s ∈ (−τ1 , τ2 ) , Sq {f (t); s} = (1 − q) s 0 s over the set of functions  A = f (t)|∃M, τ1 , τ2 > 0, |f (t)| < M Eq (|t| /τj ) , t ∈ (−1)j × [0, ∞) .

(1.7)

and

 Z ∞  1 1 Sq {f (t); s} = eq − t f (t) dq t, s ∈ (−τ1 , τ2 ) , (1 − q) s 0 s over the set of functions  B = f (t)|∃M, τ1 , τ2 > 0, |f (t)| < M eq (|t| /τj ) , t ∈ (−1)j × [0, ∞) .

(1.8)

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Durmu¸s Albayrak, Sunil Dutt Purohit and Faruk U¸car

By virtue of (1.5) and (1.6), q-Sumudu transforms can be expressed as
∞ k X q f sq k Sq {f (t) ; s} = (q; q)∞ , (q; q)k

(1.9)

k=0

and

  X s−1 1 k k
Sq {f (t) ; s} = q f (q ) − ; q . s − 1s ; q ∞ k∈Z k

(1.10)

Furthermore, the q-hypergeometric functions are defined by [11]:    X
 ∞ n s+1−r zn (a1 , a2 , . . . , ar ; q)n a1 a2 · · · ar n 2 (−1) q ; q, z = , Φ r s b1 b2 · · · bs (b1 , b2 , . . . , bs ; q)n (q; q)n n=0

where (a1 , a2 , . . . , ar ; q)n =

r Y

(ai ; q)n .

i=1

The q-Taylor’s series is defined by (see also [13]) ∞ X an (x − a)nq , f (x) = [n]!

(1.11)

n=0

where an = lim Dq(n) f (x) . x→a

The q-Bessel functions were introduced by Jackson [8] and are referred to as Jackson’s q-Bessel functions. Some q-analogues of the Bessel functions are given by
  q ν+1 ; q ∞  z ν z2 0 0 (1) Jν (z; q) = , |z| < 2, ; q, − 2 Φ1 q ν+1 (q; q)∞ 2 4
n ∞  z ν X −z 2 /4 = , (1.12) 2 (q; q)ν+n (q; q)n n=0

and Jν(2) (z; q)


  q ν+1 ; q ∞  z ν q ν+1 z 2 − ; q, − , = 0 Φ1 q ν+1 (q; q)∞ 2 4
n ∞  z ν X q n(n+ν) −z 2 /4 = . 2 (q; q)ν+n (q; q)n

|z| < 2,

n=0

The relation between these two q-Bessel functions is  2  z (2) Jν (z; q) = − ; q Jν(1) (z; q) , 4 ∞

|z| < 2.

q-Bessel functions are q-extensions of the Bessel function of the first kind since lim Jν(k) ((1 − q) z; q) = Jν (z) ,

q→1−

k = 1, 2.

(1.13)

Certain Inversion and Representation formulas for q-Sumudu Transforms

The third kind q-analogue of the Bessel function is given by following formula
  ν+1 ; q q 0 (3) 2 ∞ ν Jν (z; q) = z 1 Φ1 ; q, qz , q ν+1 (q; q)∞
n ∞ X (−1)n q n(n−1)/2 qz 2 ν =z . (q; q)ν+n (q; q)n

5

(1.14)

n=0

This third kind q-Bessel function is also known as the Hahn-Exton q-Bessel function. This is also q-extension of the Bessel function of the first kind since lim Jν(3) ((1 − q) z; q) = Jν (2z) .

q→1−

2. Main Theorems In the sequel, we need the following results: Recently, Albayrak, Purohit and U¸car [4] proved the following theorem. Theorem: Corresponding to the bounded sequence An , let f (x) be given by f (x) =

∞ X

An x n ,

n=0

then for α > 0, the following results hold: ∞ X  Sq xα−1 f (x) ; s = sα−1 An (q; q)α+n−1 sn ,

(2.1)

n=0 ∞ X  α−1 (q; q)α+n−1 n α−1 s . Sq x f (x) ; s = s An K (s; α + n)

(2.2)

n=0

If we write α = j + 1 and f (x) = 1 in the above theorem, we get  Sq xj ; s = sj (q; q)j ,  Sq xj ; s = q −j(j+1) sj (q; q)j .

(2.3) (2.4)

 q j/2   √ (1) (3) p −1 If we write α = j/2+1 and choose f (x) = a−j/2 Jj (2 ax; q) or f (x) = Jj q ax; q a in (2.1), we have   
x j/2 (1) √ Sq Jj 2 ax; q ; s = sj eq (−as) , (2.5) a     qx j/2 (3) p −1 Sq Jj q ax; q ; s = sj Eq (as) , (2.6) a

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Durmu¸s Albayrak, Sunil Dutt Purohit and Faruk U¸car

√ (2) and similarly, if we write α = j/2 + 1 and choose f (x) = q j(j+1)/2 a−j/2 Jj (2 ax; q) or  q j/2 p  (3) f (x) = Jj q j ax; q in (2.2), we have a    j/2
√ (2) j(j+1)/2 x Jj 2 ax; q ; s = sj Eq (as) , (2.7) Sq q a     qx j/2 (3) p j Jj q ax; q ; s = sj eq (−as) , (2.8) Sq a where Re (s) > 0 and Re (υ + 1) > 0. The following theorem is due to Belgacem, Karaballı and Kalla [5]. Theorem: The inverse discrete Sumudu transform, f (t), of the power series, G (u) =

∞ X

bn un ,

n=0

is given by S

−1

G(u) = f (t) =

 ∞  X 1 n=0

n!

bn tn .

In this section, we will give some theorems, which exhibit the inversion formulae for qSumudu transforms. The usefulness of the theorems are also exhibited by considering some examples.
Theorem 2.1. Let Sq {f (x) ; s} = G1 (s) and Sq {g (x) ; s} = G2 (s). If 1s G1 1s and
1 1 s G2 s are analytic outside a given circle |s| = r, r > 0 and their values are zero at ∞, then we have ∞ X xj f (x) = aj (1 − q)j n (2.9) o2 , j=0 (q; q)j where lim Dq(j) G1 (s) = aj ,

s→0

and g (x) =

∞ X

xj bj q j(j+1)/2 (1 − q)j n o2 , j=0 (q; q)j

(2.10)

where lim Dq(j) G2 (s) = bj .

s→0

Proof. We only give the proof of the
identity (2.9). Because, the proof of the identity (2.10) is similar. We suppose that 1s G1 1s is analytic in |s| > r, for some r > 0 and vanishes at ∞. Then, it can be represented as an absolutely convergent power series as follows:   X ∞ 1 1 G1 = cj s−j−1 . s s j=0

Certain Inversion and Representation formulas for q-Sumudu Transforms

7

Therefore, we can write G1 (s) =

∞ X

cj sj .

j=0

Making use of the q-Taylor formula (1.11), we have cj =

(1 − q)j lim D(j) G1 (s) . (q; q)j s→0 q

Thus, we have G1 (s) =

∞ X

aj (1 − q)j

j=0

sj , (q; q)j

where aj = lim Dq(j) G1 (s) . s→0

Taking into account (2.3), we obtain f (x) =

∞ X

xj aj (1 − q)j n o2 . j=0 (q; q)j


1 1 Theorem 2.2. Let S {f (x) ; s} = G (s) and S {f (x) ; s} = G (s). If G and q 1 q 2 1 s s
1 1 s G2 s are analytic outside a given circle |s| = r, r > 0 and their values are zero at ∞, then  r ∞ X a (1 − q)j  qx j/2 (3) x; q , (2.11) f (x) = aj Jj (q; q)j a q j=0

where aj = lim Dq(j) {eq (as) G1 (s)} . s→0

f (x) =

∞ X

bj

j=0

 (1 − q)j  qx j/2 (3) p j q ax; q , Jj (q; q)j a

(2.12)

where bj = lim Dq(j) {Eq (−as) G2 (s)} . s→0

f (x) =

∞ X j=0

bj

 (1 − q)j  x j/2 (1)  √ Jj 2 at; q , (q; q)j a

(2.13)

where bj = lim Dq(j) {Eq (−as) G2 (s)} . s→0

f (x) =

∞ X j=0


(1 − q)j j(j+1)/2  x j/2 (2) √ aj q Jj 2 ax; q , (q; q)j a

where aj = lim Dq(j) {eq (as) G1 (s)} . s→0

(2.14)

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Durmu¸s Albayrak, Sunil Dutt Purohit and Faruk U¸car

Proof. We only give the proof of the identity (2.11). Because, by using the formula (2.8), (2.5), (2.7), respectively, the proof of the identities (2.12)-(2.14) are similar. Let     1 1 1 1 = eq (a/s) G1 . F s s s s
1 1 Suppose is analytic outside the circle |s| = r and its value is zero at ∞. Therefore G 1 s s
1 1 is analytic outside the circle |s| = p and its value is zero at ∞. Thus, Theorem 2.1 F s s can be applied for F (s) and we can write F (s) =

∞ X

aj (1 − q)j

j=0

sj , (q; q)j

where aj = lim Dq(j) {F (s)} . s→0

Following the definition of F (s), we have eq (as) G1 (s) =

G1 (s) =

∞ X j=0 ∞ X

aj (1 − q)j

sj , (q; q)j

aj (1 − q)j

sj Eq (as) , (q; q)j

j=0

where aj = lim Dq(j) {eq (as) G1 (s)} . s→0

Now, on using the formula (2.6), we get f (x) =

∞ X

(3)

aj (1 − q)j

j=0

 qx j/2 Jj a

q

a q x; q

(q; q)j

 ,

where aj = lim Dq(j) {eq (as) G1 (s)} . s→0


Theorem 2.3. Let Sq {f (x) ; s} = G1 (s), Sq {f (x) ; s} = G2 (s). We suppose that 1s G1 1s
and 1s G2 1s are analytic outside a given circle |s| = r, r > 0 and their values are zero at ∞. If G1 (s) and G2 (s) have a series expansion as follow Gi (s) =

∞ X

cj (as; q)j , (i = 1, 2),

j=0

where

∞ P

|cj | convergent, then f has a series expansion as follow

j=0

f (x) =

∞ X j=0

aj q

−j(j−1)/2

j

(q − 1) a

−j 2 Φ1


q −j , 0; q; q, q j ax , (q; q)j

(2.15)

Certain Inversion and Representation formulas for q-Sumudu Transforms

9

where aj =

f (x) =

∞ X

bj q

lim

s→a−1 q −j

−j(j−1)/2 −j

a

Dq(j) G1 (s) , j 1 Φ1

(q − 1)

j=0


q −j ; q; q, −q j+1 ax , (q; q)j

(2.16)

where bj =

lim

s→a−1 q −j

Dq(j) G2 (s) .

Proof. We only give the proof of the identity (2.15). The proof of the identity (2.16), can be easily seen by making use of following known identity [4, p.419, (2.11)], 
Sq 1 Φ1 q −j ; q; q, −q j+1 ax ; s = (as; q)j .
Suppose 1s G1 1s is analytic in |s| > r, for some r > 0 and its value is zero at ∞. Then, G1 (s) has a series expansion as follow G1 (s) =

∞ X

cj (as; q)j ,

j=0

where

∞ X

|cj | is convergent. Using the q-Taylor’s formula, we get

j=0

cj =

lim

s→a−1 q −j

Dq(j) {G1 (s)} .q −j(j−1)/2 (q − 1)j a−j / (q; q)j .

Thus, we can write G1 (s) =

∞ X

aj q −j(j−1)/2 (q − 1)j a−j

j=0

(as; q)j (q; q)j

,

where aj =

lim

s→a−1 q −j

Dq(j) G1 (s) .

Furthermore, making use of formula [4, p.419, (2.10)], 
Sq 2 Φ1 q −j , 0; q; q, q j ax ; s = (as; q)j , we obtain f (x) =

∞ X

j

aj q −j(j−1)/2 (q − 1) a

j=0

−j 2 Φ1


q −j , 0; q; q, q j ax , (q; q)j

where aj =

lim

s→a−1 q −j

Dq(j) G1 (s) .

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Durmu¸s Albayrak, Sunil Dutt Purohit and Faruk U¸car


Theorem 2.4. Let Sq {f (x) ; s} = G(s). We suppose that 1s G 1s is analytic outside the circle |s| = r, r > 0 and its value is zero at ∞. Then, G (s) has the form ∞   X −1 G (s) = s , cj q (as)−1 ; q −j

j=1

where

∞ X

|cj | convergent, if and only if, f has a series expansion as follow

j=0

f (x) = eq (ax)

∞ X

cj (−1)j aj q j(j−1)/2

j=1

where

xj−1 , (q; q)j−1

∞ X j(j−1)/2 cj < ∞. q j=1

Proof. Suppose f has a series expansion as follow ∞ X xj−1 . f (x) = eq (ax) cj (−1)j aj q j(j−1)/2 (q; q)j−1 j=1

Using the series representation (1.9) of Sq -transform, we have G (s) = (q; q)∞

∞ X k=0



∞ k j−1 e asq k qk X q j j j(j−1)/2 sq . cj (−1) a q (q; q)k (q; q)j−1 j=1

Clearly, G (s) has simple poles at s = a−1 q −k , (k = 0, 1, 2, ...) which obviously all on lie or in the exterior of the circle |s| = 1/a. So, for |s| < 1/a, G (s) is absolutely convergent. Hence interchanging the order of summations in the right side of G (s), we obtain
k  ∞ ∞  j−1 X X qj j j j(j−1)/2 s G (s) = (q; q)∞ cj (−1) a q eq asq k . (q; q)j−1 (q; q)k j=1

k=0

Making use of the definitions (1.3) of eq and (1.1) of (a; q)t , we have
k ∞ ∞ j−1 X X qj 1 j j j(j−1)/2 s G (s) = (q; q)∞ cj (−1) a q (q; q)j−1 (q; q)k (asq k ; q)∞ j=1

= (q; q)∞

∞ X

cj (−1)j aj

j=1

Now on using the formula

q j(j−1)/2

(q; q)j−1 (as; q)∞

∞ X (a; q)

k k

k=0

we get G (s) =

(q; q)k

∞ X j=1

k=0 ∞ X

sj−1

z =

k=0

(as; q)k j
k q . (q; q)k

(az; q)∞ , (z; q)∞

cj (−1)j aj q j(j−1)/2

sj−1 . (as; q)j

Certain Inversion and Representation formulas for q-Sumudu Transforms

Then using (1.2), we obtain G (s) =

=

∞ X

cj (−1)j aj q j(j−1)/2

j=1 ∞ X

1 s

sj−1 (−as)j q j(j−1)/2

(q/as; q)−j

cj (q/as; q)−j .

j=1

On the other hand, let G (s) has a series expansion as follow ∞   X G (s) = s−1 , cj q (as)−1 ; q −j

j=1

where

∞ X j(j−1)/2 cj < ∞. q j=1

By the property (1.2) of (a; q)−j , we can write G (s) =

∞ X

cj (−1)j aj q j(j−1)/2

j=1

sj−1 . (as; q)j

For some s = s0 , |s0 | < a1 , ∞ j−1 X j j j(j−1)/2 s0 |G (s0 )| = . cj (−1) a q (as0 ; q)j j=1

Since

we have

s−1 −1 0 |as0 | < 1 and < s0 eq (as0 ) , (as0 ; q)j ∞ X −1 j(j−1)/2 cj . |G (s0 )| = s0 eq (as0 ) q j=1

Hence, for the absolute convergence of G(s0 ), we need ∞ X j(j−1)/2 cj < ∞. q j=1

Now on using the formula n o (q/as; q) −j Sq (−a)j q j(j−1)/2 xj−1 eq (ax) / (q; q)j−1 ; s = , s we obtain ∞ X xj−1 f (x) = eq (ax) cj (−1)j aj q j(j−1)/2 . (q; q)j−1 j=1

This completes the proof.

11

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Durmu¸s Albayrak, Sunil Dutt Purohit and Faruk U¸car


Theorem 2.5. Let Sq {f (x) ; s} = G(s) and the function 1s G 1s is analytic outside a given circle |s| = r, r > 0 and its value is zero at ∞. Then, G (s) has the form ∞   X G (s) = s−1 , cj −q (as)−1 ; q −j

j=1

where

∞ P q j(j−1)/2 cj convergent, if and only if, f has a series expansion as follow j=0

f (x) =

∞ X

cj aj q j(j−1)

j=1


xj−1 Eq aq j x . (q; q)j−1

Proof. Suppose f has a series expansion as follow ∞ X
xj−1 f (x) = cj aj q j(j−1) Eq q j ax . (q; q)j−1 j=1

Making use of series representation (1.10) of Sq -transform, we have
j−1 ∞   X  1  X qk s−1 k j j(j−1) k+j
q − . ; q c a q E aq G (s) = j q s (q; q)j−1 − 1s ; q ∞ k j=1

k∈Z

Clearly, G (s) has simple poles at s = a−1 q −k , (k = 0, 1, 2, ...) which obviously all on lie or in the exterior of the circle |s| = 1/a. So, for |s| < 1/a, G (s) is absolutely convergent. Hence, interchanging the order of summations in the right hand side of G (s), we obtain ∞   j(j−1) X X 1 
s−1 j q j k k+j
G (s) = cj a ; q . q E aq − q (q; q)j−1 − 1s ; q ∞ s k j=1

k∈Z

Making use of definitions (1.4) of Eq and (1.1) of (a; q)t , we have
−1 ∞ j(j−1) aq j ; q X s X (−1/s; q)k j
k j q ∞
G (s) = cj a q 1 (q; q)j−1 − s ; q ∞ k∈Z (aq j ; q)k j=1
−1 ∞ j(j−1) aq j ; q X
s q j j ∞
= cj aj Ψ −1/s; aq ; q, q . 1 1 (q; q)j−1 − 1s ; q ∞ j=1

Now on using the formula 1 Ψ1 (a; b; q, z)

=

(q, b/a, az, q/az; q)∞ , (b, q/a, z, b/az; q)∞

we get G (s) =

∞ X j=1

=

∞ X j=1

jq

cj a

j(j−1)

(q; q)j

cj aj q j(j−1)

s−1 q, −asq j , −q j /s, −q 1−j s; q (aq j , −qs, q j , −as; q)∞ − 1s ; q ∞

aq j ; q




∞


1 1 . s (−as; q)j (−1/s; q)j (−qs; q)−j


∞

Certain Inversion and Representation formulas for q-Sumudu Transforms

By the property (1.2) of (a; q)−j , we obtain G (s) =

=

∞ X

cj aj q j(j−1)

j=1 ∞ X

1 s

(−q/as; q)−j (−qs; q)−j 1 j s (as) q j(j−1)/2 (1/s)j q j(j−1)/2 (−qs; q)−j

cj (−q/as; q)−j .

j=1

On the other hand, let G (s) has a series expansion as follow ∞   X G (s) = s−1 , cj −q (as)−1 ; q −j

j=1

where

∞ X j(j−1)/2 cj < ∞. q j=1

By the property (1.2) of (a; q)−j , we can write G (s) =

∞ X j=1

For some s = s0 , |s0 |