1. Introduction
Throughout this paper,
and q are complex number with
. Here and in what follows, we adopt the standard q-series notation [1] . For any positive integer n,
For convenience, we use
to denote
. We will often use basic properties without reference, such as
The q-binomial coefficient is given for any nonnegative integers M and N by
Following Gasper and Rahman [1] , the bilateral basic hypergeometric series is defined by
and the unilateral basic hypergeometric series is defined by
The Ramanujan’s
sum ( [1] , Appendix (II. 29)), and the sum of a
series, ( [1] , Appendix (II.5)) are stated as follows.
(1.1)
(1.2)
Among the other formulas needed for this paper, we have separately documented the q-binomial formula and its consequence ( [1] (1.3.2)) in the sequel,
(1.3)
Setting
in (1.3), we have
(1.4)
Furthermore, Euler’s formulas (cf. [1] , Corollary 2.2)
(1.5)
Cauchy’s identity (cf. [2] , Theorem 3.3)
(1.6)
and the formula ( [1] , Exercise 1.16) which is a special case of the Bailey-Daum sum ( [1] , (1.8.1))
(1.7)
Besides, if
, the following identity was given by Ramanujan [3] ,
(1.8)
A pair of sequences
is called a Bailey pair relative to a if
(1.9)
And a conjugate Bailey pair relative to a is a pair of sequences
satisfying
(1.10)
In fact, the Bailey pair and the conjugate Bailey pair are the special cases of the following Bailey transform.
Lemma 1.1 ( [4] ). Let n be a nonnegative integer and
be sequences of complex numbers. Assuming convergence of the series, if
(1.11)
then
This terminology was first proposed by Slater [5] . Bailey transform is widely used in mathematics for a long time, especially, in the area of basic hypergeometric series. For example, Andrews [6] , Kim and Lovejoy [7] , and Lovejoy [8] established multiple sums Rogers-Ramanujan type identities and partial theta identities. Andrews and Warnaar [9] applied the Bailey transform to give another proof of false theta functions. Bailey [4] [10] , Bressoud [2] , and Slater [11] [12] used this transform to derive a number of identities of Rogers-Ramanujan type identities. Ji and Zhao [13] established the Hecke-Rogers identities for the universal mock theta functions by means of the Bailey transform.
Notice that if we take
in lemma 1.1, the pair of sequences
is a Bailey pair relative to a, and the pair of sequences
is a conjugate Bailey pair relative to a.
The main motivation for this work came from some Bailey’s transform of M. E. Bachraoui which appear in [14] , specifically the (
) is the constant sequences with value 1. Applying the ( [14] , Theorem 3, 4, 6) and choosing appropriate
, we establish some new transformation formulas for q-series.
2. Main Results
Theorem 2.1. We have
(2.1)
Corollary 2.2. There holds
(2.2)
(2.3)
(2.4)
Theorem 2.3. We have
(2.5)
Corollary 2.4. There holds
(2.6)
(2.7)
Theorem 2.5. We have
(2.8)
Theorem 2.6. There holds
(2.9)
Corollary 2.7. We have
(2.10)
(2.11)
Remark. Setting
in (2.11), we derive
Furthermore, we have
Combining the above two identities, we arrive at
The above identity is a companion to number (23) on Slater’s list [12] as follows.
Theorem 2.8. We have
(2.12)
Corollary 2.9. There holds
(2.13)
(2.14)
Theorem 2.10. We have
(2.15)
3. Proofs of Theorem 2.1 and Corollary 2.2
Proof of Theorem 2.1. Setting
and
in Lemma 1.1, we obtain
(3.1)
where the last step follows by (1.4).
Thus,
This completes the proof.
Proof of Corollary 2.2. Based on (1.4), we can rewrite (2.1) as follows
Equating terms of the corresponding powers of
, we achieve
which is (2.2) by some basic simplifications. (2.3) (2.4) follow from (2.1) upon letting
(
), respectively.
4. Proofs of Theorem 2.3 and Corollary 2.4
Proof of Theorem 2.3. We now apply Lemma 1.1 with
and
. We compute
(4.1)
where the last step follows by (1.2). Thus, we arrive at
which completes the proof.
Proof of Corollary 2.4. (2.6) follows from (2.5) upon letting
and (2.7) follows from (2.6) by letting
, which proves the desired formula.
5. Proof of Theorem 2.5
Proof of Theorem 2.5. Setting
and
is the constant sequences with value 1 in Lemma 1.1. Due to (3.1), we have
Thus,
which is (2.8).
Notice that we use (1.4) to express the left-hand side of (2.8)
(5.1)
Now we equate the terms corresponding to
in (5.1) to obtain
which gives the following identity after straightforward simplifications.
The above identity appears in ( [4] , p. 157).
6. Proofs of Theorem 2.6 and Corollary 2.7
Proof of Theorem 2.6. We apply Lemma 1.1 with
and
is the constant sequences with value 1. Then due to (4.1), we get
Thus,
which yields the desired formula.
Proof of Corollary 2.7. To prove (2.10), we first use (1.6) to express the left-hand side of (2.9) as powers series in x. Then
L. H. S. of (2.9)
Now equate the terms corresponding to
in (2.9) to obtain
which by some basic calculations down to (2.10). (2.11) follows easily from (2.9) upon letting
.
7. Proofs of Theorem 2.8 and Corollary 2.9
Proof of Theorem 2.8. Let us use Lemma 1.1 with a Bailey transform as follows:
We compute
where in the last step, we used the (1.7) with
. Then by virtue of Lemma 1.1, we get
which completes the proof.
Proof of Corollary 2.9. (2.13) and (2.14) follow from (2.12) by letting, respectively,
and
.
8. Proof of Theorem 2.10
Proof of Theorem 2.10. Let us use Lemma 1.1 with a Bailey transform as follows:
We compute
where in the last step, we used (1.8) by letting
. Then by virtue of Lemma 1.1, we get
which completes the proof.
9. Conclusion
By choosing some sequences, we can derive many identities from the Bailey transform. Furthermore, we should study the generalized Bailey transform [14] deeply to establish the multiple parameterized identities. On the other hand, we can also study the mock theta functions or the Rogers-Ramanujan identities through the Bailey transform.