Some implications of Chu's $_{10}\psi_{10}$ extension of Bailey's $_{6}\psi_{6}$ summation formula
James Mc Laughlin, Andrew V. Sills, Peter Zimmer
TL;DR
This paper extends Bailey's $_{6}\psi_{6}$ summation to Chu’s $_{10}\psi_{10}$, deriving new Bailey pairs and identities, and providing new proofs and generalizations of classical series-product identities, including Ramanujan's and Andrews' results.
Contribution
The paper introduces a generalization of Bailey pairs using Chu's $_{10}\psi_{10}$ formula, recovering Slater's pairs as special cases and discovering new Bailey pairs and identities.
Findings
Derived new Bailey pairs containing free parameters.
Proved new series-product identities related to Ramanujan and Andrews.
Established new transformation formulas and false theta series identities.
Abstract
Lucy Slater used Bailey's summation formula to derive the Bailey pairs she used to construct her famous list of 130 identities of the Rogers-Ramanujan type. In the present paper we apply the same techniques to Chu's generalization of Bailey's formula to produce quite general Bailey pairs. Slater's Bailey pairs are then recovered as special limiting cases of these more general pairs. In re-examining Slater's work, we find that her Bailey pairs are, for the most part, special cases of more general Bailey pairs containing one or more free parameters. Further, we also find new general Bailey pairs (containing one or more free parameters) which are also implied by the summation formula. Slater used the Jacobi triple product identity (sometimes coupled with the quintuple product identity) to derive her infinite products. Here we also use other summation…
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Taxonomy
TopicsAdvanced Mathematical Identities
Some implications of Chu’s
extension of
Bailey’s summation formula
James McLaughlin, Andrew V. Sills, Peter Zimmer
Abstract
Lucy Slater used Bailey’s summation formula to derive the Bailey pairs she used to construct her famous list of 130 identities of the Rogers-Ramanujan type.
In the present paper we apply the same techniques to Chu’s generalization of Bailey’s formula to produce quite general Bailey pairs. Slater’s Bailey pairs are then recovered as special limiting cases of these more general pairs.
In re-examining Slater’s work, we find that her Bailey pairs are, for the most part, special cases of more general Bailey pairs containing one or more free parameters. Further, we also find new general Bailey pairs (containing one or more free parameters) which are also implied by the summation formula.
Slater used the Jacobi triple product identity (sometimes coupled with the quintuple product identity) to derive her infinite products. Here we also use other summation formulae (including special cases of the summation formula and Jackson’s summation formula) to derive some of our infinite products.
We use the new Bailey pairs, and/or the summation methods mentioned above, to give new proofs of some general series-product identities due to Ramanujan, Andrews and others. We also derive a new general series-product identity, one which may be regarded as a partner to one of the Ramanujan identities. We also find new transformation formulae between basic hypergeometric series, new identities of Rogers-Ramanujan type, and new false theta series identities. Some of these latter are a kind of “hybrid” in that one side of the identity consists a basic hypergeometric series, while the other side is formed from a theta product multiplied by a false theta series. This type of identity appears to be new.
key words: -Series, Rogers-Ramanujan Type Identities, Bailey chains, False Theta Series
subject class[2000]: Primary: 33D15. Secondary:11B65, 05A19.
1 Introduction
Bailey’s identity [5],
[TABLE]
is probably the most important summation formula for bilateral basic hypergeometric series, and has many applications to number theory and partitions - see Andrews’ paper [2] for some examples. This summation formula was also the main tool used by Slater [16, 17] to derive Bailey pairs and, from these, her list of 130 identities of the Rogers-Ramanujan type.
Bailey’s formula was extended by Shukla [14], and Shukla’s formula was later further extended by Chu [6]. Let
[TABLE]
where, for , denotes the -th elementary symmetric function in . Then
Proposition 1.1**.**
Chu [6] For complex numbers , , , and satisfying , there holds the identity
[TABLE]
Upon letting , , we obtain Bailey’s formula (1.1), while letting recovers Shukla’s [14] generalization of Bailey’s formula. For most values of the parameters, the expression for is quite complicated, but we note in passing that the case results in considerable simplification.
Corollary 1.2**.**
For complex numbers , , and satisfying , there holds the identity
[TABLE]
Since (1.1) was used by Slater to derive her Bailey pairs, it is natural to ask if Chu’s generalization of Bailey’s formula at (1.3) can be used similarly to produce any new interesting results. The present paper is in part an investigation of that question.
One observation we make is that most of Slater’s Bailey pairs are special cases of more general Bailey pairs containing one or more free parameters - see Corollaries 2.8 and 2.13. Slater could have derived these more general pairs herself, but it would seem that she was primarily interested in the special cases which would lead to identities of the Rogers-Ramanujan type.
We also note that Slater used the Jacobi Triple Product identity to derive her infinite products. In the present paper we use other summation formulae, including special cases of the - and summation formulae, to derive some infinite products.
In the transformation at (2.5) below, Slater essentially employed three cases ( and , ) to make the series containing the sequences summable. In the present paper we also explore additional cases, in the process discovering some interesting new identities (see Section 3 below).
Our results include new proofs of some general series-product identities due to Ramanujan, Andrews and others, and also a new general series-product identity
[TABLE]
which may be regarded as a partner to an identity equivalent to one recorded by Ramanujan in his lost notebook [4, p. 99, Entry 5.3.1 with and throughout]:
[TABLE]
We also find new transformation formulae between basic hypergeometric series, new identities of Rogers-Ramanujan type, and new false theta series identities. Some of these latter are a kind of “hybrid” in that one side of the identity consists a basic hypergeometric series, while the other side is formed from a theta product multiplied by a false theta series. For example,
[TABLE]
This type of identity appears to be new.
We employ the usual notations:
[TABLE]
For later use, we recall that a pair of sequences that satisfy and
[TABLE]
is termed a Bailey pair relative to .
2 Bailey Pairs from Chu’s Extension of the summation formula
2.1 General Bailey pairs
We first consider the case and in Chu’s formula. These same substitutions were made by Slater in (1.1), and gave rise to a quite general Bailey pair (see below), which in turn led to many new identities of Rogers-Ramanujan type. Let be defined as at (1.2).
Theorem 2.1**.**
The pair of sequences is a Bailey pair with respect to , where
[TABLE]
where .
Proof.
First set , so that all the terms with negative index in the bilateral sum at (1.3) become zero. Next, for each non-negative integer , set , so that the sum on the left side of (1.3) becomes a finite sum, with the summation index running from 0 to . Simplify the resulting product on the right side of (1.3), and use the identity (see [7, (I.10), page 351])
[TABLE]
to modify the series side. The result then follows from (1.8), after some simple manipulations, noting also that , so that the requirement is satisfied. ∎
The expression for above is generally quite complicated, and thus so also is the expression for the . However, as was also the case for above, if we set then simplifies considerably. One can check, preferably using a computer algebra system, that
[TABLE]
Upon letting , in the Bailey pair in Theorem 2.1, we recover the following Bailey pair (with respect to ) due to Slater,
[TABLE]
which is implicitly contained in Equation (4.1) of [16].
It would appear that Slater’s principal motivation was to prove identities of the Rogers-Ramanujan type, so this Bailey pair, and other general pairs mentioned below, were not stated explicitly by her in [16] and [17], where she instead listed many special cases of them. However, they could all have been easily derived by her, using the same methods she used to derive the special cases.
We remark in passing that all the Bailey pairs in Slater’s B, F and H tables, as well as pairs E(3), E(6) and E(7) (see [16, page 468]), are derived from the Bailey pair at (2.4).
Slater also showed [16, Equation (1.3) on page 462] that if is a Bailey pair with respect to , then, for non-zero complex numbers and ,
[TABLE]
A finite generalization of this identity is of course implied by Bailey’s Lemma (see, for example, Theorem 12.2.3 in [3]), namely, if is a Bailey pair with respect to , and is a non-negative integer, then
[TABLE]
The identity at (2.5) is a particular case of the Bailey Transform: if and , then
[TABLE]
In the present paper, for ease of notation we will refer to (2.5) as the Bailey Transform.
If we set in the Bailey pair from Theorem 2.1, and then substitute this pair into (2.5) and (2.6), we get the following unusual basic hypergeometric identities.
Corollary 2.2**.**
Let be an integer. Then
[TABLE]
[TABLE]
Note that setting in (2.7) gives the -Pfaff-Saalschütz sum (see [7, page 355, II.12]), while setting gives the identity (for )
[TABLE]
2.2 Mod Bailey pairs
We continue to follow in Slater’s footsteps, this time making the same substitutions in Chu’s identity (1.3) that she did in (1.1) to produce the Bailey pairs in her A table. Let be as defined at (1.2).
Theorem 2.3**.**
(i) Let . Then the pair of sequences is a Bailey pair with respect to , where , , and
[TABLE]
*Let . Then
(ii) the pair of sequences is a Bailey pair with respect to , where , , and*
[TABLE]
(iii) the pair of sequences is a Bailey pair with respect to , where , , and
[TABLE]
Proof.
Set in (1.3), so that all the terms of negative index vanish. Then replace with , set , and . Then after some simple manipulations (1.3) becomes
[TABLE]
Apply (2.2) to the factor, divide both sides by to get
[TABLE]
and (2.10) follows.
For the other two pairs, replace with in (2.13), and (2.11) follows from the identity
[TABLE]
while (2.12) follows from the identity
[TABLE]
∎
Remark 2.4**.**
.
Theorem 2.5**.**
*Let and suppose . Then
(i) the pair of sequences is a Bailey pair with respect to , where and*
[TABLE]
(ii) the pair of sequences is a Bailey pair with respect to , where and
[TABLE]
(iii) the pair of sequences is a Bailey pair with respect to , where and
[TABLE]
Remark 2.6**.**
The condition is necessary to ensure that .
Proof.
Replace with in (1.3), and then set , , and . One easily checks that the right side simplifies to give
[TABLE]
On the series side, the choices for the parameters force the series to terminate above and below, and we get, after some elementary manipulations, that the series becomes
[TABLE]
Next, we apply (2.2) to the factor above and rearrange terms to get
[TABLE]
Noting that, for arbitrary non-zero and ,
[TABLE]
we get that
[TABLE]
where the last equality follows from the identities
[TABLE]
That (2.14) gives a Bailey pair now follows from the definition of a Bailey pair at (1.8), also noting that .
If, instead of using the identities at (2.18), we use the identities
[TABLE]
we get that the pair at (2.15) is a Bailey pair.
The proof of (2.16) follows from the right side of the first equality following (2.17), upon setting in the first sum and in the second sum. ∎
If we begin by setting instead of , but keep the same choices , and in (1.3), then we get Theorems 2.7 following.
Theorem 2.7**.**
*Let . Then
(i) the pair of sequences is a Bailey pair with respect to , where and*
[TABLE]
(ii) the pair of sequences is a Bailey pair with respect to , where and
[TABLE]
(iii) the pair of sequences is a Bailey pair with respect to , where , and
[TABLE]
Proof.
The proof parallels that of Theorem 2.5, except that after arriving at the identity
[TABLE]
and then separating the sum into two sums ( and ) as previously, we instead employ the identities
[TABLE]
to get (2.20). The result follows as in Theorem 2.5, except that it is necessary to re-index one of the four sums (by replacing with ).
For (2.21) we instead use the identities
[TABLE]
The proof of (2.22) is like the proof of (2.16) in Theorem 2.5, except that after separating the sum at (2.23) into two sums, according to or , and then replacing with for the sum with , we use the identities and , and finally re-index in the latter case by replacing with . ∎
The nine Bailey pairs in the next corollary derive, respectively, from the pairs in Theorems 2.3 - 2.7, by letting , in each case.
Corollary 2.8**.**
The sequences below are Bailey pairs with respect to the stated values of , where
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
All eight pairs in Slater’s A table ([16, page 463]) and all six on her J list ([17, pp. 148–149]) are derived from the five Bailey pairs (2.26), (2.29), (2.30), (2.32), (2.33) above, for particular values of . Slater did not write down these general pairs above explicitly, but she could easily have derived them by the same methods she used to derive the special cases. None of Slater’s pairs arise as special cases of (2.27), (2.28), (2.31) or (2.34), although the special case of (2.31) was given in [9]. However, specializing the parameters give Bailey pairs which then give rise to some of the series-product identities on Slater’s list, showing that different Bailey pairs may lead the same identity of Rogers-Ramanujan type.
Each of the nine Bailey pairs above also gives rise to a transformation between basic hypergeometric series, upon substituting the pair into (2.5). The pair at (2.26), for example, gives the following identity.
Corollary 2.9**.**
[TABLE]
2.3 Mod 2 Bailey Pairs
We next consider how Slater produced the Bailey pairs in the G-, C- and I tables of [16, 17].
Theorem 2.10**.**
*Let . Then
(i) the pair of sequences is a Bailey pair with respect to , where and*
[TABLE]
*Let . Then
(ii) the pair of sequences is a Bailey pair with respect to , where and*
[TABLE]
(iii) the pair of sequences is a Bailey pair with respect to , where and
[TABLE]
Proof.
The proof is quite similar to the proof of Theorem 2.3. As in the proof of that theorem, set in (1.3), so that all the terms of negative index vanish. Then replace with , set and . After some simple manipulations, (1.3) becomes
[TABLE]
Apply (2.2) to the factor, divide both sides by to get
[TABLE]
and the proof for the pair at (2.36) follows.
For (2.37) and (2.38), replace with in (2.39), and use, respectively, the identities
[TABLE]
∎
Theorem 2.11**.**
*Let . Then
(i) the pair of sequences is a Bailey pair with respect to , where and*
[TABLE]
(ii) The pair of sequences is a Bailey pair with respect to , where and
[TABLE]
(iii) The pair of sequences is a Bailey pair with respect to , where and
[TABLE]
Proof.
The proof is very similar to the proof of Theorem 2.5. This time, replace with in (1.3), and then set , , . The right side simplifies to give
[TABLE]
After some elementary manipulations, the series becomes
[TABLE]
We apply (2.2) to the factor above and rearrange terms to get
[TABLE]
After applying (2.17) to the terms of negative index in the sum above, we get that
[TABLE]
The pair at (2.40) now follows after some simple manipulations, noting that
[TABLE]
The pair at (2.41) follows from (2.43), upon absorbing the factor in the denominators there, employing the identities
[TABLE]
in the way similar to the way that the pair of identities at (2.18) was used in the proof of Theorem 2.5, and finally re-indexing one of the resulting sums (by replacing with ).
The proof for the pair at (2.42) is similar to the proof for the pair at (2.41), except we employ the identities
[TABLE]
∎
As with - , and are in general quite complicated, but simplify considerably for particular values of the parameters, leading (as was the case in Corollary 2.2) to transformations of basic hypergeometric series. We give one example.
Corollary 2.12**.**
If , , , such that and none of the denominators below vanish, then
[TABLE]
Proof.
In the Bailey pair at (2.36), set , and , so that takes the value
[TABLE]
Substitute the resulting Bailey pair into (2.5), and the result follows after some elementary -product manipulations. ∎
The Bailey pairs in Slater’s G-, C- and I tables, as well as pairs E(1), E(2) , E(4) and E(5) (see [16, pages 469 and 470]), are derived from the next six Bailey pairs. These, in turn, are derived from the pairs in Theorems 2.10 and 2.11, by letting .
Corollary 2.13**.**
The sequences below are Bailey pairs with respect to the stated values of , where and
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
2.4 Mod 4 Bailey Pairs
We continue to follow in Slater’s path [16], next considering how she produced the Bailey pairs in her K table.
Theorem 2.14**.**
Let for , and . Then (i) the pair of sequences is a Bailey pair with respect to , where , , and
[TABLE]
(ii) the pair of sequences is a Bailey pair with respect to , where , , and
[TABLE]
Proof.
The proof is initially similar to the proof of Theorem 2.3, except we replace with and set , , and . Instead of (2.13), we arrive at
[TABLE]
Next, separate the sum into terms of positive and negative index, re-index the latter sum by replacing with (and also using (2.17)), to get
[TABLE]
The proof for the Bailey pair at (2.53) then follows, upon setting , simplifying the product side, and using the identities
[TABLE]
The proof for the Bailey pair at (2.54) is similar, except we use the identities
[TABLE]
∎
Theorem 2.15**.**
Let for . Then (i) the pair of sequences is a Bailey pair with respect to , where , and
[TABLE]
(ii) the pair of sequences is a Bailey pair with respect to , where , and
[TABLE]
Proof.
Set in (2.56), simplify the product side and absorb the terms on the sum side into what were previously the and factors. The pair at (2.57) follows after using the identities
[TABLE]
The pair at (2.58) follows similarly, after employing the identities
[TABLE]
∎
Theorem 2.16**.**
Let ,
[TABLE]
*and for . Then
(i) the pair of sequences is a Bailey pair with respect to , where , and*
[TABLE]
(ii) the pair of sequences is a Bailey pair with respect to , where , and
[TABLE]
Remark 2.17**.**
The reason does not fit into the formula for for is that it is necessary to be careful with the order in which (after replacing with ) we set , , , and , in order to get the series at (1.3) to converge. It is easy to see that making those substitutions simultaneously gives , contradicting the requirement that in the series. One way around this is to first set and , causing the series to terminate above and below, after which the other replacements can be made.
Note also that, with the stated choices for the parameters, the factor in the denominator of the expression for at (1.2) vanishes, and in fact these choices also cause the numerator of to vanish. It is cancelling these “zero factors” that make the expression for different. The order of substitutions described above causes to have the value assigned to , whereas for , is independent of the order of substitutions.
We also note that a similar situation occurs in some of the other theorems above.
Proof.
The proof is similar to the proof of Theorem 2.15, except we set in (2.56). The pair at (2.59) follows after using the identities
[TABLE]
and then replacing with in the sum corresponding to the second term in the second identity above (so that this sum implicitly defines part of instead of ). The pair at (2.60) follows similarly, after using the identities
[TABLE]
∎
The six Bailey pairs in Slater’s K table [16, page 471] follow from the three theorems in this section, upon letting .
Corollary 2.18**.**
The following pairs of sequences , with , are Bailey pairs with respect to the stated value of .
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
3 Identities of the Rogers-Ramanujan-Slater type
and False Theta Series Identities
We use the Bailey pairs found in the previous section to derive several new series-product identities, and also to give new proofs of some general identities due to Ramanujan, Andrews and others. We also remark that we sometimes use a more general case of the than the Jacobi Triple Product Identity, in contrast to how Slater derived her products.
3.1 General identities containing one or more free parameters
We first give new proofs for two quite general identities of Ramanujan ([12, page 33], see also R1 and R2 on page 8 of [10]). One reason these two identities are of interest is that each leads to infinitely many identities of Rogers-Ramanujan type (set , where and are integers, and then replace with ).
Theorem 3.1**.**
For and ,
[TABLE]
Proof.
Let in the Bailey pair at (2.31), and then insert the resulting pair into (2.5), after setting . This leads to the identities
[TABLE]
The second equality above follows after some elementary -product manipulations and the last equality follows from the fact that the two series in the previous equality combine to give a special case of Bailey’s summation at (1.1) (replace with , let , set , and ). The result now follows after some further elementary manipulations. ∎
Theorem 3.2**.**
For and ,
[TABLE]
Proof.
The proof is similar to that of the theorem above. Let and set in the Bailey pair at (2.50), and then insert the resulting pair into (2.5), after setting . This leads to the identities
[TABLE]
The second equality above once again follows after some elementary -product manipulations and once again the two series on the right side in the second equality combine to give a special case of Bailey’s summation at (1.1) (replace with , let , set , , and ). The result now follows after replacing with , followed by some further elementary manipulations. ∎
Different proofs of the results in the two theorems above were also given in [4] (Entries 5.3.1 and 5.3.5) and in [11].
We now prove a new general series-product identity, one which may be regarded as a partner to Ramanujan’s result in Theorem 3.1.
Theorem 3.3**.**
For and ,
[TABLE]
Proof.
Let in the Bailey pair at (2.34), and then insert the resulting pair into (2.5), after setting . This leads to the identities
[TABLE]
The second equality above follows after some elementary -product manipulations and re-indexing the second series (replacing with ). The last equality follows from the fact that the two series in the previous equality combine to give another special case of Bailey’s summation (1.1) (replace with , let , set , and ). The result once again follows after some further elementary manipulations. ∎
Remark 3.4**.**
Several identities on Slater’s list [17] follow as special cases of (3.3). We summarize these in the following table.
[TABLE]
We next give new proofs of Andrews’ -analog of Gauss’s sum and a special case () of Heine’s [8] -analog of Gauss’s sum:
[TABLE]
We first recall Jackson’s summation formula for a very-well-poised series [7, p. 356, Eq. (II. 20)] (which follows upon setting in (1.1)):
[TABLE]
Theorem 3.5**.**
[TABLE]
[TABLE]
Proof.
Let in the Bailey pair at (2.47) to get the pair
[TABLE]
Substitute this pair into (2.5), after setting , to get
[TABLE]
The next-to-last equality follows from (3.5) (replace with , let , set and ), and (3.6) follows upon replacing with and with .
Next, substitute the pair at (3.8) once again into (2.5), this time setting . This gives
[TABLE]
The next-to-last equality follows once again from (3.5) (replace with , let , set and ), and (3.7) follows upon replacing with and with . ∎
3.2 A particular case of the Bailey Transform, I
If we set and in (2.5), the following identity results, providing both series converge:
[TABLE]
This particular case was not used by Slater [17], and may possibly be regarded as being of lesser importance for two reasons. Firstly, it is certainly the case the both series above will not converge for all Bailey pairs , and secondly, even if a series-product identity does result, the power of on the series side may not be quadratic in the exponent (and many do not consider such identities as being of Rogers-Ramanujan-Slater type), or else the resulting identity is an easy consequence of a more general identity. However we believe such identities are sufficiently interesting to include some examples.
Corollary 3.6**.**
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Proof.
For the first three identities above, insert the Bailey pairs at (2.62), (2.64) and (2.66), respectively into (3.9) (noting that , respectively), and in each case the result follows after some elementary manipulation of the resulting identity.
For (3.13), substitute the Bailey pair at (2.28) into (3.9), set , and simplify the resulting identity. ∎
3.3 A particular case of the Bailey Transform, II
If we set and let in (2.5), the following identity results:
[TABLE]
This transformation also gives rise to a number of interesting false theta series identities, and we give several examples below. See §3.5 below for an explanation of false theta series.
Corollary 3.7**.**
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Proof.
Let in the Bailey pair at (2.27), substitute the resulting pair into (3.14) (with ) and (3.15) follows after simplifying the left side and re-indexing two of the resulting series on the right side.
For (3.16), let and set in the Bailey pair at (2.52), substitute the resulting pair (Slater’s pair I(4)) into (3.14) (with ), replace with and simplify.
For (3.17), substitute pair (2.61) into (3.14) (with ), and rearrange.
To get (3.18), let and set and in Slater’s general Bailey pair at (2.4), substitute the resulting pair (Slater’s pair H(1), corrected) into (3.14) (again with ) and rearrange the resulting identity.
∎
Remark 3.8**.**
The transformation at (3.14) does lead to identities of Rogers-Ramanujan type, but those we found were either not new, or were simple linear combinations of existing identities.
3.4 Miscellaneous Identities of the Rogers-Ramanujan-Slater type
Finally, we exhibit some identities that follow from Bailey pairs not listed by Slater, pairs that do follow however from specializing the parameters in our general Bailey pairs. We use two cases of the transformation at (2.5) which were used by Slater. Firstly, let to get
[TABLE]
Secondly, let , set and then replace with and with to get
[TABLE]
Corollary 3.9**.**
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Proof.
Let and in the Bailey pair at (2.47). The identity at (3.21) follows upon substituting the resulting pair into (3.19) and simplifying.
For (3.22), set and in the pair at (2.47), substitute the resulting pair into (3.20), replace with , and rearrange.
If we set and in the pair at (2.48), we get (3.23) when this pair is inserted in (3.20), after replacing with .
The identity at (3.24) follows upon inserting the pair formed by setting and in the Bailey pair at (2.49) into (3.20), and then once again replacing with .
Let and set in the Bailey pair at (2.59). This gives the Bailey pair (with respect to ), ,
[TABLE]
Note that the expression for follows from the fact that, with the stated values for and ,
[TABLE]
The identity at (3.25) follows upon inserting this Bailey pair in (3.19), and using the Jacobi Triple Product identity to sum pairs of series. ∎
3.5 Theta-False Theta “hybrid” identities.
Rogers [13] referred to a series of the form
[TABLE]
as a theta series of order , where and . Due to the Jacobi triple product identity, the series (3.26) may be expressed as the infinite product
[TABLE]
Rogers also took interest in series of the form
[TABLE]
which he called a theta series of order . False theta series are not representable as infinite products, but nonetheless may have representations as Rogers-Ramanujan type -series, and Rogers gave a number of examples of identities of false theta series. See [10, §5, p. 35 ff] for further discussion and numerous examples of false theta series identities.
Here we give several examples of identities where one side is a basic hypergeometric series and the other side comprises a theta product multiplied by a false theta series. We believe this type of identity is new.
Corollary 3.10**.**
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Proof.
The identity at (3.28) follows upon setting and in the pair at (2.48), substituting the resulting pair (Slater’s pair I(10)) into (3.14) (with ), replacing with , and finally with .
The identity at (3.29) follows from setting and in the Bailey pair at (2.48), and substituting the resulting pair into (3.19), while the identity at (3.31) follows from substituting the same pair in (3.20), and replacing with .
Likewise, the identity at (3.30) follows as a consequence setting and in the Bailey pair at (2.49), and substituting the resulting pair into (3.19), while for (3.33) we proceed similarly, after creating a Bailey pair by instead setting (and keeping ).
The identity at (3.32) follows upon inserting the pair used in the proof of (3.30) into (3.20), after replacing with .
For (3.34), insert Slater’s pair H(12) (, and in (2.4)) into (3.14) (with ), replace with , and rearrange.
∎
4 Concluding Remarks
In a paper by the second author [15], it was shown that more than half of Slater’s identities could be derived from just three general Bailey pairs together with several limiting cases of Bailey’s lemma and an associated family of -difference equations. Here, we attempt to put Slater’s work in a broader context via general Bailey pairs, but without the use of -difference equations. Both approaches have their merits, and both yield new identities. It may well be worth exploring the combinatorial consequences of the new identities presented here.
We also note that we have given just a sample of the identities that may be produced by the methods used in Corollaries 3.6 - 3.10, and that it is likely that many similar identities may be produced by employing other Bailey pairs.
5 Addendum: May 2023
On May 8, 2023, the second author received an email from Aritram Dhar sharing the following observations:
I was going through your 2010 paper “Some implications of Chu’s extension of Bailey’s summation formula” joint with J. McLaughlin and P. Zimmer and I came across the identity in Corollary 1.2 which follows by replacing with in Proposition 1.1 which is the identity in Theorem 2 of Chu’s 2006 paper “Bailey’s very well-poised -series identity”. I have a comment regarding your Corollary 1.2 which is as follows:
If you look at Exercise 5.26 on page 152 of Gasper and Rahman’s Basic Hypergeometric Series text, the identity there is a series-product identity. As mentioned at the end of the problem, if we refer to Chu’s 1998 paper “Partial-fraction expansions and well-poised bilateral series” and look at Theorem 2 on page 500, then it is essentially the same identity as that in Gasper and Rahman. Now, if we consider and make the substitution in Gasper and Rahman’s Exercise 5.26 identity, we get exactly Corollary 1.2 of your paper. By making a similar substitution in the identity, I believe we can get Proposition 1.1 (Theorem 2 of Chu’s 2006 paper) of your paper.
6 Acknowledgment
We thank Aritram Dhar for his careful reading of our paper, and for sharing his observations now included in Section 5.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] G. E. Andrews, On the q-analog of Kummer’s theorem and applications, Duke Math. J. 40 (1973) 525–528.
- 2[2] G. E. Andrews, Applications of basic hypergeometric functions. SIAM Rev. 16 (1974) 441–484.
- 3[3] G. E. Andrews, R. Askey, and R. Roy, Special functions. Encyclopedia of Mathematics and its Applications, 71. Cambridge University Press, Cambridge, 1999. xvi+664 pp.
- 4[4] G. E. Andrews, and B. C. Berndt, Ramanujan’s lost notebook, Part II. Springer, New York, 2009. xii+418 pp.
- 5[5] W. N. Bailey, Series of hypergeometric type which are infinite in both directions, Quart. J.Math.7 (1936) 105–115.
- 6[6] W. Chu, Bailey’s very well-poised ψ 6 6 subscript subscript 𝜓 6 6 {}_{6}\psi_{6} -series identity, J. Combin. Theory Ser. A 113 (2006) 966–979.
- 7[7] G. Gasper and M. Rahman, Basic hypergeometric series , 2nd edition. Encyclopedia of Mathematics and its Applications, vol. 96. Cambridge University Press, Cambridge, 2004. xxvi+428 pp.
- 8[8] E. Heine, Untersuchungen über die Reinhe 1 + ( 1 − q α ) ( 1 − q β ) ( 1 − q ) ( 1 − q γ ) ⋅ x + ( 1 − q α ) ( 1 − q α + 1 ) ( 1 − q β ) ( 1 − q β + 1 ) ( 1 − q ) ( 1 − q 2 ) ( 1 − q γ ) ( 1 − q γ + 1 ) ⋅ x 2 + ⋯ , 1 ⋅ 1 superscript 𝑞 𝛼 1 superscript 𝑞 𝛽 1 𝑞 1 superscript 𝑞 𝛾 𝑥 ⋅ 1 superscript 𝑞 𝛼 1 superscript 𝑞 𝛼 1 1 superscript 𝑞 𝛽 1 superscript 𝑞 𝛽 1 1 𝑞 1 superscript 𝑞 2 1 superscript 𝑞 𝛾 1 superscript 𝑞 𝛾 1 superscript 𝑥 2 ⋯ 1+\frac{(1-q^{\alpha})(1-q^{\beta