{'fraction_non_alphanumeric': 0.06768599664366959, 'fraction_numerical': 0.02610479209397725, 'mean_word_length': 4.137931034482759, 'pattern_counts': {'":': 0, '<': 9, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Another proof of Bailey's 6-psi-6 summation

Adapting a method used by Cauchy, Bailey, Slater, and more recently, the second author, we give a new proof of Bailey's celebrated 6-psi-6 summation formula.

It was conjectured in [15,Remark 3.2] that any bilateral sum can be obtained from an appropriately chosen terminating identity by Cauchy's method (without appealing to analytic continuation). However, at the same place it was also pointed out that it was already not known whether Bailey (again, see the end of this introduction for the notation), where |q| < 1 and |a 2 q/bcde| < 1, would follow from such an identity. It is maybe interesting to mention that Bailey's 6 ψ 6 summation (1.2), although it contains more parameters than Ramanujan's 1 ψ 1 summation (1.1), does not include the latter as a special case.
While the conjecture of [15,Remark 3.2] remains open, this paper features a new derivation of Bailey's 6 ψ 6 summation formula using a variant of Cauchy's method. After explaining some notation in the end of this introduction, the proof from [15,Sec. 3] of Ramanujan's 1 ψ 1 summation is being reviewed in Section 2, for illustration. The starting point for the derivation of Bailey's 6 ψ 6 summation in Section 3 is Bailey's terminating very-well-poised 10 φ 9 transformation (3.2). After appropriately applying Cauchy's method to both sides of this identity, a specific transformation of 6 ψ 6 series is obtained. The resulting transformation is then iterated, after which a suitable specialization yields the desired 6 ψ 6 summation.
The authors plan to give an account of Cauchy's method applied to terminating quadratic, cubic and quartic identities [10, Sec. 3.8] elsewhere.
Notation: It is appropriate to recall some standard notation for q-series and basic hypergeometric series.
Let q be a fixed complex parameter (the "base") with 0 < |q| < 1. The q-shifted factorial is defined for any complex parameter a by where k is any integer. Since the same base q is used throughout this paper, it may be readily omitted (in notation) which will not lead to any confusion. For brevity, write (a 1 , · · · , a m ) k := (a 1 ) k · · · (a m ) k , where k is an integer or infinity. Further, recall the definition of basic hypergeometric series, and of bilateral basic hypergeometric series, See Gasper and Rahman's text [10] for a comprehensive study on the theory of basic hypergeometric series. In particular, the computations in this paper rely on some elementary identities for q-shifted factorials, listed in [10, Appendix I].
In Jackson's [11] q-Pfaff-Saalschütz summation (cf. [10, Appendix (II.12)]), first replace n by 2n and then shift the index of summation by n such that the new sum runs from −n to n: Next, replace a by aq −n , and c by cq −n , respectively, to get Now, one may let n → ∞, assuming |c/ab| < 1 and |b| < 1, while appealing to Tannery's theorem [7] for being allowed to interchange limit and summation. This gives where |c/a| < |c/ab| < 1. Finally, replacing b by c/az and then c by b gives (1.1).
Remark 2.1. Cauchy [8] had applied the above method to a special case of (2.1).
(After all, (2.1) was not available to him yet.) His starting point was the terminating q-binomial theorem, an identity already known to Euler [9], which can be derived from (2.1) by first doing the substitution b → c/z, then c → b, and then letting a → ∞ and b → 0. As a result of "bilateralizing" (2.2) by the above procedure, Cauchy recovered Jacobi's [13] well-known triple product identity, it becomes apparent that at least one parameter is missing here for the given purpose. As a next step, one might consider Watson's transformation formula of a terminating 8 φ 7 into a multiple of a 4 φ 3 [10, Appendix (III.18)], which involves an additional parameter, and apply Cauchy's method (to both sides of the transformation). In fact, this was undertaken by Bailey [6] who obtained by this procedure, and some symmetry argument, a transformation for 2 ψ 2 series (see also [10,Ex. 5.11]). The next level object in the hierarchy of identities for (very-well-poised) basic hypergeometric series is Bailey's [4]  Note that for b, c or d → ∞, (3.2) reduces to Watson's transformation formula. By a further specialization one obtains Jackson's 8 φ 7 summation formula which may also be derived directly from (3.2) by letting b = aq/c (thus λ = a/d).
Now, one may let n → ∞, assuming |qa 2 /cdef | < 1 while appealing to Tannery's theorem, which yields the following transformation formula: where λ = qa 2 /bcd. In this identity, the right-hand side involves one more parameter (namely b) than the left-hand side. Note that when b = aq/cd (whence λ = a), this identity is trivial. It should also be pointed out that (3.3) is not Slater's [16] transformation [10, Eq. (5.5. 3)], the latter involving more symmetric parameters. A possibility, of course, would be to directly specialize the extra parameter b in (3.3) such that the 6 ψ 6 on the right-hand side reduces to a 6 φ 5 series, which can be summed by the n → ∞ case of (3.1). However, this was not the idea of the above derivaton of the 6 ψ 6 transformation in (3.3). Indeed, several proofs of Bailey's 6 ψ 6 summation which make use of the nonterminating 6 φ 5 summation already exist, see e. g. Slater and Lakin [18], Andrews [1], Askey and Ismail [3], and the second author [14].