NEW CURIOUS BILATERAL q-SERIES IDENTITIESFRÉDÉRIC JOUHET AND MICHAEL J. SCHLOSSER∗

Abstract. By applying a classical method, already employed by Cauchy, to aterminating curious summation by one of the authors, a new curious bilateralq-series identity is derived. We also apply the same method to a quadraticsummation by Gessel and Stanton, and to a cubic summation by Gasper,
respectively, to derive a bilateral quadratic and a bilateral cubic summation
formula.

1. Introduction
In [11, Thm. 7.29] one of the authors derived the following curious summation:(qa/b2 c; q)n=nX(c + 1 − (a + b))(bc + a + b)
(ac − (a + bq −k )2 )(c + 1 − (a + bq −k )) (bc + a + bq −k ) (ac − (a + b)(a + bq −k ))
k=0

aq
−k

− c); q n aqb2 c (a + bq
×
−
;
q

−k )
b(a + bq −k )naq c−(a+bq;q nb(a+bq −k )1(q −n ; q)k (a + bq −k − c; q)k − bc
(a + bq −k ); q k× qk .
−n−k − c); q(q; q)k − bqa (a + bq −k ); q k baq
2 c (a + bq
k(1)

Here (α; q)k denotes the q-shifted factorial, defined byY(α; q)∞(α; q)∞ :=(1 − αq j ),and(α; q)k :=,(αq k ; q)∞j≥0where q (the base) is a fixed complex parameter with 0 < |q| < 1, α is a complex
parameter and k is any integer. For brevity, we shall also use the compact notation(α1 , · · · , αm ; q)k := (α1 ; q)k · · · (αm ; q)k .
The summation in (1) was derived by application of inverse relations to the qPfaff–Saalschütz summation (cf. [6, Appendix (II.12)]). In [11] also several other“curious summations” (involving series which themselves do not belong to the respective hierarchies of hypergeometric and basic hypergeometric series) were derived
by utilizing various summation formulae for hypergeometric and basic hypergeometric series. Similar identities were derived by the same means in [8]. Special
Date: June 25, 2012.
2010 Mathematics Subject Classification. 33D15.
Key words and phrases. bilateral basic hypergeometric series, q-series, curious summations.
∗ The second author was partly supported by FWF Austrian Science Fund grant S9607 (whichis part of the Austrian national Research Network “Analytic Combinatorics and probabilistic
Number Theory”).
1

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FRÉDÉRIC JOUHET AND MICHAEL J. SCHLOSSER

cases of two of the summations were even extended there to bilateral summations
by means of analytic continuation.
Another method to obtain bilateral summations from terminating ones was employed in [12] to give a new proof of Ramanujan’s 1 ψ1 summation formula and to
derive (for the first time) Abel–Rothe type extensions of Jacobi’s triple productidentity. Actually, the method of [12] was already utilized by Cauchy [4] in his
second proof of Jacobi’s [9] triple product identity. The very same method (which
we shall refer to as “Cauchy’s method of bilateralization”) had also been exploited
by Bailey [1, Secs. 3 and 6], [2], and Slater [13, Sec. 6.2]. In [10] the current authors
used a variant of Cauchy’s method to give a new derivation of Bailey’s [1, Eq. (4.7)]
very-well-poised 6 ψ6 summation (cf. [6, Appendix (II.33)]), 2 k∞X(b, c, d, e; q)kqa(1 − aq 2k )(1 − a) (aq/b, aq/c, aq/d, aq/e; q)k bcdek=−∞=(q, aq, q/a, aq/bc, aq/bd, aq/be, aq/cd, aq/ce, aq/de; q)∞,(q/b, q/c, q/d, q/e, aq/b, aq/c, aq/d, aq/e, a2 q/bcde; q)∞(2)where |q| < 1 and |qa2 /bcde| < 1.
In Section 2, we apply Cauchy’s method of bilateralization to the curious summation in (1). (This possibility, which appears to be applicable to (1) but, to thebest of our knowledge, not to any of the other curious summations of [11, Sec. 7],
was missed so far.) As a result, we obtain the new curious bilateral summation in
Proposition 2.1. In the same section we explicitly display some noteworthy special
cases of the new curious bilateral identity. In Section 3 we apply Cauchy’s method
to a terminating quadratic summation by Gessel and Stanton [7], and to a terminating cubic summation by Gasper [5]. Hereby we obtain a bilateral quadratic anda bilateral cubic summation, both which evaluate to zero, see Propositions 3.1 and
3.2, respectively.
For a comprehensive treatise on basic hypergeometric series, see Gasper and Rahman’s text [6]. Several of the computations in this paper rely on various elementary
identities for q-shifted factorials, listed in [6, Appendix I].
2. A new curious bilateral summation
To apply Cauchy’s method to the terminating summation in (1), we first replacen by 2n and then shift the index of summation by n such that the new sum runs
from −n to n. Further, we replace b by bq n . In total, we thus obtain(q 1−2n a/b2 c; q)2n=nX
(c + 1 − (a + bq n )) (bcq n + a + bq n )(ac − (a + bq −k )2 )−kn−k(c + 1 − (a + bq )) (bcq + a + bq ) (ac − (a + bq n )(a + bq −k ))k=−n(q; q)2n(q; q)n+k (q; q)n−k 
aq 1−2n−k− c); q 2n
aq 1−nb2 c (a + bq
×
;
q
−

−k )
b(a + bq −k )2naq 1−n c−(a+bq; q 2nb(a+bq −k )−n(a + bq −k − c; q)n+k − qbc (a + bq −k ); q n+k
×
.


−n
1−2n
− bqa (a + bq −k ); q n+k aqb2 c (a + bq −k − c); q n+knk2

× (−1)n+k q ( 2 )+(2)−nk−2n +n+k

NEW CURIOUS BILATERAL q-SERIES IDENTITIES

32nNow, after multiplying both sides by (q; q)n (b2 c/a)2n q ( 2 ) we may let n → ∞,assuming max(|q|, |b2 /a|, |a − c|) < 1, while appealing to Tannery’s theorem [3]for being allowed to interchange limit and summation. This results, after someelementary manipulations of q-shifted factorials, in the following curious bilateralsummation:

Proposition 2.1. Let a, b, c be indeterminates, let |q| < 1 and max(|q|, |b2 /a|, |a −c|) < 1. Then∞Xa(ac − (a + bq −k )2 )(c + 1 − a)(q, b2 c/a; q)∞ =−k−k(c + 1 − (a + bq )) (a + bq ) (ac − a(a + bq −k ))
k=−∞


bcq
b2 c
; q ∞ − a+bq
−k ; q ∞a(a+bq −k −c)
×


−k )
b(a+bq −k ); q ∞ a(c−(a+bq
aq c−(a+bq−k )) ; q ∞b(a+bq −k ) b
× (a + bq −k − c; q)∞ − (a + bq −k )q k ; q
a
∞
k



2
b c
1
.
(3)
× − (a + bq −k ); qbca(a + bq −k − c)kRemark 2.2. We checked the validity of the identity in (3) by Mathematica. In particular, by replacing a, b, c by aq, bq, cq, respectively, the identitycan be interpreted1
as a power series identity in q (valid for |q| < min ba2 , a−c, 1 , in particular, forq around zero). Only a finite number of terms contribute to the coefficient of q n
for each n ≥ 0.
We write out some noteworthy special cases of Proposition 2.1. The first one is
obtained by replacing (a, b, c) by (at, bt, ct) and then taking the limit t → 0 (which,again, is justified by Tannery’s theorem [3]).
Corollary 2.3. Let a, b and c be indeterminates and |q| < 1. Then
k−1 
k∞X(ac − (a + bq −k )2 ) a + bq −kb(q; q)∞ =(ac − a(a + bq −k ))aa + bq −k − ck=−∞ c − (a + bq −k ) −1  b(a + bq −k )
−1k× q (2) aq
;
q
;
q
. (4)
b(a + bq −k )∞ a(c − (a + bq −k ))∞This turns out to be a generalization of Jacobi’s triple product identity (thec → 0, b → −az case of (4)).
If instead, we directly take c → 0 in (3), then we obtain another generalizationof Jacobi’s triple product identity, a special case of a curious bilateral summationconsidered in [12].
It is also interesting to take the c → a case of (3). The result, after someelementary manipulations, isCorollary 2.4. Let a and b be indeterminates, let |q| < 1 and |b2 /a| < 1. Then∞
1
X(q, b2 ; q)∞(2a + bq −k ) (1/b; q)k − ab (a + bq −k ); q k=(−1)k b2k
(b, bq; q)∞(a + bq −k )(b; q)k
k=−∞


k
abq
− bqa (a + bq −k ); q ∞ − a+bq
k+1
−k ; q ∞× q( 2 )
. (5)


−k
aq 1−kk a+bq− a+bq;q ∞−k ; q ∞ − qa4FRÉDÉRIC JOUHET AND MICHAEL J. SCHLOSSER

If we now let a → ∞, we obtain after some elementary manipulations of q-shiftedfactorials the following summation for a bilateral 1 ψ2 series.
Corollary 2.5. Let b be an indeterminate and |q| < 1. Then∞X2
(q 2 , b2 q; q 2 )∞(1/b2 ; q 2 )k=(−1)k q k b2k .
2
2
2
2
2
(q, b q ; q )∞(b ; q )k

(6)

k=−∞

As a matter of fact, the identity in (6) is not a special case of the bilateral qKummer summation [6, Appendix (II.30)], the latter which is an easy consequenceof Bailey’s 6 ψ6 summation formula (2). Nevertheless, also Corollary 2.5 can bederived from Bailey’s 6 ψ6 summation formula. Indeed, note that by replacing thesummation index k by 1 − k in (6), the right-hand side becomes∞X2(1/b2 ; q 2 )k(−1)k q k −2k+1 b2k .
(b2 ; q 2 )kk=−∞

It follows that∞X
(1/b2 ; q 2 )kk=−∞(b2 ; q 2 )k=2(−1)k q k b2k∞2
1 + q X 1 + q 2k−1 (1/b2 ; q 2 )k(−1)k q k −2k b2k .
−1
2
2
2
1+q
(b ; q )kk=−∞

But this can be evaluated by the (a, b, c, d, e, q) → (−q −1 , 1/b, −1/b, ∞, ∞, q) limit
case of (2), after which one readily obtains the product side of (6).
3. A bilateral quadratic and a bilateral cubic summation
First we apply Cauchy’s method of bilateralization to the following quadraticsummation formula due to Gessel and Stanton [7, Equation (1.4), q → q 2 ]:nX
(1 − aq 3k )k=0(a, b, q/b; q)k (d, a2 q 1+2n /d, q −2n ; q 2 )kqk
(1 − a) (aq/d, dq −2n /a, aq 1+2n ; q)k (q 2 , aq 2 /b, abq; q 2 )k=(aq; q)2n (abq/d, aq 2 /bd; q 2 )n.
(aq/d; q)2n (aq 2 /b, abq; q 2 )n(7)

We replace n by 2n and then shift the index of summation by n such that the newsum runs from −n to n. We also replace a by aq −3n , and b by bq −n , respectively.
After some elementary manipulations of q-shifted factorials we thus obtain theidentitynX(1 − aq 3k )k=−n(aq −2n , b, q 1+2n /b; q)k (dq 2n , a2 q/d, q −2n ; q 2 )kqk
(1 − a) (aq 1−2n /d, d/a, aq 1+2n ; q)k (q 2+2n , aq 2 /b, abq 1−2n ; q 2 )kn(aq, q/a; q)2n (q 2 , q 2 , aq 2 /bd, bd/a; q 2 )n (dq/ab; q 2 )2nd=.
(q/b, d/a; q)2n (d, aq 2 /b, q/ab, dq/a2 ; q 2 )n (q 2 ; q 2 )2nabq

Now, under the assumption |q| < 1 and |d/abq| < 1 we may let n → ∞, while
appealing to Tannery’s theorem for being allowed to interchange limit and summation. Finally, we perform the substitution d 7→ a2 q/c and arrive at the following
bilateral quadratic summation formula:

NEW CURIOUS BILATERAL q-SERIES IDENTITIES

5

Proposition 3.1. Let a, b, c be indeterminates, let |q| < 1 and |a/bc| < 1. Then∞X
(1 − aq 3k ) (b; q)k(c; q 2 )k  a k
= 0.
(8)
(1 − a) (aq/c; q)k (aq 2 /b; q 2 )k bc
k=−∞Next, we apply Cauchy’s method of bilateralization to the following cubic summation formula due to Gasper [5, Equation (5.22), c → q −3n ]:nX(1 − aq 4k )k=0

(1 − a)(a, b; q)k (q/b; q)2k (a2 bq 3n , q −3n ; q 3 )k
qk
1+3n
(aq
, q 1−3n /ab; q)k (ab; q)2k (q 3 , aq 3 /b; q 3 )k=(aq; q)3n (ab2 ; q 3 )n.
(ab; q)3n (aq 3 /b; q 3 )n(9)

We replace n by 2n and then shift the index of summation by n such that the newsum runs from −n to n. We also replace a by aq −4n , and b by bq −n , respectively.
Then, under the assumption |q| < 1 and |1/ab2 | < 1 we let n → ∞, while appealing
to Tannery’s theorem for being allowed to interchange limit and summation. We
eventually arrive at the following bilateral cubic summation formula:
Proposition 3.2. Let a, b be indeterminates, let |q| < 1 and |1/ab2 | < 1. Thenk∞X
(a2 b; q 3 )k1
(1 − aq 4k ) (b; q)k
= 0.
(10)
(1 − a) (q/ab; q)k (aq 3 /b; q 3 )k ab2
k=−∞

It seems worth checking whether Cauchy’s method could be applied to other quadratic, cubic and quartic summation formulae appearing in the literature, leading
to other interesting bilateral summations.
References[1] W. N. Bailey, Series of hypergeometric type which are infinite in both directions, Quart. J.
Math. (Oxford) 7 (1936), 105–115.
[2] W. N. Bailey, On the basic bilateral hypergeometric series 2 ψ2 , Quart. J. Math. (Oxford) (2)1 (1950), 194–198.
[3] T. J. l’A. Bromwich, An introduction to the theory of infinite series, 2nd ed., Macmillan,
London, 1949.
[4] A.-L. Cauchy, Mémoire sur les fonctions dont plusieurs valeurs sont liées entre elles par uneéquation linéaire, et sur diverses transformations de produits composés d’un nombre indéfinide facteurs, C. R. Acad. Sci. Paris 17 (1843), 523; reprinted in Oeuvres de Cauchy, Ser. 1 8,
Gauthier-Villars, Paris (1893), 42–50.
[5] G. Gasper, Summation, transformation, and expansion formulas for bibasic series, Trans.
Amer. Math. Soc. 312 (1989), 257–277.
[6] G. Gasper and M. Rahman, Basic Hypergeometric Series, second edition, Encyclopedia of
Mathematics And Its Applications 96, Cambridge University Press, Cambridge, 2004.
[7] I. M. Gessel and D. Stanton, Application of q-Lagrange inversion to basic hypergeometricseries, Trans. Amer. Math. Soc. 277 (1983), 173-203.
[8] V. J. W. Guo and M. J. Schlosser, Curious extensions of Ramanujan’s 1 ψ1 summation formula,
J. Math. Anal. Appl. 334 (1) (2007), 393–403.
[9] C. G. J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum, Regiomonti. Sumptibus fratrum Bornträger, 1829; reprinted in Jacobi’s Gesammelte Werke, vol. 1, (Reimer,
Berlin, 1881–1891), pp. 49–239; reprinted by Chelsea (New York, 1969); now distributed by
the Amer. Math. Soc., Providence, RI.
[10] F. Jouhet and M. J. Schlosser, Another proof of Bailey’s 6 ψ6 summation, Aequationes Math.
70 (1–2) (2005), 43–50.
[11] M. J. Schlosser, Some new applications of matrix inversions in Ar , Ramanujan J. 3 (1999),
405–461.

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FRÉDÉRIC JOUHET AND MICHAEL J. SCHLOSSER[12] M. J. Schlosser, Abel–Rothe type generalizations of Jacobi’s triple product identity, in “Theory and Applications of Special Functions. A volume dedicated to Mizan Rahman” (M. E. H. Ismail and E. Koelink, eds.), Dev. Math. 13 (2005), 383–400.
[13] L. J. Slater, Generalized hypergeometric functions, Cambridge University Press, London/New
York, 1966.
Institut Camille Jordan, Université Claude Bernard Lyon 1, 69622 Villeurbanne
Cedex, France
E-mail address: jouhet@math.univ-lyon1.fr
URL: http://math.univ-lyon1.fr/~jouhet
Fakultät für Mathematik der Universität Wien, Nordbergstraße 15, A-1090 Wien,
AustriaE-mail address: michael.schlosser@univie.ac.at
URL: http://www.mat.univie.ac.at/~schlosse