Ramanujan’s 1 ψ1 summation
S. Ole Warnaar
Acknowledgements I thank Dick Askey, Bruce Berndt, Susanna Fishel, Jeff Lagarias and Michael
Schlosser for their helpful correspondence.
Notation. It is impossible to give an account of the 1 ψ1 summation without introducing some
q-series notation. To keep the presentation as simple as possible, weQassume that 0 < q < 1.
∞
Suppressing q-dependence, we define two q-shifted factorials: (a)∞ := k=0 (1 − aq k ) and (a)z :=
z
(a)∞ /(aq )∞ for z ∈ C. Note that 1/(q)n = 0 if n is a negative integer. For x ∈ C − {0, −1, . . . },
the q-gamma function is defined as Γq (x) := (q)x−1 /(1 − q)x−1 .
Ramanujan’s 1 ψ1 summation. Ramanujan recorded his now famous 1 ψ1 summation as item 17
of Chapter 16 in the second of his three notebooks [13, p. 32], [46]. It was brought to the attention
of the wider mathematical community in 1940 by Hardy, who included it in his twelfth and final
lecture on Ramanujan’s work [31]. Hardy remarked that the result constituted “a remarkable
formula with many parameters”. Instead of presenting the 1 ψ1 sum as given by Ramanujan and
Hardy, we will state its modern form:
∞
X (a)n n (az)∞ (q/az)∞ (b/a)∞ (q)∞
(1) z = , |b/a| < |z| < 1,
n=−∞
(b)n (z)∞ (b/az)∞ (q/a)∞ (b)∞
where it is understood that a, q/b 6∈ {q, q 2 , . . . }. Characteristically, Ramanujan did not provide
a proof of (1). Neither did Hardy, who however remarked that it could be “deduced from one
which is familiar and probably goes back to Euler”. The result to which Hardy was referring is
another
P∞ nfamous identity—known as the q-binomial theorem—corresponding to (1) with b = q:
n=0 z (a)n /(q)n = (az)∞ /(z)∞ and valid for |z| < 1. Although not actually due to Euler, the
q-binomial theorem is certainly classic. It seems to have appeared first and without proof (for
a = q −N ) in Rothe’s 1811 book “Systematisches Lehrbuch der Arithmetik ”, and in the 1840s many
mathematicians of note, such as Cauchy (1843), Eisenstein (1846), Heine (1847) and Jacobi (1847)
published proofs. The first proof of the 1 ψ1 sum is due to Hahn in 1949 [30] and, as hinted by Hardy,
uses the q-binomial theorem. After Hahn a large number of alternative proofs of (1) were found,
including one probabilistic and three combinatorial proofs [2,3,5,16–20,23,24,32,34,35,44,48,50,53].
The proof from the book, which again relies on the q-binomial theorem, was discovered by Ismail
[32] and is short enough to include here. Assuming |z| < 1 and |b| < min{1, |az|}, both sides
are analytic functions of b. Moreover, they coincide when b = q k+1 with k = 0, 1, 2, . . . by the
q-binomial theorem with a 7→ aq −k . Since 0 is the accumulation point of this sequence of b’s the
proof is done.
Apart from the q-binomial theorem, the 1 ψ1 sum generalises another classic identity, known as
n
n n (2)
P∞
the Jacobi triple-product identity: n=−∞ (−1) z q = (z)∞ (q/z)∞ (q)∞ =: θ(z). This result
plays a central role in the theory of theta and elliptic functions.
The 1 ψ1 sum as discrete beta integral. As pointed out by Askey [8, 9], the 1 ψ1 summation
may
R c·∞ be viewed as a discrete
P∞ analoguen of Euler’s beta integral. First define the Jackson or q-integral
n
0
f (t)dq t := (1 − q) n=−∞ f (cq )cq . Replacing (a, b, z) 7→ (−c, −cq α+β , q α ) in (1) then gives
Z c·∞
tα−1 θ(−cq α ) Γq (α)Γq (β)
(2) dq t = cα ,
0 (−t)α+β θ(−c) Γq (α + β)
where Re(α), Re(β) > 0. For real, positive c the limit q → 1 can be taken, resulting in the beta
integral modulo the substitution t 7→ t/(1 − t). Askey further noted in [8] that the specialisation
(α, β) 7→ (x, 1 − x) in (2) (so that 0 < Re(x) < 1) may be viewed as a q-analogue of Euler’s
reflection formula.
Simple applications of the 1 ψ1 sum. There are numerous easy applications of the 1 ψ1 sum. For
example, Jacobi’s well-known four- and six-square theorems as well as a number of similar results
1
2
readily follow from (1), see e.g., [1, 14, 15, 21, 22, 25]. To give a flavour of how the 1 ψ1 implies these
types of results we shall sketch a proof of the four-square theorem. Let rs (n) be P the number of
representations of n as the sum of s squares. The generating function Rs (q) := n≥0 rs (n)(−q)n
m m2 s
P∞ s
is given by m=−∞ (−1) q . By the triple-product identity this is also (q)∞ /(−q)∞ . Any
identity that allows the extraction of the coefficient of (−q)n results in an explicit formula for
rs (n). Back to (1), replace (b, z) 7→ (aq, b) and multiply both sides by (1 − b)/(1 − ab). By the
geometric series this yields
∞
(1 − a)(1 − b) X kn k n (abq)∞ (q/ab)∞ (q)2∞
(3) 1+ q (a b − a−k b−n ) = ,
1 − ab (aq)∞ (q/a)∞ (bq)∞ (q/b)∞
k,n=1
which may also be found in Kronecker’s 1881 paper “Zur Theorie der elliptischen Functionen”.
For a, b → −1 the right side gives R4 (q) whereas the left side becomes
∞
X ∞
X ∞
X X
1−8 qm n(−1)n+k = 1 + 8 (−q)m d.
m=1 n,k=1 m=1 d≥1
nk=m 4-d|m
P
Hence r4 (n) = 8 d≥1; 4-d|n d. This result of Jacobi implies Lagrange’s theorem that every positive
integer is a sum of four squares. By taking a, b2 → −1 in (3) the reader will have little trouble
showing that r2 (n) = 4(d1 (n) − d3 (n)), with dk (n) the number of divisors of n of the form 4m + k.
This is a result of Gauss and Lagrange which implies Fermat’s two-square theorem.
Other simple but important applications of the 1 ψ1 sum concern orthogonal polynomials. In [11]
it was employed by Askey and Wilson to compute a special case—corresponding to the continuous
q-Jacobi polynomials—of the Askey–Wilson integral, and in [10] Askey gave an elementary proof of
the full Askey–Wilson integral using the 1 ψ1 sum. The sum also implies the norm evaluation of the
weight functions of the q-Laguerre polynomials [45]. These are a family of orthogonal polynomials
with discrete measure µ on [0, c · ∞) given by dq µ(t) = tα /(−t)∞ dq t. The normalisation dq µ(t)
R
thus follows from the q-beta integral (2) in the limit of large β.
Generalisations in one dimension. There exist several generalisations of Ramanujan’s sum
containing one additional parameter. In his work on partial theta functions Andrews [4] obtained
a generalisation in which each product of four infinite products on the right-hand side is replaced
by six such products. Another example is the curious identity of Guo and Schlosser, which is no
longer hypergeometric in nature [27]:
∞
X (a)k (1 − ack q k )(ck q)∞ (b/ack )∞ k 1 (q)∞ (b/a)∞
ck = ,
(b)k (1 − azq k )(ac
k ∞
(q/ack ∞
(1 − z) (q/a)∞ (b)∞
k=−∞
where ck := z(1 − aczq k )/(1 − azq k ) and |b/ac| < |z| < 1. For c = 1 this is (1).
As discovered by Schlosser [49], a quite different extension of the 1 ψ1 sum arises by considering
non-commutative variables. Let R be a unital Banach algebra with identity 1, central Qn elements b
and q, and norm k · k. Write a−1 for the inverse of an invertible element a ∈ R. Let i=m ai stand
for 1 if n = m − 1, am · · · an if n ≥ m and a−1 −1
m−1 · · · an+1 if n < m − 1, and define
± Y Y r
a1 , . . . , ar i−1 i−1 −1
;z := z (1 − as q )(1 − bs q ) ,
b1 , . . . , b r k i s=1
Q Qk Q Q1
where k ∈ Z ∪ {∞}, a1 , . . . , ar , b1 , . . . , br ∈ R, i = i=1 in the + case and i = i=k in the −
case. Subject to max{kqk, kzk, kba−1 z −1 k} < 1, the following non-commutative 1 ψ1 sum holds:
∞ + − −1 −1 + −
bza−1 z −1 , q
X a za qa z
;z = ;1 −1 z −1
; 1 ; 1 .
b k z ∞ qza ∞ ba−1 z −1 , b ∞
k=−∞
3
Higher-dimensional generalisations. Various authors have generalised (1) to multiple 1 ψ1
sums. Below we state a generalisation due to Gustafson and Milne [28, 41] which is labelled by the
A-type root system. Similar such 1 ψ1 sums are given in [6, 7, 29, 43, 47]. More involved multiple
1 ψ1 sums with a Schur or Macdonald polynomial argument can be found in [12, 36, 42, 52]. For
r = (r1 , . . . , rn ) ∈ Zn denote |r| := r1 + · · · + rn . Then
n n
X Y xi q ri − xj q rj Y (aj xij )ri (az)∞ (q/az)∞ Y (bj xij /ai )∞ (qxij )∞
z |r| = ,
xi − xj (bj xij )ri (z)∞ (b/az)∞ i,j=1 (qxij /ai )∞ (bj xij )∞
r∈Zn 1≤i<j≤n i,j=1
where a := a1 · · · an , b := q 1−n b1 · · · bn , xij := xi /xj and |b/a| < |z| < 1. Milne first proved this
for b1 = · · · = bn [41] and shortly thereafter Gustafson established the full result [28]. We have
(1)
already seen that the 1 ψ1 sum implies the Jacobi triple-product identity. The latter is the A1
case of Macdonald’s generalised Weyl denominator identities for affine root systems [38]. To obtain
further Macdonald identities from the Gustafson–Milne sum one replaces z → z/a before letting
a1 , . . . , an → ∞ and b1 , . . . , bn → 0. Extracting the coefficient of z 0 (on the right this requires the
(1)
triple-product identity) results in the Macdonald identity for An−1 .
Higher-dimensional generalisations of a special case of the 1 ψ1 sum can be given for all affine
root systems. A full description is beyond this note, and we will only sketch the simplest case.
The reader is referred to [26, 38–40] for the full details. In [39] Macdonald gave the following
multivariable extension of the product formula for the Poincaré polynomial of a Coxeter group
X Y 1 − tα ew(α) Y 1 − tα tht(α)
(4) W (t) := = .
w∈W α∈R+
1 − ew(α) 1 − tht(α)
α∈R+
Here R is a reduced, irreducible finite root system in a Euclidean space V , R+ the set of positive
roots, W the Weyl group and tα for α ∈ R+ a set of formal variables constant along Weyl orbits.
(β,α)/kαk2
The symbol tht(α) stands for β∈R+ tβ
Q
with (·, ·) the W -invariant positive definite bilinear
ht(α)
form on V . If all tα are set to t then t = tht(α) with ht(α) the usual height function on R,
in which case W (t) reduces to the classical Poincaré polynomial W (t). Now let S be a reduced,
irreducible affine root system of type S = S(R) [38]. In analogy with the finite case, assume that
ta for a ∈ S is constant along orbits of the affine Weyl group W of S. Then Macdonald generalised
(4) to [40]
X Y 1 − ta ew(a) Y (tα tht(α) )∞ (tht(α) q χ(α∈B) /tα )∞
(5) w(a)
= ,
w∈W +
1−e
a∈S + α∈R
(tht(α) )2∞
Q
where B is a base for R. The parameter q on the right is fixed by q = a∈B(S) exp(na a), where
B(S) is a basis for S and the na are the labels of the extended Dynkin diagrams given in [38]. If
(1)
R is simply-laced then ta = t. In the case of S(R) = A1 , q = exp(a0 + a1 ) so that after replacing
exp(a1 ) by x we obtain the 1 ψ1 sum (1) with (a, b, z) → (x/t, tx, t). This is not the end of the
story concerning root systems and the 1 ψ1 sum. Identity (5) can be rewritten as [40]
X Y (q eα )(α,γ) Y (tα tht(α) q)∞ (tht(α) q χ(α∈B) /tα )∞ (q eα )∞ (q e−α )∞
(6) = ,
∨
(tα q eα )(α,γ) + (tht(α) q)∞ (tht(α) )∞ (tα q eα )∞ (tα q e−α )∞
γ∈Q α∈R α∈R
where Q∨ is the coroot lattice. Interestingly, for tα = t this was also found by Fishel, Grojnowski
and Teleman [26] by computing the generating function of the q-weightedPn Euler characteristics
of certain Dolbeault cohomologies. For R = An−1 , Q∨ = Q = i=1 ri i with |r| = 0, R =
{i − j : 1 ≤ i 6= j ≤ n} and tht(i −j ) = tj−i . By fairly elementary manipulations the identity
4
(6) may then be transformed into the multiple 1 ψ1 sum
X (a)|r| Y xi q ri − xj q rj (t−1 xij )ri −rj ri −rj −rj
z |r| t q
(b)|r| xi − xj (tqxij )ri −rj
r∈Zn 1≤i<j≤n
n−1 n
(az)∞ (q/az)∞ (b/a)∞ (tq)∞ Y (ti+1 q)∞ Y (qxij )∞
= ,
(z)∞ (b/az)∞ (q/a)∞ (b)∞ i=1 (ti )∞ i,j=1 (tqxij )∞
for |b/a| < |z| < 1 and |t| < 1. This is the only result in this survey that is new.
We finally remark that all higher-dimensional 1 ψ1 sums admit representations as discrete Selberg-
type integrals. The most important such integrals are due to Aomoto [6, 7] and Ito [33], and are
closely related to (5). Further examples may be found in [37, 51].
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School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia
Work supported by the Australian Research Council
