On Some Ramanujan Identities for the Ratios of Eta-Functions
We give direct proofs of some of Ramanujan’s P-Q modular equations based on simply proved elementary identities from Chapter 16 of his Second Notebook.
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| author | Bhargava, S. Rajanna, K. R. Vasuki, K. R. Бхаргава, С. Райана, К. Р. Васюкі, К. Р. |
| author_facet | Bhargava, S. Rajanna, K. R. Vasuki, K. R. Бхаргава, С. Райана, К. Р. Васюкі, К. Р. |
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| description | We give direct proofs of some of Ramanujan’s P-Q modular equations based on simply proved elementary identities from Chapter 16 of his Second Notebook. |
| first_indexed | 2026-03-24T02:20:30Z |
| format | Article |
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UDC 511.19
S. Bhargava, K. R. Vasuki (Univ. Mysore, India),
K. R. Rajanna (MVJ College Engineering, Bangalore, India)
ON SOME RAMANUJAN IDENTITIES FOR THE RATIOS OF ETA-FUNCTIONS
ПРО ДЕЯКI ТОТОЖНОСТI РАМАНУДЖАНА
ДЛЯ ВIДНОШЕНЬ ЕТА-ФУНКЦIЙ
The purpose of this paper is to provide direct proofs of some of Ramanujan’s P-Q modular equations based on simply
proved elementary identities of Chapter 16 of his Second Notebook.
Наведено прямi доведення деяких P-Q модульних рiвнянь Рамануджана на пiдставi елементарних тотожностей з
глави 16 його Другого зошита, що просто доводяться.
1. Introduction. In the unorganized pages of his second notebook [11], Ramanujan recorded 23
identities involving ratios of Dedekind’s eta-function all of which have been proved by B. C. Berndt
and L.-C. Zhang [7] by employing Ramanujan’s modular identities of various degrees, or via his
mixed modular equations, or via the theory of modular forms. Similar 14 identities involving ratios
of Dedekind’s eta-function found on page 55 of Ramanujan’s lost notebook [12] have been proved
by Berndt [6] employing the above methods.
The purpose of this paper, consistent with Berndt’s often made call for continued efforts to
discern Ramanujan’s thinking (see, for example, his book [5, p. 1]), is to demonstrate amenability of
10 of the above mentioned identities proved in [3, 4] via modular and mixed modular equations, to
more direct proof based on simply proved identities of Chapter 16 of Ramanujan’s second notebook
[11], including his celebrated, so called, “remarkable identity with several parameters” [9], or “1ψ1
summation”. In the remainder of this section, we find it convenient, for our later use, to give a brief
account of relevant definitions and results of Chapter 16 of the second notebook [11] as well as some
results easily deducible there from. Significantly, the 1ψ1 summation, stated below as (1.1), is not
only the first of the entries (Entry 17, Chapter 16, Second Notebook [11]) with which Ramanujan
begins his development of classical theory of theta and elliptic functions but also a very important
tool all through his work in the classical theory as well as his own alternative theories:
1 +
∞∑
n=1
(1/α; q2)n(−αq)n
(βq2; q2)n
zn +
∞∑
n=1
(1/β; q2)n(−βq)n
(αq2; q2)n
z−n =
=
(−qz; q2)∞(−q/z; q2)∞(q2; q2)∞(αβq2; q2)∞
(−αqz; q2)∞(−βq/z; q2)∞(αq2; q2)∞(βq2; q2)∞
, (1.1)
where |q| < 1, |βq| < |z| < 1/|αq| and, as is customary,
(a; q)n :=
n−1∏
k=0
(1− aqk)
c© S. BHARGAVA K. R. VASUKI, K. R. RAJANNA, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8 1011
1012 S. BHARGAVA K. R. VASUKI, K. R. RAJANNA
and
(a; q)∞ :=
∞∏
k=0
(1− aqk).
For instance, (1.1) contains as its special cases the well known Jacobi’s triple product identity and
the q-binomial theorem of Euler and Cauchy which are ubiquitous in theory of numbers and in
theory of special functions. More over, K. Venkatachaliengar [13] has given an elementary and self
contained proof of (1.1) giving it its pride of place in literature above its aforementioned special cases.
Venkatachalingar’s novel proof consists in first making the simple observation that the product side,
say f(z), satisfies a certain functional relationship which then readily yields a recurrence relation for
the coefficients in the power series expansion of f(z) in neighbourhood of z = 0. The recurrence
relation, in turn, gives all the coefficients except the constant term. Lastly, the constant term is
determined by an application of Abel’s theorem. Ramanujan’s equivalent of Jacobi’s theta function is
f(a, b) :=
∞∑
n=−∞
an(n+1)/2bn(n−1)/2, |ab| < 1. (1.2)
Note that Jacobi’s θ3(q, z)is same as f(qz, q/z) and that Jacob’s triple product identity
f(qz, q/z) =
∞∑
n=−∞
qn
2
= (−qz; q2)∞(−q/z; q2)∞(q2; q2)∞, (1.3)
is the special case α = 0 = β of (1.1). Also, the Euler – Cauchy q-binomial theorem, in the form
(−qz; q2)∞
(−αqz; q2)∞
= 1 +
∞∑
n=1
(1/α; q2)
(q2; q2)∞
(−αqz)n (1.4)
is the special case β = 1 of (1.1). All through his work Ramanujan employs the following restrictions
ϕ(q) , ψ(q) and f(−q) of (1.2):
ϕ(−q) := f(−q,−q) = 1 + 2
∞∑
k=1
(−1)kqk2 =
(q; q)∞
(−q; q)∞
, (1.5)
ψ(q) := f(q, q3) =
∞∑
k=0
qk(k+1)/2 =
(q2; q2)∞
(q; q2)∞
, (1.6)
and
f(−q) := f(−q,−q2) =
∞∑
n=−∞
(−1)nqn(3n−1)/2 = (q; q)∞. (1.7)
He also employs the functions
χ(q) := (−q; q2)∞ (1.8)
and
v(q) = q1/3
χ(−q)
χ3(−q3)
. (1.9)
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ON SOME RAMANUJAN IDENTITIES FOR THE RATIOS OF ETA-FUNCTIONS 1013
The series representations in (1.5) – (1.7) including the first equation in (1.6) follow by simple
manipulations of the terms in the respective defining series. Similarly, the product forms in (1.5) –
(1.7) are obtained on employing (1.3) followed by manipulations of the factors involved. A simple
but often used identity obtained by such manipulation of factors is due to Euler, namely:
(−q; q)∞ =
1
(q; q2)∞
. (1.10)
We find it convenient to gather in the the following Lemmas some of the elementary results of
Chapters 16 – 20 in Ramanujan’s Second Notebook [11] and briefly sketch their proofs. The proofs
are elementary and follow from some simple manipulations of series and products. One may see
C. Adiga’s doctoral thesis [1] for many of these proofs in the spirit of Ramanujan. Though Lemmas 1.2
and 1.3 are more general than we need, we feel that it is desirable to record them here because they
seem new and because the proofs are as elementary as those of their special cases.
Lemma 1.1. If |q| < 1 and |ab| < 1, then
(i) ϕ(−q) = f2(−q)
f(−q2)
, ψ(q) =
f2(−q2)
f(−q)
, χ(−q) = f(−q)
f(−q2)
,
(1.11)
ϕ(q)ψ(q2) = ψ2(q), ϕ(q)ϕ(−q) = ϕ2(−q2);
(ii) ψ(q) = f(q3, q6) + qψ(q9) and ϕ(q) = 2qf(q3, q15) + ϕ(q9); (1.12)
(iii) 1 +
1
v(q)
=
ψ(q1/3)
q1/3ψ(q3)
and 1− 2v(q) =
ϕ(−q1/3)
ϕ(−q3)
; (1.13)
(iv) 1 +
1
v3(q)
=
ψ4(q)
qψ4(q3)
or v3(q) =
χ3(−q)ψ4(q)
χ9(−q3)ψ4(q3)
− 1, (1.14)
1− 8v3(q) =
ϕ4(−q)
ϕ4(−q3)
or
1
v3(q)
=
χ9(−q3)ϕ4(−q)
qχ3(−q)ϕ4(−q3)
+ 8; (1.15)
(v) f(a, b) = af(a(ab), b(ab)−1); (1.16)
(vi) f(a, ab2)f(b, a2b) = f(a, b)ψ(ab),
(1.17)
f(a, b)f(−a,−b) = f2(−a2,−b2)ϕ(−ab);
(vii) f2(a, b)− f2(−a,−b) = 4af
(
b
a
,
a
b
a2b2
)
ψ(a2b2); (1.18)
(viii) if ab = cd, then
f(a, b)f(c, d) + f(−a,−b)f(−c,−d) = 2f(ac, bd)f(ad, bc) (1.19)
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
1014 S. BHARGAVA K. R. VASUKI, K. R. RAJANNA
and
f(a, b)f(c, d)− f(−a,−b)f(−c,−d) = 2af
(
b
c
,
c
b
abcd
)
f
(
b
d
,
d
b
abcd
)
. (1.20)
Proof. The identities in (1.11) and (1.16) – (1.20) merely follow from (1.5) – (1.8) on simple
manipulations of the series or products involved. For instance, the factors in the product rep-
resentation for ϕ(−q) in (1.5) can be regrouped as (q; q2)∞(q2; q2)∞/(−q; q)∞ which becomes
(q2; q2)∞/(q; q)∞(−q; q)∞ on using (1.10). Recombining the factors in the denominator and em-
ploying the product representation in (1.7) twice we have the first of (1.11). Identity (1.16) is the well
known quasiperiodicity. To simply establish it, we have, from (1.2), the right-hand side of (1.16) to
be equal to
a
∞∑
−∞
{a(ab)}n(n+1)/2{b(ab)}n(n−1)/2
or, what is the same
∞∑
−∞
a(n+1)(n+2)/2bn(n+1)/2,
which is f(a, b), as per (1.2) again, with n + 1 change to n. The identity (1.17) is true, since its
right-hand side can be written as, with p = ab,
(−a; p)∞(−b; p)∞(p; p)∞(p2; p2)∞/(p; p
2)∞
or, what is the same, on regrouping various factors,
(−a; p2)∞(−ap; p2)∞(−b; p2)∞(−bp; p2)∞(p2; p2)2∞
or
f(a, bp)f(b, ap).
But this is the same as the left-hand side of (1.17). For (1.18), we need realize firstly that, by
virtue of (1.2), the left-hand side equals
∑∞
−∞
∑∞
−∞
(ab)(m
2+n2)/2(a/b)(m+n)/2 with m+n = odd.
Transforming the indices to (s, t) by means of m+n = 2s+1,m−n = 2t+1, the double sum can be
rewritten as the product 2a
∑∞
−∞
(ab)s
2
(a)2s
∑∞
−∞
(ab)t
2
(ab)t or, 2af
(
b
a
,
a
b
a2b2
)
f(1, a2b2). But
this is the same as right-hand side of (1.18). Proofs of (1.19) and (1.20) follow in the same vein. For
instance, with p = ab = cd, the left-hand side of (1.19) equals 2
∑∞
−∞
p
m2+n2
2
(
a
b
)m
2
(
c
d
)n
2
, with
m+ n =even. Transforming (m,n) to (s, t) by means of m+ n = 2s and m− n = 2t, the double
sum becomes the product 2
∑∞
−∞
ps
2
(
ac
bd
) s
2 ∑∞
−∞
pt
2
(
ad
bc
) t
2
, or 2f(ac, bd)f(ad, bc). But this is
the right-hand side of (1.19).
Proofs of (1.12) – (1.15) are slightly different, but equally simple. For the first of (1.12), we may
start with the defining series 2ψ(q) = f(1, q) =
∑∞
−∞
q
n(n+1)
2 and regroup the terms according as
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
ON SOME RAMANUJAN IDENTITIES FOR THE RATIOS OF ETA-FUNCTIONS 1015
n ≡ 0,+1,−1 (mod 3). Similarly for the other identity of (1.12), we need only regroup the terms in
the series ϕ(q) =
∑∞
−∞
qn
2
according as n ≡ 0,+1,−1 (mod 3).
Manipulating the factors appearing in the right-hand side of the definition (1.9) and employing the
product forms appearing in (1.5) – (1.7) and the first of (1.12) (respectively the second of (1.12)) one
gets the first of (1.13) (respectively the second). Now, following Berndt [3, p. 346], since changing
q1/3 to q1/3ω, q1/3ω2 changes v(q) respectively to ω2v(q) and ωv(q), as is clear from (1.8) and (1.9),
we have from (1.13)
1 +
1
v3
=
(
1 +
1
v
)(
1 +
ω
v
)(
1 +
ω2
v3
)
=
ψ(q1/3)ψ(q1/3ω)ψ(q1/3ω2)
q3ω(q3)
.
This reduces to the first of (1.14), on repeated application of (1.6). Similarly, the second of (1.13)
and (1.5) yield the first of (1.15). That the second form of (1.14) is equivalent to the first form can
be seen by employing the definition (1.9) to eliminate the χ-ratio involved. The second of (1.15)
follows similarly from the first.
Following lemma seems new to literature and simply follows from a few applications of (1.1)
and series and product manipulations. It contains Entry 10 (iv) of Chapter 19 of [11] as special case
(z = 1).
Lemma 1.2.
ϕ2(−q2) f(qz, q/z)
f(−qz,−q/z)
− ϕ2(−q10) f(q5z, q5/z)
f(−q5z,−q5/z)
= 2qϕ2(−q10)f(−q2,−q18)×
×
[
zf(−q14z2,−q6/z2)
f(−qz,−q9/z)f(−q3z,−q7/z)
+
z−1f(−q6z2,−q14/z2)
f(−q9z,−q/z)f(−q7z,−q3/z)
]
. (1.21)
Proof. Let
P (z, q) := ϕ2(−q2) f(qz, q/z)
f(−qz,−q/z)
(1.22)
which is the right-hand side of (1.1) when α = β = −1, and let
P ∗(z, q) := ϕ2(−q10)2qzf(−q
2,−q18)f(−q14z2,−q6/z2)
f(−qz,−q9/z)f(−q3z,−q7/z)
. (1.23)
Then (1.21) is the same as
P (z, q)− P (z, q5) = P ∗(z, q) + P ∗(z−1, q). (1.24)
On putting α = β = −1 in (1.1), we have
P (z, q) = S(z, q) := 1 + 2
∞∑
1
qnzn
1 + q2n
+ 2
∞∑
1
qn/zn
1 + q2n
. (1.25)
Converting each series in this into double series by expanding each of the summands by geometric
series, interchanging the order of summation and summing the inner geometric series, we obtain
P (z, q) = S(z, q) = S∗(z, q) := 1 + 2
∞∑
0
(−1)nzq2n+1
1− zq2n+1
+ 2
∞∑
0
(−1)nz−1q2n+1
1− z−1q2n+1
. (1.26)
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
1016 S. BHARGAVA K. R. VASUKI, K. R. RAJANNA
Repeating the same procedure on each of the series obtained by grouping the terms in (1.26) according
as n ≡ 0, 1, 2, 3, 4 (mod 5) we get
P (z, q)− P (z, q5) = S∗(z, q)− S∗(z, q5) =
=
∞∑
1
(qn + q9n)(zn + z−n)
1 + q10n
−
∞∑
1
(q3n + q7n)(zn + z−n)
1 + q10n
=
=
[
S(q−4z, q5)− S(q−2z, q5)
]
−
[
S(q−4z−1, q5)− S(q−2z−1, q5)
]
. (1.27)
Now, from (1.25) and (1.22),
S(q−4z, q5)− S(q−2z, q5) = P (zq−4, q5)− P (q−2z, q5) =
= ϕ2(−q10)
[
f(qz, q9/z)
f(−qz,−q9/z)
− f(q3z, q7/z)
f(−q3z,−q7/z)
]
= P ∗(z, q), (1.28)
on using the identity (1.20) with a = qz, b = q9z, c = −q3z and d = −q7/z; P ∗ being as in (1.23).
Now substituting (1.28) in (1.27) we have (1.24) which is the same as (1.21) by virtue of (1.22)
and (1.23).
Corollary 1.1 (Entry 10 (iv), Chapter 19, [11]).
ϕ2(q)− ϕ2(q5) = 4qf(q, q9)f(q3, q7). (1.29)
Proof. Putting z = 1 in (1.21), we obtain
ϕ2(−q2)ϕ(q)
ϕ(−q)
− ϕ2(−q10)ϕ(q5)
ϕ(−q5)
=
4qϕ2(−q10)f(−q2,−q18)f(−q6,−q14)
f(−q,−q9)f(−q3,−q7)
.
This reduces to (1.29) on applying the last of the identities in (1.11) twice and also the second of
(1.17) twice.
Following lemma also seems new to literature and follows from few applications of (1.1). Its
proof is similar to that of pervious lemma but slightly more involved. It contains Entry 10 (v) of
Chapter 19 of [11] as special case (z = 1).
Lemma 1.3.
ψ2(q)
f(z, q2/z)
f(qz, q/z)
− q2ψ2(q5)
f(z, q10/z)
f(q5z, q5/z)
= ψ2(q5)f(q, q4)×
×
[
f(q2/z, q3z)
f(q/z, q9z)f(q3/z, q7z)
+
zf(q2z, q3/z)
f(qz, q9/z)f(q3z, q7/z)
]
. (1.30)
Proof. Let
P̂ (q, z) := ψ2(q)
f(z, q2/z)
f(qz, q/z)
, (1.31)
which is indeed (1 − q)−1 times the right-hand side of (1.1) with α = q = 1/β and z changed to
z/q, and let
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ON SOME RAMANUJAN IDENTITIES FOR THE RATIOS OF ETA-FUNCTIONS 1017
P̂ ∗(z, q) :=
ψ2(q5)f(q, q4)f(q2/z, q3z)
f(q/z, q9z)f(q3/z, q7z)
. (1.32)
Then (1.30) is the same as
P̂ (z, q)− q2P̂ (z, q5) = P̂ ∗(z, q) + zP̂ ∗(1/z, q). (1.33)
On changing z to z/q and putting α = q, β = 1/q in (1.1) and then dividing throughout by (1− q),
we have
P̂ (z, q) = Ŝ(z, q) :=
∞∑
0
(−1)nqnzn
1− q2n+1
+
∞∑
0
(−1)nqn/zn
1− q2n+1
. (1.34)
Converting each series in this into double series by expanding the summands by geometric series,
interchanging the order of summation and summing the inner sums, we obtain
P̂ (z, q) = Ŝ(z, q) = Ŝ∗(z, q) :=
∞∑
0
zqn
1 + zq2n+1
+
∞∑
0
qn
1 + z−1q2n+1
. (1.35)
Repeating the same procedure on each of the series obtained by regrouping the terms in (1.35)
according as n ≡ 0, 1, 2, 3, 4 (mod 5) we get
P̂ (z, q)− q2P̂ (z, q5) = Ŝ∗(z, q)− q2Ŝ∗(z, q5) =
=
∞∑
0
(−1)n(qn + q9n+4)(zn+1 + z−n)
1− q10n+5
+
∞∑
0
(−1)n(q3n+1 + q7n+3)(zn+1 + z−n)
1− q10n+5
(1.36)
Now, we have from (1.25) and (1.22)
P (iz, iq)− P (iz−1, iq) = S(iz, iq)− S(iz−1, iq),
or
ϕ2(q2)
[
f(−qz, q/z)
f(qz,−q/z)
− f(qz,−q/z)
f(−qz, q/z)
]
= 4
∞∑
0
q2n+1(z−2n−1 − z2n+1)
1− q2(2n+1)
,
or
ϕ(q2)ψ(q4)
f(−1/z2,−z2q4)
f(−q2z2,−q2/z2)
=
∞∑
0
q2n(z2n − z2(n+1))
1− q2(2n+1)
,
or, changing q2 to q5 and z2 to −z−1, and using fourth of the equations in (1.11)
ψ2(q5)
f(z, q10/z)
f(q5z, q5/z)
=
∞∑
0
(−1)nq5n(zn+1 + z−n)
1− q10n+5
.
This gives, on changing z to q−4z,
ψ2(q5)
f(q−4z, q14/z)
f(qz, q9/z)
=
∞∑
0
(−1)n(qnzn+1 + q9n+1z−n)
1− q10n+5
.
In turn, this yields, on changing z to z−1 and then multiplying throughout by z,
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
1018 S. BHARGAVA K. R. VASUKI, K. R. RAJANNA
ψ2(q5)
zf(q−4z−1, q14z)
f(qz−1, q9z)
=
∞∑
0
(−1)n(qnz−n + q9n+4zn+1)
1− q10n+5
.
Similarly, we have
q3ψ2(q5)
f(q−2z, q12/z)
f(q3z, q7/z)
=
∞∑
0
(−1)n(q3n+1zn+1 + q7n+3z−n)
1− q10n+5
,
q3ψ2(q5)
zf(q−2z−1, q12z)
f(q3z−1, q7z)
=
∞∑
0
(−1)n(q3n+1z−n + q7n+3zn+1)
1− q10n+5
.
Adding the last four identities and making four applications of (1.16) to rewrite q4f(q−4z, q14/z),
q4f(q−4/z, q14z), q3f(q−2z, q12/z), and q3f(q−2/z, q12z) respectively as zf(q6z, q4/z), z−1f(q6/z,
q4z), qzf(q8z, q2/z) and qz−1f(q8/z, q2z), we get
ψ2(q5)
[
zf(q6z, q4/z)
f(qz, q9/z)
+
f(q6/z, q4z)
f(q/z, q9z)
+
qzf(q8z, q2/z)
f(q3z, q7/z)
+
qf(q8/z, q2z)
f(q3/z, q7z)
]
=
=
∞∑
0
(qn + q9n+4 + q3n+1 + q7n+3)(zn+1 + z−n)
1− q10n+5
. (1.37)
or, on using the sum of (1.19) twice with (a, b, c, d) = (q, q4, q2/z, q3z) and (a, b, c, d) = (q, q4,
q2z, q3/z), we obtain
ψ2(q5)f(q, q4)
[
f(q2/z, q3z)
f(q/z, q9z)f(q3/z, q7z)
+
zf(q2z, q3/z)
f(qz, q9/z)f(q3z, q7/z)
]
=
=
∞∑
0
(qn + q9n+4 + q3n+1 + q7n+3)(zn+1 + z−n)
1− q10n+5
.
Using this in (1.36) gives (1.33) or what is the same (1.30).
Corollary 1.2 (Entry 10 (v), Chapter 19, [11]).
ψ2(q)− qψ2(q5) = f(q, q4)f(q2, q3). (1.38)
Proof. Putting z = 1 in (1.30), we have, on employing the fourth of the equations in (1.11) and
(1.3) several times and (1.6),
ψ2(q2)− q2ψ2(q10) =
ψ2(q5)f(q, q4)f(q2, q3)
f(q, q9)f(q3, q7)
=
=
[
(q10; q10)2∞(q5; q5)2∞
(q5; q10)2∞(q10; q10)2∞
][
(−q; q5)∞(−q4; q5)∞(−q2; q5)∞(−q3; q5)∞
(−q; q10)∞(−q9; q10)∞(−q3; q10)∞(−q7; q10)∞
]
=
= (q10; q10)2∞
(−q; q)∞
(q5; q5)∞
×
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ON SOME RAMANUJAN IDENTITIES FOR THE RATIOS OF ETA-FUNCTIONS 1019
×
[
(−q2; q10)∞(−q4; q10)∞(−q5; q10)∞(−q6; q10)∞(−q8; q10)∞(−q10; q10)∞
(−q; q)∞
]
=
= (q10; q10)2∞(−q2; q10)∞(−q4; q10)∞(−q6; q10)∞(−q8; q10)∞ =
= f(q2, q8)f(q4, q6).
Changing q2 to q in this we have (1.38).
The first two identities in the following lemma are due to Ramanujan (Chapter 19, [11] and [3])
which we obtain as special cases of our general results in Lemmas 1.2 and 1.3. The other two results
are due to S.-Y. Kang [10] and our proofs are slightly different from hers. In what follows we also
employ repeatedly some additional notations, for brevity, without further mention.
Lemma 1.4. Let
λn := ϕ2(−qn), µn := ψ2(qn), sn := χ(−qn) and tn := f(−qn). (1.39)
Then
λ1 − λ5 = −4qf(−q,−q9)f(−q3,−q7) = −4q
s1
s5
t210 = −4qs1s5µ5, (1.40)
µ1 − qµ5 = f(q, q4)f(q2, q3) =
s5
s1
t25 =
λ5
s1s5
, (1.41)
λ1 − 5λ5 = −
4s5
s1
t22 = −4s1s5µ1 (1.42)
and
µ1 − 5qµ5 =
s1
s5
t22 =
λ1
s1s5
. (1.43)
Proof. Identities in (1.40) follow from (1.21) on putting z = 1, changing q to −q, employing
(1.11), (1.3) repeatedly and suitably manipulating the factors involved. Similarly, (1.41) follows
from (1.30).
Identities in (1.42) follow by simply eliminating µ5 between (1.40) and (1.41) and then by
manipulating factors involved. Similarly, identities in (1.43) follow by eliminating λ5 between (1.40)
and (1.41).
The following lemma seems new.
Lemma 1.5. We have
λ1 − 5λ25 = −4(s1s5µ1 + 5q5s5s25µ5) (1.44)
and
µ1 − 5q6µ25 =
λ1
s1s5
+
5qλ25
s5s25
. (1.45)
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1020 S. BHARGAVA K. R. VASUKI, K. R. RAJANNA
Proof. Changing q to q5 in (1.40) we get λ5 − λ25 = −4q5s5s25µ25. Adding this to (1.40) and
using (1.43) we obtain
λ1 − λ25 = −4q5s5s25µ25 −
4
5
s1s5
(
µ1 −
λ1
s1s5
)
.
But this reduces to (1.44). Proof of (1.45) is similar. We need change q to q5 in (1.41) and add the
resulting identity to (1.41) and lastly use (1.42).
2. Main results. Theorem 2.1 below establishes four of Ramanujan’s P -Q identities easily from
the v-identities (1.13), (1.14) and (1.15).
Theorem 2.1. In the notations of (1.39) of Lemma 1.4 we have the following:
(i) ([11, p. 327], [4, p. 204], Entry 51). Let
P :=
t21
q1/6t23
and Q :=
t22
q1/3t26
.
Then
PQ+
9
PQ
=
(
Q
P
)3
+
(
P
Q
)3
. (2.1)
(ii) ([11, p. 327], [4, p. 205], Entry 52). Let
P :=
t2
q1/24t3
and Q :=
t1
q5/24t6
.
Then
(PQ)2 − 9
(PQ)2
=
(
Q
P
)3
− 8
(
P
Q
)3
. (2.2)
(iii) (Equivalent of Entry 5 (xii) of [11, p. 231] and [3, p. 230]). Let
P :=
t1
q1/24t2
and Q :=
t3
q1/8t6
.
Then
(PQ)3 +
8
(PQ)3
=
(
Q
P
)6
−
(
P
Q
)6
. (2.3)
(iv) ([11, p. 327], [4, p. 210], Entry 56). If
P :=
t1
q1/3t9
and Q :=
t2
q2/3t18
,
then
P 3 +Q3 = P 2Q2 + 3PQ. (2.4)
Proof. (i) Eliminating v between the first of (1.14) and the first of (1.15) we have,(
µ21
9µ23
− 1
)(
1− λ21
λ23
)
= 8,
or
λ21µ
2
1
qλ23µ
2
3
+ 9 =
µ21
qµ23
+
λ21
λ23
,
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ON SOME RAMANUJAN IDENTITIES FOR THE RATIOS OF ETA-FUNCTIONS 1021
or, identically,
(PQ)2 + 9 = (PQ)
(
Q
P
)3
+ (PQ)
(
P
Q
)3
,
on routinely employing the definitions of P, Q, λ1, µ1, λ3 and µ3 and (1.11) and (1.39). This, on
dividing throughout by PQ, gives (2.1).
(ii) Eliminating v between the second of (1.14) and the second of (1.15) we get(
s31µ
2
1
s93µ
2
3
− 1
)(
s93λ
2
1
qs31λ
2
3
+ 8
)
= 1,
or
1
q
(
λ1µ1
λ3µ3
)2
− 9 =
s93λ
2
1
qs31λ
2
3
− 8
s31µ
2
1
s93µ
2
3
,
or, identically,
(PQ)4 − 9 = (PQ)2
(
Q
P
)3
− 8(PQ)2
(
P
Q
)3
,
on routinely employing the definitions of P, Q, λ1, µ1, λ3 and µ3 and (1.11) and (1.39). This
becomes (2.2) on dividing throughout by (PQ)2.
(iii) From the first of (1.14) and the first of (1.15), we immediately have(
1 +
1
v3
)
ϕ4(−q)
ϕ4(−q3)
− (1− 8v3)
ψ4(q)
qψ2(q3)
= 0,
or, on using the definition (1.9) of v,
s93λ
2
1
qs31λ
2
3
+ 8
s31µ
2
1
s93µ
2
3
=
µ21
qµ23
− λ21
λ23
,
or
s183 λ
2
1µ
2
3
qs61λ
2
3µ
2
1
+ 8 =
s93
qs31
− s93λ
2
1µ
2
3
s31λ
2
3µ
2
1
,
or, identically,
(PQ)6 + 8 = (PQ)3
(
P
Q
)3
− (PQ)3
(
P
Q
)6
,
on using the definitions P, Q, λ1, µ1, λ3 and µ3 and (1.11) and (1.39).
(iv) Eliminating v(q3) from the two equations in (1.13) and expanding we obtain
ψ(q)ϕ(−q)
qψ(q9)ϕ(−q9)
+ 3 =
ψ(q)
qψ(q9)
− ϕ(−q)
ϕ(−q9
,
or, identically,
PQ+ 3 =
1
(PQ)
P 2 +
1
(PQ)
Q2,
on using the definitions of P and Q and (1.11). This becomes (2.4) on multiplying throughout
by (PQ).
Theorem 2.1 is proved.
The following corollary is needed later in the proof of Theorem 2.4.
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1022 S. BHARGAVA K. R. VASUKI, K. R. RAJANNA
Corollary 2.1. If
u :=
q1/3t1t15
t3t5
and v :=
q2/3t2t30
t6t10
,
x :=
(v
u
)3
+
(u
v
)3
and w := uv +
1
uv
,
Bn :=
t2nt3n
qn/12tnt6n
and Dn :=
tnt3n
qn/6t2nt6n
,
y1 := (B1B5)
2 +
(
1
B1B5
)2
and y2 := (D1D5)
2 +
(
4
D1D5
)2
,
then
(y31 − 3y1 − 9w2 + 18)2 = (y32 − 48y2 + 128).
Proof. Setting
Cn :=
tnt2n
qn/4t3nt6n
, y := (C1C5)
2 +
(
9
C1C5
)2
, y4 := (D1D5)
3 +
(
4
D1D5
)3
and
y5 :=
(
D1
D5
)3
+
(
D5
D1
)3
,
we can write (2.1) as
C2
1 +
9
C2
1
= B6
1 +
1
B6
1
,
from which follows, on changing q to q5,
C2
5 +
9
C2
5
= B6
5 +
1
B6
5
.
Multiplying the last two equations, we obtain
y3 = y31 − 3y1 − 9w2 + x2 + 16, (2.5)
since, as can be easily shown,
x =
(
B1
B5
)3
+
(
B5
B1
)3
and w =
C1
C5
+
C5
C1
.
Similarly (2.2) and (2.3) respectively yield
y4 − 8y5 = y3 − 9w2 + 18 (2.6)
and
y4 + 8y5 = y31 − 3y1 − x2 + 2. (2.7)
Adding (2.5), (2.6) and (2.7), we have
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ON SOME RAMANUJAN IDENTITIES FOR THE RATIOS OF ETA-FUNCTIONS 1023
y4 = y31 − 3y1 − 9w2 + 18.
From our definitions of y2 and y4 we get
y24 = y32 − 48y2 + 128.
Eliminating y4 between the last two equations, we have the required result.
The following theorem establishes some P −Q identities of Ramanujan that simply follow from
the results of Ramanujan [12] and Kang [10], which we have recollected in Lemma 1.4.
Theorem 2.2. In the notations (1.39) we have the following:
(i) ([11, p. 325], [4, p. 206], Entry 53). Let
P :=
t1
q1/6t5
and Q :=
t2
q1/3t10
.
Then
PQ+
5
PQ
=
(
Q
P
)3
+
(
P
Q
)3
. (2.8)
(ii) ([12, p. 55], [6]). Let
P :=
t1
q1/24t2
and Q :=
t5
q5/24t10
.
Then
(PQ)2 +
4
(PQ)2
=
(
Q
P
)3
−
(
P
Q
)3
. (2.9)
(iii) ([11, p. 327], [4, p. 207], Entry 54). Let
P :=
t2
q1/8t5
and Q :=
t1
q3/8t10
.
Then
PQ− 5
PQ
=
(
Q
P
)2
− 4
(
P
Q
)2
. (2.10)
Proof. (i) Eliminating s1s5 by multiplying (1.40) and (1.41) we get
λ1µ1
qλ5µ5
+ 5 =
λ1
µ5
+
µ1
qµ5
.
But this is precisely, identically,
(PQ)2 + 5 = (PQ)
(
Q
P
)3
+ (PQ)
(
P
Q
)3
,
on routinely using the definitions of P, Q, λ1, µ1, λ5 and µ5 (1.11) and (1.39). Dividing this
throughout by PQ we have (2.8).
Proof. (ii) Dividing (1.42) by λ5 and (1.43) by qµ5 and subtracting the resulting equations one
from the other, we have
µ1
qµ5
− λ1
λ5
=
s1t
2
1
qs5µ5
+ 4
s5t
2
2
s1λ5
,
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1024 S. BHARGAVA K. R. VASUKI, K. R. RAJANNA
or
1
q
(
s1t1
s5t2
)
+ 4 =
s1µ1λ5
s5t22µ5
− s1λ1
s5t22
,
or, identically,
(PQ)4 + 4 = (PQ)2
(
Q
P
)3
− (PQ)2
(
P
Q
)3
,
on routinely using the definitions of P, Q, λ1, µ1, λ5 and µ5 (1.11) and (1.39). Dividing this
throughout by (PQ)2, we have (2.9).
(iii) Dividing (1.40) by λ5 and (1.43) by µ1 and adding the resulting equations, we obtain
λ1
λ5
− 5qµ5
µ1
=
s1t
2
1
s5µ1
− 4qs1s5µ5
λ5
,
or
λ1µ1
qλ5µ5
− 5 =
s1t
2
1
qs5µ5
− 4
s1s5µ1
λ5
,
or, identically,
(PQ)2 − 5 = (PQ)
(
Q
P
)2
− 4PQ
(
P
Q
)2
,
on routinely using the definitions of P, Q, λ1, µ1, λ5 and µ5 (1.11) and (1.39). Dividing this
throughout by (PQ), we have (2.10).
Theorem 2.2 is proved.
The following corollary, along with Corollary 2.1, is useful in the proof of Theorem 2.4.
Corollary 2.2. In the notations of Corollary 2.1, we have
4y1 = 5w − x
and
8y2 = 9w2 − 40w − x2 − 16.
Proof. Setting
En :=
tnt2n
qn/2t5nt10n
, Fn :=
t2nt5n
qn/6tnt10n
and
y6 := E1E3 +
25
E1E3
,
we can rewrite (2.8) as
E1 +
5
E1
= F 3
1 +
1
F 3
1
.
Changing q to q3 in this, we get
E3 +
5
E3
= F 3
3 +
1
F 3
3
.
Multiplying the last two equations we have, employing the notations of Corollary 2.1,
y6 + 5w = y5 + x, (2.11)
since, as is easily verified,
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ON SOME RAMANUJAN IDENTITIES FOR THE RATIOS OF ETA-FUNCTIONS 1025
w =
E1
E3
+
E3
E1
, x =
(
F1
F3
)3
+
(
F3
F1
)3
and y5 = (F1F3)
3 +
1
(F1F3)3
.
Similarly, (2.9) and (2.10) respectively yield
y2 + 4y1 = y5 − x (2.12)
and
y6 + 4y1 = y2 + 5w. (2.13)
Subtracting (2.11) from the sum of (2.12) and (2.13), we obtain the first of the required results.
Subtracting (2.7) from the sum of (2.5) and (2.6) yields
8y5 = 9w2 − x2 − 16.
Using this in the sum of (2.11) and (2.12) and then subtracting (2.13) from the resulting equation
gives the second of the required results.
The following theorem establishes a P −Q identity of Ramanujan simply from our Lemma 1.5.
Theorem 2.3 ([11, p. 325], [4, p. 212], Entry 58). Let
P :=
f(−q1/5)
q1/5f(−q5)
and Q :=
f(−q2/5)
q2/5f(−q10)
.
Then
PQ+
25
PQ
=
(
Q
P
)3
− 4
(
Q
P
)2
− 4
(
P
Q
)2
+
(
P
Q
)3
. (2.14)
Proof. Multiplying (1.44) and (1.45) and then dividing throughout by qλ25µ25 and then expanding
we have
λ1µ1
q6λ25µ25
+ 25 =
µ1
q6µ25
− 4
s1µ1
q5s25µ25
− 4
s25λ1
qs1λ25
+
λ1
λ25
,
or, identically,
(PQ)2 + 25 = PQ
(
Q
P
)3
− 4PQ
(
Q
P
)2
− 4PQ
(
P
Q
)2
+ PQ
(
P
Q
)3
,
on using the definitions of P, Q, λ1, µ1, λ25 and µ25 (1.11) repeatedly. This becomes (2.14) on
dividing throughout by PQ.
The proof of the following theorem is elementary and is so devised as to circumvent any temptation
to use computer packages.
Theorem 2.4. (i) ([11, p. 330], [4, p. 314], Entry 59). If
P :=
t3t5
q1/3t1t15
and Q :=
t6t10
q2/3t2t30
,
then
PQ+
1
PQ
=
(
Q
P
)3
+
(
P
Q
)3
+ 4. (2.15)
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1026 S. BHARGAVA K. R. VASUKI, K. R. RAJANNA
(ii) ([11, p. 330], [4, p. 218], Entry 61). If
P :=
t6t5
q1/4t2t15
and Q :=
t3t10
q3/4t1t30
,
then
PQ+ 1 +
1
PQ
=
(
Q
P
)2
+
(
P
Q
)2
. (2.16)
(iii) ([11, p. 213], [4, p. 230], Entry 65). If
P :=
t1t2
q1/2t5t10
and Q :=
t3t6
q3/2t15t30
,
then
PQ+
25
PQ
=
(
Q
P
)2
+
(
P
Q
)2
− 3
(
Q
P
+
P
Q
+ 2
)
. (2.17)
Proof. (i) Substituting in the result of Corollary 2.1, the expressions for y1 and y2 obtained in
Corollary 2.2 we obtain{
8(9w2 − 40w − x2 − 16)3 − 6.84(9w2 − 40w − x2 − 16) + 2.82
}
−
−
{
(5w − x)3 − 48(5w − x)− 9.82w2 + 18.82
}2
= 0.
Or, in terms of the analytic functions
W := qw and X := qx, |q| < 1,
with W (q)→ 1 and X(q)→ 1, as can be seen from our definitions of w and x in Corollary 2.1, we
have
F (W,X) := 8(9W 2 − 40Wq −X2 − 16q2)3 − 6.84(9W 2 − 40Wq −X2 − 16q2)q4+
+2.86.q6 −
{
(5W −X)3 − 48(5W −X)q2 − 9.82Wq2 + 18.82q3
}2
= 0.
Now, for W =W0 := X + 4q, we have F (W0, X) = 0 as can be easily verified. In fact, after slight
simplification, we get
F (W0, X) = 84
[
(X2 + 4Xq − 4q2)3 − 48(X2 + 4Xq − 4q2)q4 + 128q6
]
.
This in turn is seen to be identically 0 by further simplification.
Thus we can write
F (W,X) = (W −W0)G(W,X),
whereG(W (q), X(q)) is analytic in |q| < 1. In fact, we can realizeG(W,X) as, by applying Taylor’s
formula to F (W,X), or otherwise,
G(W,X) =
∂F
∂W0
+
6∑
k=2
(W −W0)
k−1
k!
∂kF
∂W k
0
.
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ON SOME RAMANUJAN IDENTITIES FOR THE RATIOS OF ETA-FUNCTIONS 1027
Further, from this form of G(W,X) and the definition of F (W,X), and since W (q), X(q) and
W(q)→ 1 as q → 0, we have
lim
q→0
G(W (q), X(q)) = lim
q→0
∂F
∂W0
=
= lim
q→0
[
3.8(9W 2 −X2)2(18W )− 30(5W −X)5 + ◦(q)
]
= −3072 6= 0.
This implies, because of continuity of G(W (q), X(q)) in |q| < 1, that there exists a neighborhood
N of q = 0, where G(W (q), X(q)) 6= 0. This in turn gives, since F (W (q), X(q)) = (W (q) −
−W0(q))G(W (q), X(q)) is identically 0 in |q| < 1, that
W (q)−W0(q) = 0
identically inN. SinceW (q)−W0(q) is analytic in all of |q| < 1, this implies by analytic continuation
0 =W (q)−W0(q) = q(w(q)− x(q)− 4) throughout |q| < 1,
or
w(q) = x(q) + 4, in 0 < q < 1, (2.18)
or
uv +
1
uv
=
(v
u
)3
+
(u
v
)3
+ 4, in 0 < |q| < 1.
This is the same as the required result (2.15) since P = 1/u and Q = 1/v.
(ii) In the notations of Corollaries 2.1 and 2.2, the required result is easily seen to be the same as
C5
C1
+
C1
C5
+ 1 = (B1B5)
2 +
1
(B1B5)2
or
w + 1 = y1.
But this at once follows by adding (2.18) to the first of the results of Corollary 2.2.
(iii) In the notations of Corollaries 2.1 and 2.2, the required result is seen to be the same as
E1E3 +
25
E1E3
=
(
E3
E1
)3
+
(
E1
E3
)3
− 3
(
E3
E1
+
E1
E3
)
− 6
or
y6 = w2 − 3w − 8.
To obtain this, we first rewrite, on using (2.18), the results of Corollary 2.2 as
y1 = w + 1
and
y2 = w2 − 3w − 4.
It now suffices to use the last two equations in (2.17).
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
1028 S. BHARGAVA K. R. VASUKI, K. R. RAJANNA
1. Adiga C. A study of some identities stated by Srinivasa Ramanujan in His “Lost” note book and earlier works:
Doctoral Thesis. – Univ. Mysore, 1983.
2. Adiga C., Berndt B. C., Bhargava S, Watson G. N. Chapter 16 of Ramanujan’s second notebook: Theta functions and
q-series // Mem. Amer. Math. Soc. – 1985. – № 315.
3. Berndt B. C. Ramanujan’s notebooks. – New York: Springer-Verlag, 1991. – Pt III.
4. Berndt B. C. Ramanujan’s notebooks. – New York: Springer-Verlag, 1994. – Pt IV.
5. Berndt B. C. Ramanujan’s Notebooks. – New York: Springer-Verlag, 1998. – Pt V.
6. Berndt B. C. Modular equations in Ramanujan’s lost notebook // Number Theory / Eds R. P. Bambah, V. C. Dumir,
R. J. Hans-Gill. – New Delhi: Hindustan Book Agency, 2000. – P. 55 – 74.
7. Berndt B. C., Zhang L.-C. Ramanujan’s identities for eta functions // Math. Ann. – 1992. – 292. – P. 561 – 573.
8. Bhargava S., Adiga C. Simple proofs of Jacobi’s two and four square theorems // Int. J. Math. Ed. Sci. Tech. – 1988.
– 19. – P. 779 – 782.
9. Hardy G. H. Ramanujan. – 3 rd ed. – New York: Chelsea, 1978.
10. Kang S. Y. Some theorems on the Roger’s – Ramanujan continued fraction and associated theta function identities in
Ramanujan’s lost notebook // Ramanujan J. – 1999. – 3. – P. 91 – 111.
11. Ramanujan S. Notebooks. – Bombay: Tata Inst. Fundam. Res., 1957. – Vols 1, 2.
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Kamaraj Univ., 1988. – 2.
Received 15.08.12
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 8
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| id | umjimathkievua-article-2196 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:20:30Z |
| publishDate | 2014 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/7c/de590334a9b3a769d45deaa80272d77c.pdf |
| spelling | umjimathkievua-article-21962019-12-05T10:26:14Z On Some Ramanujan Identities for the Ratios of Eta-Functions Про деякі тотожності Рамануджана для відношень Eta-функцій Bhargava, S. Rajanna, K. R. Vasuki, K. R. Бхаргава, С. Райана, К. Р. Васюкі, К. Р. We give direct proofs of some of Ramanujan’s P-Q modular equations based on simply proved elementary identities from Chapter 16 of his Second Notebook. Наведено прямі доведення деяких P-Q модульних РІВНЯНЬ Рамануджана на підставі елементарних тотожностей з глави 16 його Другого зошита, що просто доводяться. Institute of Mathematics, NAS of Ukraine 2014-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2196 Ukrains’kyi Matematychnyi Zhurnal; Vol. 66 No. 8 (2014); 1011–1028 Український математичний журнал; Том 66 № 8 (2014); 1011–1028 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2196/1393 https://umj.imath.kiev.ua/index.php/umj/article/view/2196/1394 Copyright (c) 2014 Bhargava S.; Rajanna K. R.; Vasuki K. R. |
| spellingShingle | Bhargava, S. Rajanna, K. R. Vasuki, K. R. Бхаргава, С. Райана, К. Р. Васюкі, К. Р. On Some Ramanujan Identities for the Ratios of Eta-Functions |
| title | On Some Ramanujan Identities for the Ratios of Eta-Functions |
| title_alt | Про деякі тотожності Рамануджана для відношень Eta-функцій |
| title_full | On Some Ramanujan Identities for the Ratios of Eta-Functions |
| title_fullStr | On Some Ramanujan Identities for the Ratios of Eta-Functions |
| title_full_unstemmed | On Some Ramanujan Identities for the Ratios of Eta-Functions |
| title_short | On Some Ramanujan Identities for the Ratios of Eta-Functions |
| title_sort | on some ramanujan identities for the ratios of eta-functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2196 |
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