Modular Identities and Explicit Evaluations of a Continued Fraction of Ramanujan

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Volume 2012, Article ID 694251,10pages doi:10.1155/2012/694251

Research Article

Modular Identities and Explicit Evaluations of a Continued Fraction of Ramanujan

Nipen Saikia

Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh-791112, Arunachal Pradesh, India

Correspondence should be addressed to Nipen Saikia,[email protected] Received 23 March 2012; Accepted 11 June 2012

Academic Editor: Stefaan Caenepeel

Copyrightq2012 Nipen Saikia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study a new continued fraction of Ramanujan. We prove its modular identities and give some explicit evaluations.

1. Introduction

Throughout the paper, we assume|q|<1. As usual, for positive integersnand any complex numbera, we write

a_n: a;q

n:ⁿ⁻¹

j0

1−aq^j

, a_∞: a;q

∞:^∞

n0

1−aqⁿ

. 1.1

Ramanujan’s general theta-functionfa, bis defined by

fa, b: ^∞

k−∞

a^kk1/2b^kk−1/2, 1.2

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where|ab|<1. After Ramanujan, we define

φ q

:f q, q

12 ∞ k1

q^k²

−q;q²

∞

q²;q² ∞

q;q²

∞

−q²;q²

∞

, 1.3

ψ q

:f q, q³

^∞

k0

q^kk1/2

q²;q² ∞

q;q²

∞

, 1.4

f

−q :f

−q,−q² ^∞

k0

−1^kq^k3k−1/2^∞

k1

−1^kq^k3k1/2 q;q

∞, 1.5

χ q

f q f

−q²

−q;q²

∞. 1.6

Ramanujan recorded many q-continued fractions and some of their explicit values in his second notebook1and in his lost notebook2. The following beautiful continued fraction identity was recorded by Ramanujan in his second notebook and can be found in3, p. 11, Entry 11:

−a_∞b_∞−a_∞−b_∞

−a_∞b_∞ a_∞−b_∞ a−b 1−q

a−bq aq−b 1−q³ q

a−bq²

aq²−b

1−q⁵ · · ·, 1.7 where eitherq,a, andbare complex numbers with|q|<1, orq,a, andbare complex numbers withabq^mfor some integerm. Several elegantq-continued fractions have representations asq-products and some of them can be expressed in terms of Ramanujan’s theta-functions.

An account of this can be found in in Chapter 32 of Berndt’s book4 also see5. The most famous one, of course, is the Rogers-Ramanujan continued fractionRqdefined by

R q

: q^1/5 1 q

1 q² 1 q³

1 · · ·. 1.8

The continued fractionRqhas a very beautiful and extensive theory almost all of which was developed by Ramanujan. In particular, his lost notebook2contains several results on the Rogers-Ramanujan continued fraction. We refer to the paper by Berndt et al.6, Kang 7,8for proofs of many of these theorems.

In this paper, we examine another continued fractionTqof Ramanujan arising from 1.7and is defined by

T q

: q

1−q² q⁴

1−q⁶ q⁸

1−q¹⁰ · · ·. 1.9 Note that, replacingqbyq²and then settingaqandb0 in1.7, we obtain1.9.

InSection 2, we record some preliminary results.Section 3is devoted to prove some modular identities for the continued fractionTq. Finally, inSection 4, we give some explicit evaluations ofTq.

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We complete this introduction by defining Ramanujan’s modular equation from Berndt’s book3. The complete elliptic integral of the first kindKkis defined by

Kk: _π/2

0

dφ 1−k²sin²φ

π 22F₁

1 2,1

2; 1;k²

, 1.10

where 0< k <1,₂F₁denotes the ordinary or Gaussian hypergeometric function. The number k is called the modulus ofK, andk : √

1−k² is called the complementary modulus. Let K, K, L, andL denote the complete elliptic integrals of the first kind associated with the modulik,k,l, andl, respectively. Suppose that the equality

nK K L

L 1.11

holds for some positive integern. Then, a modular equation of degreenis a relation between the modulikandlwhich is implied by1.11. If we set

qexp

−πK K

, qexp

−πL L

, 1.12

we see that1.11is equivalent to the relationqⁿq. Thus, a modular equation can be viewed as an identity involving theta-functions at the argumentsqandqⁿ. Ramanujan recorded his modular equations in terms ofαandβ, whereαk² andβl². We say thatβhas degreen overα. The multipliermconnectingαandβis defined by

m K

L, 1.13

wherez_r φ²q^r.

2. Preliminary Results

In this section, we record some results that will be used in the subsequent sections.

Lemma 2.1see3, p. 124, Entry 12iandii. One has f

q √

z₁2^−1/6{α1−α}^1/24q^−1/24, f

−q √

z₁2^−1/61−α^1/6α^1/24q^−1/24. 2.1 Lemma 2.2see3, p. 214, Entry 24iii. Ifβhas degree 2 overα, then

m√

α−1 β1, m²√

α−1β1.

2.2

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Lemma 2.3see3, p. 230, Entry 5ii. Ifβhas degree 3 overα, then αβ1/4

1−α

1−β1/4 1. 2.3

Lemma 2.4see3, p. 215,24.22. Ifβhas degree 4 overα, then

β

1−1−α^1/4 1 1−α^1/4

₂

. 2.4

Lemma 2.5see3, p. 280-281, Entry 13vandvi. Ifβhas degree 5 overα, then

m 1 1−β₅

/1−α1/8

1

1−α³

1−β^1/8 , 5

m 1−

1−α⁵/

1−β^1/8 1−

1−α

1−β₃1/8 .

2.5

Lemma 2.6see3, p. 314, Entry 19i. Ifβhas degree 7 overα, then αβ_1/8

1−α

1−β_1/8

1. 2.6

3. Modular Identitites for T q

In this section, we use Ramanujan’s modular equations to prove certain modular identities forTq.

Theorem 3.1. One has

T q

f q

−f

−q f

q f

−q. 3.1

Proof. Replacingqbyq²and the settingaqandb0 in1.7and simplifying, we obtain −q;q²

∞− q;q² ∞

−q;q²

∞ q;q²

∞

q

1−q² q⁴

1−q⁶ q⁸

1−q¹⁰ · · · . 3.2 Employing1.6and1.9in3.2and simplifying, we complete the proof.

Corollary 3.2. One has

1T q 1−T

q f q f

−q. 3.3

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Proof. Dividing numerator and denominator on right-hand side of the identity in Theorem 3.1byf−qand simplifying, we complete the proof.

Theorem 3.3. One has iα1−

1−T q 1T

q ₈

, iiβ1−

1−Tqⁿ 1Tqⁿ

8

, 3.4

whereβhas degree n overα.

Proof. We employLemma 2.1inCorollary 3.2to complete the proof.

Theorem 3.4. LetuTqandvT−q. Then,

uv0. 3.5

Proof. Replacingqby−qinCorollary 3.2, we obtain 1T

−q 1−T

−q f

q . 3.6

Now, eliminatingfq/f−qbetween3.6andCorollary 3.2and simplifying, we complete the proof.

Theorem 3.5. LetuTqandvTq². Then,

u²−v−2u²v−u⁴v6u²v²−v³−2u²v³−u⁴v³u²v⁴0. 3.7 Proof. Eliminatingmin2.2and then simplifying, we deduce that

1β

β−1√ 1−α₂

−4β0. 3.8

FromTheorem 3.3i, we have

√1−α

1−T q 1T

q ₄

. 3.9

Now, employing Theorem 3.3ii with n 2 and 3.9 in 3.8 and factorizing using Mathematica, we obtain

−1v⁸

u²−v−2u²v−u⁴v6u²v²−v³−2u²v³−u⁴v³u²v⁴

×

12u²u⁴−16u²v6v²12u²v²6u⁴v²−16u²v³v⁴2u²v⁴u⁴v⁴ 0.

3.10

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It can be seen that the first and the last factors in3.10 do not vanish for|q| → 0. So, by identity theorem, we have

u²−v−2u²v−u⁴v6u²v²−v³−2u²v³−u⁴v³u²v⁴0. 3.11

Theorem 3.6. LetuTqandvTq³. Then,

u³−v−3u²v3uv²3u³v²−3u²v³−u⁴v³uv⁴0. 3.12

Proof. FromLemma 2.3, we obtain

αβ−

1−1−α^1/4

1−β_1/44

0. 3.13

FromTheorem 3.3, we deduce that

α1− 1−u

1u ₈

, β1− 1−v

1v ₈

,

1−α^1/4 1−u

1u ₂

,

1−β_1/4

1−v 1v

₂ ,

3.14

whereβhas degree 3 overα.

Employing3.14in3.13and factorizing using Mathematica, we arrive at −u³v3u²v−3uv²−3u³v²3u²v³u⁴v³−uv⁴

×

−u3u²vu⁴v−3uv²−3u³v²v³3u²v³−u³v⁴ 0.

3.15

It can be seen that the second factor of3.15 does not vanish for|q| → 0, so by identity theorem, we have

u³−v−3u²v3uv²3u³v²−3u²v³−u⁴v³uv⁴0. 3.16

Theorem 3.7. LetuTqandvTq⁴. Then,

u⁴−v−4u²v2u⁴v−4u⁶v−u⁸v28u⁴v²−7v³−28u²v³14u⁴v³−28u⁶v³

−7u⁸v³70u⁴v⁴−7v⁵−28u²v⁵14u⁴v⁵−28u⁶v⁵−7u⁸v⁵28u⁴v⁶−v⁷−4u²v⁷ 2u⁴v⁷−4u⁶v⁷−u⁸v⁷u⁴v⁸0.

3.17

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Proof. Squaring the modular equation inLemma 2.4and simplifying, we obtain

β−

1−1−α^1/4 1 1−α^1/4

₄

0. 3.18

FromTheorem 3.3i, we have

1−α^1/4

1−T q 1T

q ₂

. 3.19

Now, employingTheorem 3.3iiwithn4 and3.19in3.18and simplifying, we complete the proof.

Theorem 3.8. LetuTqandvTq⁵. Then,

u⁵−v−5u²v10u³v²5u⁵v²−10u²v³−10u⁴v³5uv⁴10u³v⁴−5u⁴v⁵−u⁶v⁵uv⁶0.

3.20 Proof. FromTheorem 3.3, we obtain

c: 1−α^1/8 1−u

1u

, d:

1−β_1/8

1−v 1v

, 3.21

Employing3.21in2.5, we find that

m cd⁵

c1c³d, 3.22

5

m d−c⁵

d1−cd³, 3.23

respectively.

Eliminatingmbetween3.22and3.23and simplifying, we deduce that 5cd

1c³d

1−cd³

−

cd⁵ d−c⁵

0. 3.24

Substituting forcanddfrom3.21in3.24and simplifying, we arrive at

u⁵−v−5u²v10u³v²5u⁵v²−10u²v³−10u⁴v³5uv⁴10u³v⁴−5u⁴v⁵−u⁶v⁵uv⁶0.

3.25

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Theorem 3.9. LetuTqandvTq⁷. Then,

u⁸−uv−7u³v−7u⁵v7u⁷v28u⁶v²−7uv³−49u³v³7u⁵v³−7u⁷v³70u⁴v⁴

−7uv⁵7u³v⁵−49u⁵v⁵−7u⁷v⁵28u²v⁶7uv⁷−7u³v⁷−7u⁵v⁷−u⁷v⁷v⁸0.

3.26

Proof. FromLemma 2.6, we obtain

αβ−

1−1−α^1/4

1−β1/4₈

0. 3.27

Again, fromTheorem 3.3, we deduce that

α1− 1−u

1u 8

, β1− 1−v

1v 8

,

1−α^1/8 1−u

1u

,

1−β_1/8

1−v 1v

,

3.28

Employing3.28in3.27and simplifying using Mathematica, we arrive at u⁸−uv−7u³v−7u⁵v7u⁷v28u⁶v²−7uv³−49u³v³7u⁵v³−7u⁷v³70u⁴v⁴

−7uv⁵7u³v⁵−49u⁵v⁵−7u⁷v⁵28u²v⁶7uv⁷−7u³v⁷−7u⁵v⁷−u⁷v⁷v⁸0.

3.29

4. Explicit Evaluations of T q

In this section, we establish some general theorems for the explicit evaluations of the continued fractionTqand give examples.

Forq:e^−π^√ⁿ, Ramanujan’s two class invariantsG_nandg_nare defined by G_n2^−1/4q^−1/24χ

q

, g_n2^−1/4q^−1/24χ

−q

. 4.1

The class invariantsG_nandg_nare connected by the relation4, p. 187, Entry 2.1:

g_4n2^1/4g_nG_n. 4.2

The singular modulusαn is defined byαn : αe^−π^√ⁿ, wherenis a positive integer and unique positive number between 0 and 1 satisfying

√n ²F₁1/2,1/2; 1; 1−αn

2F₁1/2,1/2; 1;α_n . 4.3

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Class invariants and singular moduli are intimately related by the equalities4, p. 185, 1.6:

G_n 4α_n1−α_n^−1/24, g_n

4α_n1−α_n⁻²_−1/24

. 4.4

An account of Ramanujan’s class invariants and singular moduli can be found in Chapter 34 of Berndt’s book4.

Theorem 4.1. One has

T e^−π^√ⁿ

1−1−α_n^1/8

1 1−αn^1/8. 4.5

Proof. We setq: e^−π^√ⁿ inTheorem 3.3iand use the definition of singular moduliαnand simplifying, we complete the proof.

In the scattered places of his first notebook1, Ramanujan calculated over 30 singular moduliα_n. See Chapter 34 of Berndt’s book4for details. Thus, one can useTheorem 4.1to find the values ofTe^−π^√ⁿif the corresponding values ofαnare known. For example, from 4, p. 281, Theorem 9.2, we note that

α2√ 2−12

. 4.6

Employing4.6inTheorem 4.1, we calculate

T e^−π^√²

1−

−22√ 2_1/8 1

−22√

2_1/8. 4.7

Many other values ofTe^−π^√ⁿcan be computed by using the known values ofαn. Theorem 4.2. One has

T e^−π^√ⁿ

g_4n−2^1/4g_n² g_4n−2^1/4g_n²

. 4.8

Proof. Dividing numerator and denominator of right-hand side ofTheorem 3.1and employ- ing1.6, we obtain

T q

χ q

−χ

−q χ

q χ

−q. 4.9

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Settingq:e^−π^√ⁿ, employing the definitions ofGnandgnfrom4.1in4.9and simplifying, we obtain

T e^−π^√ⁿ

G_n−g_n

G_ng_n. 4.10

Substituting forGnfrom4.2in4.10and simplifying, we complete the proof.

Theorem 4.2implies that if we know the values ofgnandg4nfor any positive number n, then corresponding values ofTe^−π^√ⁿcan easily be calculated. Saikia9evaluated several values ofg_n andg_4nfor positive numbern. For example, noting from9, Theorem 3.5, we have

g32^−1/6 2√

31/8

, g122^1/6 2√

31/8

. 4.11

Employing4.11inTheorem 4.2, we obtain

T e^−π^√³

2−2^3/4 2√

3_1/8 22^3/4

2√

31/8. 4.12

Many other values ofTe^−π^√ⁿcan be determined by using the values ofgnandg4nevaluated in9.

References

1 S. Ramanujan, Notebooks, vol. 1,2, Tata Institute of Fundamental Research, Bombay, India, 1957.

2 S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, India, 1988.

3 B. C. Berndt, Ramanujan’s Notebooks, part 3, Springer, New York, NY, USA, 1991.

4 B. C. Berndt, Ramanujan’s Notebooks, Part 4, Springer, New York, NY, USA, 1998.

5 B. C. Berndt, “Flowers which we cannot yet see growing in Ramanujan’s garden of hypergeometric series, elliptic functions, andq’s,” in Special Functions 2000: Current Perspective and Future Directions, J.

Bustoz, M. E. H. Ismail, and S. K. Suslov, Eds., vol. 30, pp. 61–85, Kluwer Academic, Dordrecht, The Netherlands, 2001.

6 B. C. Berndt, S. S. Huang, J. Sohn, and S. H. Son, “Some theorems on the Rogers-Ramanujan continued fraction in Ramanujan’s lost notebook,” Transactions of the American Mathematical Society, vol. 352, no.

5, pp. 2157–2177, 2000.

7 S.-Y. Kang, “Some theorems on the Rogers-Ramanujan continued fraction and associated theta function identities in Ramanujan’s lost notebook,” The Ramanujan Journal, vol. 3, no. 1, pp. 91–111, 1999.

8 S.-Y. Kang, “Ramanujan’s formulas for the explicit evaluation of the Rogers-Ramanujan continued fraction and theta-functions,” Acta Arithmetica, vol. 90, no. 1, pp. 49–68, 1999.

9 N. Saikia, “Ramanujan’s modular equations and Weber-Ramanujan’s class invariants Gn and gn ,”

Bulletin of Mathematical Sciences, vol. 2, no. 1, pp. 205–223, 2012.

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Modular Identities and Explicit Evaluations of a Continued Fraction of Ramanujan

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