Modular Identities and Explicit Evaluations of a Continued Fraction of Ramanujan

Abstract

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 integers 𝑛 and any complex number 𝑎, we write (𝑎)_𝑛∶=𝑎;𝑞_𝑛∶=^𝑛−1_𝑗=01−𝑎𝑞^𝑗,(𝑎)_∞∶=𝑎;𝑞_∞∶=^∞_𝑛=01−𝑎𝑞^𝑛. (1.1)

Ramanujan's general theta-function 𝑓(𝑎,𝑏) is defined by 𝑓(𝑎,𝑏)∶=^∞_𝑘=−∞𝑎^{𝑘(𝑘+1)/2}𝑏^{𝑘(𝑘−1)/2}, (1.2) where |𝑎𝑏|<1. After Ramanujan, we define 𝜙𝑞∶=𝑓𝑞,𝑞=1+2^∞_𝑘=1𝑞^𝑘2=−𝑞;𝑞²_∞𝑞²;𝑞²_∞𝑞;𝑞²_∞−𝑞²;𝑞²_∞,𝜓𝑞∶=𝑓𝑞,𝑞³=^∞_𝑘=0𝑞^{𝑘(𝑘+1)/2}=𝑞²;𝑞²_∞𝑞;𝑞²_∞,𝑓−𝑞∶=𝑓−𝑞,−𝑞²=^∞_𝑘=0(−1)^𝑘𝑞^{𝑘(3𝑘−1)/2}+^∞_𝑘=1(−1)^𝑘𝑞^{𝑘(3𝑘+1)/2}=𝑞;𝑞_∞,𝜒𝑞=𝑓𝑞𝑓−𝑞²=−𝑞;𝑞²_∞. (1.3)(1.4)(1.5)(1.6) Ramanujan recorded many 𝑞-continued fractions and some of their explicit values in his second notebook [1] and in his lost notebook [2]. The following beautiful continued fraction identity was recorded by Ramanujan in his second notebook and can be found in [3, p. 11, Entry 11]: (−𝑎)_∞(𝑏)_∞−(𝑎)_∞(−𝑏)_∞(−𝑎)_∞(𝑏)_∞+(𝑎)_∞(−𝑏)_∞=𝑎−𝑏+1−𝑞𝑎−𝑏𝑞𝑎𝑞−𝑏1−𝑞³+𝑞𝑎−𝑏𝑞²𝑎𝑞²−𝑏1−𝑞⁵+⋯, (1.7) where either 𝑞, 𝑎, and 𝑏 are complex numbers with |𝑞|<1, or 𝑞, 𝑎, and 𝑏 are complex numbers with 𝑎=𝑏𝑞^𝑚 for some integer 𝑚. Several elegant 𝑞-continued fractions have representations as 𝑞-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 book [4] (also see [5]). The most famous one, of course, is the Rogers-Ramanujan continued fraction 𝑅(𝑞) defined by 𝑅𝑞𝑞∶=^1/51+𝑞1+𝑞²1+𝑞³1+⋯. (1.8) The continued fraction 𝑅(𝑞) has a very beautiful and extensive theory almost all of which was developed by Ramanujan. In particular, his lost notebook [2] contains several results on the Rogers-Ramanujan continued fraction. We refer to the paper by Berndt et al. [6], Kang [7, 8] for proofs of many of these theorems.

In this paper, we examine another continued fraction 𝑇(𝑞) of Ramanujan arising from (1.7) and is defined by 𝑇𝑞𝑞∶=1−𝑞²+𝑞⁴1−𝑞⁶+𝑞⁸1−𝑞¹⁰+⋯. (1.9) Note that, replacing 𝑞 by 𝑞² and then setting 𝑎=𝑞 and 𝑏=0 in (1.7), we obtain (1.9).

In Section 2, we record some preliminary results. Section 3 is devoted to prove some modular identities for the continued fraction 𝑇(𝑞). Finally, in Section 4, we give some explicit evaluations of 𝑇(𝑞).

We complete this introduction by defining Ramanujan’s modular equation from Berndt’s book [3]. The complete elliptic integral of the first kind 𝐾(𝑘) is defined by 𝐾(𝑘)∶=₀^𝜋/2𝑑𝜙1−𝑘²sin²𝜙=𝜋2²𝐹₁12,12;1;𝑘², (1.10) where 0<𝑘<1,₂𝐹₁ denotes the ordinary or Gaussian hypergeometric function. The number 𝑘 is called the modulus of 𝐾, and 𝑘^′∶=√1−𝑘² is called the complementary modulus. Let 𝐾,𝐾^′,𝐿, and 𝐿^′ denote the complete elliptic integrals of the first kind associated with the moduli 𝑘, 𝑘^′, 𝑙, and 𝑙^′, respectively. Suppose that the equality 𝑛𝐾′𝐾=𝐿′𝐿 (1.11) holds for some positive integer 𝑛. Then, a modular equation of degree 𝑛 is a relation between the moduli 𝑘 and 𝑙 which is implied by (1.11). If we set 𝐾𝑞=exp−𝜋^′𝐾,𝑞^′𝐿=exp−𝜋^′𝐿, (1.12) we see that (1.11) is equivalent to the relation 𝑞^𝑛=𝑞^′. Thus, a modular equation can be viewed as an identity involving theta-functions at the arguments 𝑞 and 𝑞^𝑛. Ramanujan recorded his modular equations in terms of 𝛼 and 𝛽, where 𝛼=𝑘² and 𝛽=𝑙². We say that 𝛽 has degree 𝑛 over 𝛼. The multiplier 𝑚 connecting 𝛼 and 𝛽 is defined by 𝐾𝑚=𝐿, (1.13) where 𝑧_𝑟=𝜙²(𝑞^𝑟).

2. Preliminary Results

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

Lemma 2.1 (see [3, p. 124, Entry 12(i) and (ii)]). One has 𝑓𝑞=√𝑧₁2^−1/6{𝛼(1−𝛼)}^1/24𝑞^−1/24,𝑓−𝑞=√𝑧₁2^−1/6(1−𝛼)^1/6𝛼^1/24𝑞^−1/24. (2.1)

Lemma 2.2 (see [3, p. 214, Entry 24(iii)]). If 𝛽 has degree 2 over 𝛼, then 𝑚√𝛼−1+𝑚𝛽=1,₂√𝛼−1+𝛽=1. (2.2)

Lemma 2.3 (see [3, p. 230, Entry 5(ii)]). If 𝛽 has degree 3 over 𝛼, then 𝛼𝛽^1/4+(1−𝛼)1−𝛽^1/4=1. (2.3)

Lemma 2.4 (see [3, p. 215, (24.22)]). If 𝛽 has degree 4 over 𝛼, then 𝛽=1−(1−𝛼)^1/41+(1−𝛼)^1/42. (2.4)

Lemma 2.5 (see [3, p. 280-281, Entry 13(v) and (vi)]). If 𝛽 has degree 5 over 𝛼, then 𝑚=1+1−𝛽⁵/(1−𝛼)^1/81+(1−𝛼)³1−𝛽^1/8,5𝑚=1−(1−𝛼)⁵/1−𝛽^1/81−(1−𝛼)1−𝛽^31/8. (2.5)

Lemma 2.6 (see [3, p. 314, Entry 19(i)]). If 𝛽 has degree 7 over 𝛼, 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 for 𝑇(𝑞).

Theorem 3.1. One has 𝑇𝑞=𝑓𝑞−𝑓−𝑞𝑓𝑞+𝑓−𝑞. (3.1)

Proof. Replacing 𝑞 by 𝑞² and the setting 𝑎=𝑞 and 𝑏=0 in (1.7) and simplifying, we obtain −𝑞;𝑞²_∞−𝑞;𝑞²_∞−𝑞;𝑞²_∞+𝑞;𝑞²_∞=𝑞1−𝑞²+𝑞⁴1−𝑞⁶+𝑞⁸1−𝑞¹⁰+⋯. (3.2) Employing (1.6) and (1.9) in (3.2) and simplifying, we complete the proof.

Corollary 3.2. One has 1+𝑇𝑞𝑞=𝑓𝑞1−𝑇𝑓−𝑞. (3.3)

Proof. Dividing numerator and denominator on right-hand side of the identity in Theorem 3.1 by 𝑓(−𝑞) and simplifying, we complete the proof.

Theorem 3.3. One has (𝑖)𝑞𝛼=1−1−𝑇1+𝑇𝑞⁸,(𝑖𝑖)𝛽=1−1−𝑇(𝑞^𝑛)1+𝑇(𝑞^𝑛)⁸, (3.4) where 𝛽 has degree n over 𝛼.

Proof. We employ Lemma 2.1 in Corollary 3.2 to complete the proof.

Theorem 3.4. Let 𝑢=𝑇(𝑞) and 𝑣=𝑇(−𝑞). Then, 𝑢+𝑣=0. (3.5)

Proof. Replacing 𝑞 by −𝑞 in Corollary 3.2, we obtain 1+𝑇−𝑞=𝑓1−𝑇−𝑞−𝑞𝑓𝑞. (3.6) Now, eliminating 𝑓(𝑞)/𝑓(−𝑞) between (3.6) and Corollary 3.2 and simplifying, we complete the proof.

Theorem 3.5. Let 𝑢=𝑇(𝑞) and 𝑣=𝑇(𝑞²). Then, 𝑢²−𝑣−2𝑢²𝑣−𝑢⁴𝑣+6𝑢²𝑣²−𝑣³−2𝑢²𝑣³−𝑢⁴𝑣³+𝑢²𝑣⁴=0. (3.7)

Proof. Eliminating 𝑚 in (2.2) and then simplifying, we deduce that 1+𝛽+𝛽−1√1−𝛼²−4𝛽=0. (3.8) From Theorem 3.3(i), we have √𝑞1−𝛼=1−𝑇1+𝑇𝑞⁴. (3.9) Now, employing Theorem 3.3(ii) with 𝑛=2 and (3.9) in (3.8) and factorizing using Mathematica, we obtain (−1+𝑣)⁸𝑢²−𝑣−2𝑢²𝑣−𝑢⁴𝑣+6𝑢²𝑣²−𝑣³−2𝑢²𝑣³−𝑢⁴𝑣³+𝑢²𝑣⁴×1+2𝑢²+𝑢⁴−16𝑢²𝑣+6𝑣²+12𝑢²𝑣²+6𝑢⁴𝑣²−16𝑢²𝑣³+𝑣⁴+2𝑢²𝑣⁴+𝑢⁴𝑣⁴=0. (3.10) It can be seen that the first and the last factors in (3.10) do not vanish for |𝑞|→0. So, by identity theorem, we have 𝑢²−𝑣−2𝑢²𝑣−𝑢⁴𝑣+6𝑢²𝑣²−𝑣³−2𝑢²𝑣³−𝑢⁴𝑣³+𝑢²𝑣⁴=0. (3.11)

Theorem 3.6. Let 𝑢=𝑇(𝑞) and 𝑣=𝑇(𝑞³). Then, 𝑢³−𝑣−3𝑢²𝑣+3𝑢𝑣²+3𝑢³𝑣²−3𝑢²𝑣³−𝑢⁴𝑣³+𝑢𝑣⁴=0. (3.12)

Proof. From Lemma 2.3, we obtain 𝛼𝛽−1−(1−𝛼)^1/41−𝛽^1/44=0. (3.13) From Theorem 3.3, we deduce that 𝛼=1−1−𝑢1+𝑢⁸,𝛽=1−1−𝑣1+𝑣⁸,(1−𝛼)^1/4=1−𝑢1+𝑢²,1−𝛽^1/4=1−𝑣1+𝑣², (3.14) where 𝛽 has degree 3 over 𝛼.
Employing (3.14) in (3.13) and factorizing using Mathematica, we arrive at −𝑢³+𝑣+3𝑢²𝑣−3𝑢𝑣²−3𝑢³𝑣²+3𝑢²𝑣³+𝑢⁴𝑣³−𝑢𝑣⁴×−𝑢+3𝑢²𝑣+𝑢⁴𝑣−3𝑢𝑣²−3𝑢³𝑣²+𝑣³+3𝑢²𝑣³−𝑢³𝑣⁴=0. (3.15) It can be seen that the second factor of (3.15) does not vanish for |𝑞|→0, so by identity theorem, we have 𝑢³−𝑣−3𝑢²𝑣+3𝑢𝑣²+3𝑢³𝑣²−3𝑢²𝑣³−𝑢⁴𝑣³+𝑢𝑣⁴=0. (3.16)

Theorem 3.7. Let 𝑢=𝑇(𝑞) and 𝑣=𝑇(𝑞⁴). Then, 𝑢⁴−𝑣−4𝑢²𝑣+2𝑢⁴𝑣−4𝑢⁶𝑣−𝑢⁸𝑣+28𝑢⁴𝑣²−7𝑣³−28𝑢²𝑣³+14𝑢⁴𝑣³−28𝑢⁶𝑣³−7𝑢⁸𝑣³+70𝑢⁴𝑣⁴−7𝑣⁵−28𝑢²𝑣⁵+14𝑢⁴𝑣⁵−28𝑢⁶𝑣⁵−7𝑢⁸𝑣⁵+28𝑢⁴𝑣⁶−𝑣⁷−4𝑢²𝑣⁷+2𝑢⁴𝑣⁷−4𝑢⁶𝑣⁷−𝑢⁸𝑣⁷+𝑢⁴𝑣⁸=0. (3.17)

Proof. Squaring the modular equation in Lemma 2.4 and simplifying, we obtain ()𝛽−1−1−𝛼^1/41+(1−𝛼)^1/44=0. (3.18) From Theorem 3.3(i), we have (1−𝛼)^1/4=1−𝑇𝑞1+𝑇𝑞². (3.19) Now, employing Theorem 3.3(ii) with 𝑛=4 and (3.19) in (3.18) and simplifying, we complete the proof.

Theorem 3.8. Let 𝑢=𝑇(𝑞) and 𝑣=𝑇(𝑞⁵). Then, 𝑢⁵−𝑣−5𝑢²𝑣+10𝑢³𝑣²+5𝑢⁵𝑣²−10𝑢²𝑣³−10𝑢⁴𝑣³+5𝑢𝑣⁴+10𝑢³𝑣⁴−5𝑢⁴𝑣⁵−𝑢⁶𝑣⁵+𝑢𝑣⁶=0. (3.20)

Proof. From Theorem 3.3, we obtain 𝑐∶=(1−𝛼)^1/8=1−𝑢1+𝑢,𝑑∶=1−𝛽^1/8=1−𝑣,1+𝑣 (3.21) where 𝛽 has degree 5 over 𝛼.
Employing (3.21) in (2.5), we find that 𝑚=𝑐+𝑑⁵𝑐1+𝑐³𝑑,5𝑚=𝑑−𝑐⁵𝑑1−𝑐𝑑³, (3.22)(3.23) respectively.
Eliminating 𝑚 between (3.22) and (3.23) and simplifying, we deduce that 5𝑐𝑑1+𝑐³𝑑1−𝑐𝑑³−𝑐+𝑑⁵𝑑−𝑐⁵=0. (3.24) Substituting for 𝑐 and 𝑑 from (3.21) in (3.24) and simplifying, we arrive at 𝑢⁵−𝑣−5𝑢²𝑣+10𝑢³𝑣²+5𝑢⁵𝑣²−10𝑢²𝑣³−10𝑢⁴𝑣³+5𝑢𝑣⁴+10𝑢³𝑣⁴−5𝑢⁴𝑣⁵−𝑢⁶𝑣⁵+𝑢𝑣⁶=0. (3.25)

Theorem 3.9. Let 𝑢=𝑇(𝑞) and 𝑣=𝑇(𝑞⁷). Then, 𝑢⁸−𝑢𝑣−7𝑢³𝑣−7𝑢⁵𝑣+7𝑢⁷𝑣+28𝑢⁶𝑣²−7𝑢𝑣³−49𝑢³𝑣³+7𝑢⁵𝑣³−7𝑢⁷𝑣³+70𝑢⁴𝑣⁴−7𝑢𝑣⁵+7𝑢³𝑣⁵−49𝑢⁵𝑣⁵−7𝑢⁷𝑣⁵+28𝑢²𝑣⁶+7𝑢𝑣⁷−7𝑢³𝑣⁷−7𝑢⁵𝑣⁷−𝑢⁷𝑣⁷+𝑣⁸=0. (3.26)

Proof. From Lemma 2.6, we obtain 𝛼𝛽−1−(1−𝛼)^1/41−𝛽^1/48=0. (3.27) Again, from Theorem 3.3, we deduce that 𝛼=1−1−𝑢1+𝑢⁸,𝛽=1−1−𝑣1+𝑣⁸,(1−𝛼)^1/8=1−𝑢,1+𝑢1−𝛽^1/8=1−𝑣,1+𝑣 (3.28) where 𝛽 has degree 7 over 𝛼.
Employing (3.28) in (3.27) and simplifying using Mathematica, we arrive at 𝑢⁸−𝑢𝑣−7𝑢³𝑣−7𝑢⁵𝑣+7𝑢⁷𝑣+28𝑢⁶𝑣²−7𝑢𝑣³−49𝑢³𝑣³+7𝑢⁵𝑣³−7𝑢⁷𝑣³+70𝑢⁴𝑣⁴−7𝑢𝑣⁵+7𝑢³𝑣⁵−49𝑢⁵𝑣⁵−7𝑢⁷𝑣⁵+28𝑢²𝑣⁶+7𝑢𝑣⁷−7𝑢³𝑣⁷−7𝑢⁵𝑣⁷−𝑢⁷𝑣⁷+𝑣⁸=0. (3.29)

4. Explicit Evaluations of 𝑇(𝑞)

In this section, we establish some general theorems for the explicit evaluations of the continued fraction 𝑇(𝑞) and give examples.

For 𝑞∶=𝑒^{√−𝜋𝑛}, Ramanujan's two class invariants 𝐺_𝑛 and 𝑔_𝑛 are defined by 𝐺_𝑛=2^−1/4𝑞^−1/24𝜒𝑞,𝑔_𝑛=2^−1/4𝑞^−1/24𝜒−𝑞. (4.1) The class invariants 𝐺_𝑛 and 𝑔_𝑛 are connected by the relation [4, p. 187, Entry 2.1]: 𝑔_4𝑛=2^1/4𝑔_𝑛𝐺_𝑛. (4.2)

The singular modulus 𝛼_𝑛 is defined by 𝛼_𝑛∶=𝛼(𝑒^{√−𝜋𝑛}), where 𝑛 is a positive integer and unique positive number between 0 and 1 satisfying √𝑛=²𝐹₁1/2,1/2;1;1−𝛼_𝑛2𝐹₁1/2,1/2;1;𝛼_𝑛. (4.3)

Class invariants and singular moduli are intimately related by the equalities [4, 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 book [4].

Theorem 4.1. One has 𝑇𝑒^{√−𝜋𝑛}=1−1−𝛼_𝑛^1/81+1−𝛼_𝑛^1/8. (4.5)

Proof. We set 𝑞∶=𝑒^{√−𝜋𝑛} in Theorem 3.3(i) and use the definition of singular moduli 𝛼_𝑛 and simplifying, we complete the proof.

In the scattered places of his first notebook [1], Ramanujan calculated over 30 singular moduli 𝛼_𝑛. See Chapter 34 of Berndt's book [4] for details. Thus, one can use Theorem 4.1 to find the values of 𝑇(𝑒^{√−𝜋𝑛}) if the corresponding values of 𝛼_𝑛 are known. For example, from [4, p. 281, Theorem 9.2], we note that 𝛼₂=√2−1². (4.6) Employing (4.6) in Theorem 4.1, we calculate 𝑇𝑒^√−𝜋2=1−−2+2√2^1/81+−2+2√2^1/8. (4.7) Many other values of 𝑇(𝑒^{√−𝜋𝑛}) can be computed by using the known values of 𝛼_𝑛.

Theorem 4.2. One has 𝑇𝑒^{√−𝜋𝑛}=𝑔_4𝑛−2^1/4𝑔²_𝑛𝑔_4𝑛−2^1/4𝑔²_𝑛. (4.8)

Proof. Dividing numerator and denominator of right-hand side of Theorem 3.1 and employing (1.6), we obtain 𝑇𝑞=𝜒𝑞−𝜒−𝑞𝜒𝑞+𝜒−𝑞. (4.9) Setting 𝑞∶=𝑒^{√−𝜋𝑛}, employing the definitions of 𝐺_𝑛 and 𝑔_𝑛 from (4.1) in (4.9) and simplifying, we obtain 𝑇𝑒^{√−𝜋𝑛}=𝐺_𝑛−𝑔_𝑛𝐺_𝑛+𝑔_𝑛. (4.10) Substituting for 𝐺_𝑛 from (4.2) in (4.10) and simplifying, we complete the proof.

Theorem 4.2 implies that if we know the values of 𝑔_𝑛 and 𝑔_4𝑛 for any positive number 𝑛, then corresponding values of 𝑇(𝑒^{√−𝜋𝑛}) can easily be calculated. Saikia [9] evaluated several values of 𝑔_𝑛 and 𝑔_4𝑛 for positive number 𝑛. For example, noting from [9, Theorem 3.5], we have 𝑔₃=2^−1/62+√3^1/8,𝑔₁₂=2^1/62+√3^1/8. (4.11) Employing (4.11) in Theorem 4.2, we obtain 𝑇𝑒^√−𝜋3=2−2^3/42+√3^1/82+2^3/42+√3^1/8. (4.12) Many other values of 𝑇(𝑒^{√−𝜋𝑛}) can be determined by using the values of 𝑔_𝑛 and 𝑔_4𝑛 evaluated in [9].