A Product of Theta-Functions Analogous to Ramanujan's Remarkable Product of Theta-Functions and Applications

Abstract

We define a product 𝑙_𝑘,𝑛 for any positive real numbers 𝑘 and 𝑛 involving Ramanujan's theta-functions 𝜙(𝑞) and 𝜓(𝑞) which is analogous to Ramanujan's remarkable product of theta-functions recorded by Ramanujan (1957) and study its several properties. We prove general theorems for the explicit evaluations of 𝑙_𝑘,𝑛 and find some explicit values. As application of the product 𝑙_𝑘,𝑛, we also offer explicit formulas for explicit values of Ramanujan's continued fraction 𝑉(𝑞) in terms of 𝑙_𝑘,𝑛 and give examples.

1. Introduction

Ramanujan’s theta-functions 𝜙(𝑞), 𝜓(𝑞), and 𝑓(−𝑞) are defined as 𝜙𝑞∶=^∞_𝑛=−∞𝑞^𝑛2=−𝑞;−𝑞_∞𝑞;−𝑞_∞,𝜓𝑞∶=^∞_𝑛=0𝑞^{𝑛(𝑛+1)/2}=𝑞²;𝑞²_∞𝑞;𝑞²_∞,𝑓−𝑞∶=𝑓−𝑞,−𝑞²=^∞_𝑛=−∞(−1)^𝑛𝑞^{𝑛(3𝑛−1)/2}=𝑞;𝑞_∞, (1)(2) where (𝑎;𝑞)_∞∶=∏^∞_𝑘=0(1−𝑎𝑞^𝑘).

On page 338 of his first notebook, Ramanujan [1] defined the remarkable product of theta-functions as 𝑎_𝑚,𝑛=𝑛𝑒^{−(𝜋/4)(𝑛−1)𝑚/𝑛}𝜓²𝑒^{√−𝜋𝑚𝑛}𝜙²−𝑒^{√−2𝜋𝑚𝑛}𝜓²𝑒^{−𝜋𝑚/𝑛}𝜙²−𝑒^{−2𝜋𝑚/𝑛}, (3) where 𝑚 and 𝑛 are positive real numbers. He then, on pages 338 and 339, offered a list of eighteen particular values of the product 𝑎_𝑚,𝑛. All these eighteen values are proved by Berndt et al. [2]. An account of these can be found in Berndt’s book [3]. Naika and Dharmendra [4] also established some general theorems for explicit evaluations of the product 𝑎_𝑚,𝑛 and found some new explicit values therefrom. Further results on 𝑎_𝑚,𝑛 can be found in [5, 6].

In [7], Mahadeva Naika et al. defined the product 𝑏_𝑚,𝑛=𝑛𝑒^{−(𝜋/4)(𝑛−1)𝑚/𝑛}𝜓²−𝑒^{√−𝜋𝑚𝑛}𝜙²−𝑒^{√−2𝜋𝑚𝑛}𝜓²−𝑒^{−𝜋𝑚/𝑛}𝜙²−𝑒^{−2𝜋𝑚/𝑛}. (4) They established general theorems for explicit evaluations of 𝑏_𝑚,𝑛 and obtained some particular values. Mahadeva Naika et al. [8] established general formulas for explicit values of Ramanujan’s cubic continued fraction 𝑉(𝑞) in terms of the products 𝑎_𝑚,𝑛 and 𝑏_𝑚,𝑛 defined above, where 𝑉𝑞𝑞∶=^1/31+𝑞+𝑞²1+𝑞²+𝑞⁴1+𝑞³+𝑞⁶1+⋯,𝑞<1. (5) and found some particular values of 𝑉(𝑞).

Motivated by the above work, in this paper, we define the product of theta-functions 𝑙_𝑘,𝑛 as 𝑙_𝑘,𝑛=𝜓−𝑞𝜙𝑞^𝑘𝑞^(𝑘−1)/8𝜓−𝑞^𝑘𝜙𝑞;𝑞∶=𝑒^{−𝜋𝑛/𝑘}, (6) where 𝑛 and 𝑘 are positive real numbers. We establish several properties of the product 𝑙_𝑘,𝑛. We prove general formulas for explicit evaluations of 𝑙_𝑘,𝑛 and find its explicit values. As application of the product 𝑙_𝑘,𝑛, we also offer explicit formulas for explicit values of Ramanujan’s cubic continued fraction 𝑉(𝑞) in terms of 𝑙_𝑘,𝑛 and give examples.

In Section 2, we collect some preliminary results. In Section 3, we prove several properties of the product 𝑙_𝑘,𝑛. Section 4, is devoted to find explicit values of 𝑙_𝑘,𝑛. Finally in Section 5, we offer explicit formulas for explicit evaluations of continued fraction 𝑉(𝑞) in terms of 𝑙_𝑘,𝑛 with examples.

To end this introduction, we define Ramanujan’s modular equation. 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 (7) 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 (7). 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. By denoting 𝑧_𝑟=𝜙²(𝑞^𝑟), where 𝑞=exp(−𝜋𝐾′/𝐾),  |𝑞|<1, the multiplier 𝑚 connecting 𝛼 and 𝛽 is defined by 𝑚=𝑧₁/𝑧_𝑛.

2. Preliminary Results

Lemma 1 (see [9, p. 43, Entry 27(i)]). If 𝛼𝛽=𝜋 then √𝛼𝜙𝑒^−𝛼2=𝛽𝜙𝑒^−𝛽2. (8)

Lemma 2 (see [10, p. 1049, (1.13)]). If 𝛼𝛽=𝜋² then 𝑒√^−𝛼/84𝛼𝜓−𝑒^−𝛼=𝑒^−𝛽/84𝛽𝜓−𝑒^−𝛽. (9)

Lemma 3 (see [9, p. 122, Entry 10(i)]). One has 𝜙𝑞=√𝑧₁. (10)

Lemma 4 (see [9, p. 123, Entry 11(ii)]). One has 𝜓−𝑞=𝑧₁2{𝛼(1−𝛼)}^1/8𝑞^−1/8. (11)

Lemma 5 (see [9, p. 124, Entry 12(v), (vi)]). One has 𝜒𝑞=2^1/6𝛼(1−𝛼)𝑞^−1/24,𝜒−𝑞=2^1/6(1−𝛼)^1/12𝛼𝑞^−1/24. (12) One also notes that if we replace 𝑞 by 𝑞^𝑛 in the Lemmas 3 and 4, then 𝑧₁ and 𝛼 will be replaced by 𝑧_𝑛 and 𝛽, respectively, where 𝛽 has degree 𝑛 over 𝛼.

Lemma 6 (see [9, p. 345, Entry 1(i), (ii)]). If 𝑉(𝑞) is as defined in (5), then 11+𝑉𝑞=𝜓𝑞^1/3𝑞^1/3𝜓𝑞³,11+𝑉³𝑞=𝜓⁴𝑞𝑞𝜓⁴𝑞³,𝑞𝜙2𝑉=1−−𝑞^1/3𝜙−𝑞³. (13)(14)(15)

Lemma 7 (see [9, p. 347]). If 𝑉(𝑞) is as defined in (5), then 1−8𝑉³𝑞=𝜙⁴−𝑞𝜙⁴−𝑞³. (16)

Lemma 8 (see [9, p. 231, Entry 5(xii)]). Let 𝑃={16𝛼𝛽(1−𝛼)(1−𝛽)}^1/8 and 𝑄=(𝛽(1−𝛽)/𝛼(1−𝛼))^1/4, then 𝑄+𝑄⁻¹+2√2𝑃−𝑃⁻¹=0, (17) where 𝛽 has degree 3 over 𝛼.

Lemma 9 (see [9, p. 282, Entry 13(xiv)]). Let 𝑃={16𝛼𝛽(1−𝛼)(1−𝛽)}^1/12 and 𝑄=(𝛽(1−𝛽)/𝛼(1−𝛼))^1/8, then 𝑄+𝑄⁻¹+2𝑃−𝑃⁻¹=0, (18) where 𝛽 has degree 5 over 𝛼.

3. Some Properties of 𝑙_𝑘,𝑛

In this section, we study some properties of the product 𝑙_𝑘,𝑛.

Theorem 10. For all positive real numbers 𝑘 and 𝑛, one has (i)𝑙_𝑘,1=1, (ii)𝑙_𝑘,1/𝑛=1/𝑙_𝑘,𝑛, (iii)𝑙_𝑘,𝑛=𝑙_𝑛,𝑘.

Proof. Using the definition of 𝑙_𝑘,𝑛 and Lemmas 1 and 2, we easily arrive at (i). Replacing 𝑛 by 1/𝑛 in 𝑙_𝑘,𝑛 and using Lemmas 1 and 2, we find that 𝑙_𝑘,𝑛𝑙_𝑘,1/𝑛=1 which completes the proof of (ii). To prove (iii), we interchange 𝑛 and 𝑘 in 𝑙_𝑘,𝑛.

Remark 11. By using the definitions of 𝜙(𝑞), 𝜓(𝑞), and 𝑙_𝑘,𝑛, it can be seen that 𝑙_𝑘,𝑛 has positive real value and that the values of 𝑙_𝑘,𝑛 increases as 𝑛 increases when 𝑘>1. Thus, by Theorem 10(i), 𝑙_𝑘,𝑛>1 for all 𝑛>1 if 𝑘>1.

Theorem 12. For all positive real numbers k, m, and n, one has 𝑙_{𝑘,𝑛/𝑚}=𝑙_{𝑚𝑘,𝑛}𝑙_{𝑛𝑘,𝑚}. (19)

Proof. Using the definition of 𝑙_𝑘,𝑛, we find that 𝑙_{𝑚𝑘,𝑛}𝑙_{𝑛𝑘,𝑚}=𝑒^{−𝜋(𝑛/𝑚𝑘−𝑚/𝑛𝑘)/8}𝜓−𝑒^{−𝜋𝑛/𝑚𝑘}𝜙𝑒^{−𝜋𝑚/𝑛𝑘}𝜓−𝑒^{−𝜋𝑚/𝑛𝑘}𝜙𝑒^{−𝜋𝑛/𝑚𝑘}. (20) Applying Lemmas 1 and 2 in the denominator of right hand side of (20) and simplifying using Theorem 10(ii) and (iii), we complete the proof.

Corollary 13. For all positive real numbers k and n, one has 𝑙_𝑘²_,𝑛=𝑙_{𝑛𝑘,𝑛}𝑙_{𝑘,𝑛/𝑘}. (21)

Proof. Setting 𝑘=𝑛 in Theorem 12 and simplifying using Theorem 10(ii), we obtain 𝑙_𝑘²_,𝑚=𝑙_{𝑚𝑘,𝑘}𝑙_{𝑘,𝑚/𝑘}. (22) Replacing 𝑚 by 𝑛, we complete the proof.

Theorem 14. Let k, a, b, c, and d be positive real numbers such that 𝑎𝑏=𝑐𝑑. Then 𝑙_𝑎,𝑏𝑙_{𝑘𝑐,𝑘𝑑}=𝑙_{𝑘𝑎,𝑘𝑏}𝑙_𝑐,𝑑. (23)

Proof. From the definition of 𝑙_𝑘,𝑛 and using 𝑎𝑏=𝑐𝑑 for positive real numbers 𝑘,𝑎,𝑏,𝑐, and 𝑑, we deduce that 𝑙_{𝑘𝑎,𝑘𝑏}𝑙⁻¹_𝑎,𝑏=𝑙_{𝑘𝑐,𝑘𝑑}𝑙⁻¹_𝑐,𝑑. (24) Rearranging the terms in (24) we arrive at the desired result.

Corollary 15. For any positive real numbers 𝑛 and 𝑝, one has 𝑙_{𝑛𝑝,𝑛𝑝}=𝑙_{𝑛𝑝2,𝑛}𝑙_𝑝,𝑝. (25)

Proof. The result follows immediately from Theorem 14 with 𝑎=𝑝²,𝑏=1,𝑐=𝑑=𝑝 and 𝑘=𝑛.

Theorem 16. For all positive real numbers 𝑎, 𝑏, 𝑐, and 𝑑, one has 𝑙_{𝑎/𝑏,𝑐/𝑑}=𝑙_{𝑎𝑑,𝑏𝑐}𝑙_{𝑎𝑐,𝑏𝑑}. (26)

Proof. Applying Theorem 10(iii) in Theorem 12, we deduce that, for all positive real numbers 𝑎, 𝑏, and 𝑛𝑙_{𝑎/𝑏,𝑛}=𝑙_{𝑎,𝑏𝑛}𝑙⁻¹_{𝑏,𝑎𝑛}. (27) Now setting 𝑛=𝑐/𝑑 and again employing Theorems 12 and 10(iii) in (27), we complete the proof.

Theorem 17. For all positive real numbers 𝑎 and 𝑏, one has (i)𝑙_{𝑎/𝑏,𝑎/𝑏}=𝑙_𝑏,𝑏𝑙_{𝑎,𝑎/𝑏}², (ii)𝑙_𝑎,𝑎𝑙_𝑎,𝑏²_/𝑎=𝑙_𝑏,𝑏𝑙_{𝑏,𝑎2/𝑏}, (iii)𝑙_𝑎,𝑎𝑙_{𝑏,𝑎2𝑏}=𝑙_{𝑏,𝑏𝑎,𝑎𝑏}².

Proof. By using Theorems 10(ii) and 16, we find that 𝑙_{𝑎/𝑏,𝑎/𝑏}=𝑙_𝑏,𝑏𝑙_{𝑎,𝑎/𝑏}². (28) So we complete the proof of (i). Setting 𝑛=𝑎/𝑏 in (27), we find that 𝑙_{𝑎/𝑏,𝑎/𝑏}=𝑙_𝑎,𝑎𝑙⁻¹_{𝑏,𝑎2/𝑏}. (29) Now (ii) follows from (28) and (29) and Theorem 10(ii).
By using Theorems 10(ii) and 12, we find that 𝑙_{𝑎/𝑏,𝑎/𝑏}=𝑙_𝑏,𝑏𝑙_𝑎𝑏²_,𝑎𝑙_𝑎⁻¹_2,𝑏2. (30) Similarly, we find that 𝑙_{𝑏/𝑎,𝑏/𝑎}=𝑙_𝑎,𝑎𝑙_{𝑎2𝑏,𝑏}𝑙_𝑏⁻¹_2,𝑎2. (31) From (30), (31), and Theorem 10(ii) and (iii), we complete the proof of (iii).

4. Explicit Values of 𝑙_𝑘,𝑛

In this section, we prove general theorems for explicit evaluations of 𝑙_𝑘,𝑛 and find some explicit values. First we define Ramanujan’s class invariants. Ramanujan’s two class invariants 𝐺_𝑛 and 𝑔_𝑛 which are defined by 𝐺_𝑛=2^−1/4𝑞^−1/24𝜒𝑞,𝑔_𝑛=2^−1/4𝑞^−1/24𝜒−𝑞,𝑞∶=𝑒^{√−𝜋𝑛}, (32) where 𝜒(𝑞)=(−𝑞;𝑞²)_∞ and 𝑛 is a positive rational number. Employing Lemma 5 in (32) it follows that 𝐺_𝑛={4𝛼(1−𝛼)}^−1/24,𝑔_𝑛=2^−1/12(1−𝛼)^1/12𝛼^−1/24. (33) Also, if 𝛽 has degree 𝑟 over 𝛼, then 𝐺_𝑟2𝑛=4𝛽1−𝛽^−1/24,𝑔_𝑟2𝑛=2^−1/121−𝛽^1/12𝛽^−1/24. (34) In his notebooks [1] and paper [11], Ramanujan recorded a total of 116 class invariants or monic polynomials satisfied by them. An account of these can also be found in Berndt’s book [3]. For further references, see [2, 12–17].

Theorem 18. One has 𝑙_𝑘,𝑛=𝐺_𝑛𝑘𝐺_𝑛/𝑘³. (35)

Proof. Employing Lemmas 3 and 4 in the definition of 𝑙_𝑘,𝑛, we find that 𝑙_𝑘,𝑛=𝛼(1−𝛼)𝛽1−𝛽^1/8. (36) Again from (32) and (33) with 𝑞∶=𝑒^{−𝜋𝑛/𝑘}, we find that 𝐺_𝑛𝑘𝐺_𝑛/𝑘=𝛼(1−𝛼)𝛽1−𝛽^1/24. (37) Combining (36) and (37), we complete the proof.

From Theorem 18, it is obvious that if we know (𝐺_𝑛𝑘/𝐺_𝑛/𝑘) or (𝐺_𝑛𝑘𝑎𝑛𝑑𝐺_𝑛/𝑘) then explicit values of 𝑙_𝑘,𝑛 can easily be determined. For example, we find some values of 𝑙_𝑘,𝑛 in next theorem by employing the values of (𝐺_𝑛𝑘/𝐺_𝑛/𝑘) or (𝐺_𝑛𝑘𝑎𝑛𝑑𝐺_𝑛/𝑘) from literature.

Theorem 19. One has(i)𝑙_3,5=(𝐺₁₅𝐺_5/3)³=3+√52, (ii)𝑙_3,7=(𝐺₂₁/𝐺_7/3)³=(3√3+2√7)^1/2, (iii)𝑙_3,11=(𝐺₃₃/𝐺_11/3)³=(3+√11/√2), (iv)𝑙_3,13=(𝐺₃₉/𝐺_13/3)³=1/2^3/2((3+√13)/2+(3+3√13)/2^3/2),(v)𝑙_5,11=(𝐺₅₅/𝐺_11/5)³=1/2((9+5√5)/2+(95+45√5)/2), (vi)𝑙_5,13=(𝐺₆₅/𝐺_13/5)³=((7+√65)/8+(√65−1)/8)³, (vii)𝑙_3,23=(𝐺₆₉/𝐺_23/3)³=(24+5√23)^1/4(25+3√69/2)^3/8, (viii)𝑙_3,47=(𝐺₁₄₁/𝐺_47/3)³=(48+7√47)^3/12(95+8√141)^3/8, (ix)𝑙_3,71=(𝐺₂₁₃/𝐺_71/3)³=(3480+41371)^1/4((73+5√213)/2)^3/8.

Proof. For (i)–(vi), the corresponding values of (𝐺_𝑛𝑘/𝐺_𝑛/𝑘) can be found in equations (4.10), (4.20), (4.29), (4.37), (4.51), and (4.60), respectively of [12]. For (vii) to (ix), the corresponding values of can be found in equations [p.227,(5.8)], [p.229,(5.15)], and [p.233,(5.25)], respectively of [3].

Corollary 20. One has 𝑙_𝑛,𝑛=𝐺_𝑛2³. (38)

Proof. We set 𝑘=𝑛 in Theorem 18 and use the result 𝐺₁=1, to complete the proof.

From Corollary 20, it is obvious explicit values of 𝑙_𝑛,𝑛 can easily be determined if the corresponding values of the class invariants 𝐺_𝑛2 are known. We give some examples in the next theorem.

Theorem 21. One has(i)𝑙_2,2=𝐺³₄=(1+√2)^3/4/2^9/16, (ii)𝑙_4,4=𝐺³₁₆=2^−3/4(16+152^1/4+12√2+92^3/4)³, (iii)𝑙_3,3=𝐺³₉=(1+√3)/√2, (iv)𝑙_5,5=𝐺³₂₅=((7^1/4+4+√7)/(2))³, (v)𝑙_7,7=𝐺³₄₉=((1+√5)/(2))³, (vi)𝑙_9,9=𝐺³₈₁=(³2(√3+1)+1)/(³2(√3−1)+1).

Proof. For (i) and (ii), we use the values of 𝐺₄ and 𝐺₁₆ from [18, p. 114-115, Theorem 6.2.2(ii) (vi)]. For (iii)–(vi), we use the corresponding values of 𝐺_𝑛 from [3, p. 189-193].

Ramanujan’s Schalfli type modular equations of prime degree can also be used to explicit values of the product 𝑙_𝑘,𝑛. We offer two theorems as examples.

Theorem 22. One has 𝑙⁻²_3,𝑛+𝑙²_3,𝑛+2√2𝐺_3𝑛𝐺_𝑛/3⁻³−𝐺_3𝑛𝐺_𝑛/3³=0. (39)

Proof. We use the definitions of class invariants 𝑙_𝑘,𝑛 and 𝐺_𝑛 in Lemma 8, to complete the proof.

Theorem 23. One has 𝑙_5,𝑛+𝑙⁻¹_5,𝑛+2𝐺_5𝑛𝐺_𝑛/5⁻²−𝐺_5𝑛𝐺_𝑛/5²=0. (40)

Proof. We use the definitions of class invariants 𝑙_𝑘,𝑛 and 𝐺_𝑛 in Lemma 9, to complete the proof.

5. Evaluations of Ramanujan’s Cubic Continued Fraction

In this section, we prove two formulas of explicit evaluations Ramanujan’s cubic continued fraction 𝑉(𝑞) in terms of the product 𝑙_𝑘,𝑛. We also give examples.

Theorem 24. One has 𝑉³−𝑒^{−𝜋𝑛/3}=𝑙⁴_3,𝑛+1−𝑙⁸_3,𝑛+34𝑙⁴_3,𝑛+116𝑙⁴_3,𝑛. (41)

Proof. Replacing 𝑞 by −𝑞 in (14) and Lemma 7, we obtain −11+𝑉³𝑞=𝜓⁴−𝑞𝑞𝜓⁴−𝑞³,1−8𝑉³=𝜙−𝑞⁴𝑞𝜙⁴𝑞³. (42)(43) Dividing (42) by (43), setting 𝑞∶=𝑒^{−𝜋𝑛/3} and simplifying using the definition of 𝑙_𝑘,𝑛, we obtain 8𝑙⁴_3,4𝑉⁶−𝑒^{−𝜋𝑛/3}−𝑙⁴_3,4+1𝑉³−𝑒^{−𝜋𝑛/3}−1=0. (44) Solving (2) for 𝑉³(−𝑒^{−𝜋𝑛/3}) and noting that 𝑉(−𝑒^{−𝜋𝑛/3})<0 and the fact in Remark 11, we arrive at the desired result.

From Theorem 24, it is clear that to find explicit values of 𝑉³(−𝑒^{−𝜋𝑛/3}) it is enough to know the explicit values of 𝑙_3,𝑛. For example, noting 𝑙_3,1=1 from Theorem 10(i) we evaluate 𝑉³−𝑒^√−𝜋/3=−1/4. (45)

Theorem 25. One has 𝑉−𝑒^{√−𝜋𝑛}=𝑙_9,𝑛+1−𝑙²_9,𝑛+10𝑙_9,𝑛+14𝑙_9,𝑛. (46)

Proof. Replacing 𝑞 by 𝑞³ and then replacing 𝑞 by −𝑞 in (13) and (15), we obtain −11+𝑉−𝑞³=𝜓−𝑞𝑞𝜓−𝑞⁹,1−2𝑉−𝑞³=𝜙𝑞𝜙𝑞⁹. (47)(48) Dividing (47) by (48), setting 𝑞∶=𝑒^{−𝜋𝑛/9} and simplifying by using the definition of 𝑙_𝑘,𝑛, we arrive at 2𝑙_9,𝑛𝑉²−𝑒^{√−𝜋𝑛}−𝑙_9,𝑛+1𝑉−𝑒^{√−𝜋𝑛}−1=0. (49) Solving (49) for 𝑉(−𝑒^{√−𝜋𝑛}) and noting 𝑉(−𝑒^{√−𝜋𝑛})<0 and the fact in Remark 11, we complete the proof.

From Theorem 25, it is obvious that if we know the values of 𝑙_9,𝑛 the corresponding values of 𝑉(−𝑒^{√−𝜋𝑛}) can easily be evaluated. For example, by noting 𝑙_9,1=1 from Theorem 10(i) we calculate 𝑉−𝑒^−𝜋=1−√32. (50) Similarly, one can find explicit values of 𝑉(−𝑒^−3𝜋) by using the value of 𝑙_9,9 from Theorem 21(vi).

Acknowledgments

The author is thankful to University Grants Commission, New Delhi, India for partially supporting the research work under the grant no. F. No. 41-1394/2012(SR).