ON EVALUATIONS OF THE CUBIC CONTINUED FRACTION BY A MODULAR EQUATION OF DEGREE 9

Abstract

We show how to evaluate the cubic continued fraction $G(e^{-{\pi}\sqrt{n}})$ and $G(-e^{-{\pi}\sqrt{n}})$ for n = 4^m, 4^−m, 2 · 4^m, and 2⁻¹ · 4^−m for some nonnegative integer m by using modular equations of degree 9. We then find some explicit values of them.

1. INTRODUCTION

The cubic continued fraction G(q) is defined by

Ramanujan investigated G(q) and claimed that there are many results which are similar to those for the Rogers-Ramanujan continued fraction F(q), where

Motivated by Ramanujan’s claim, there has been interest by number theorists in evaluating the numerical values of and for some positive real number n. Using Ramanujan’s class invariants, Berndt, Chan, and Zhang [3] evaluated the values of for n = 2, 10, 22, 58 and for n = 1, 5, 13, 37. In addition, Chan [4] established the values of for n = 1, 2, 4, and for n = 1, 5 by using some reciprocity theorems for G(q). Yi [6] found the values of for n = 2, 3, 4, 6, 7, 8, 10, 12, 16, 28, , , , , , , and for n = 1, 2, 3, 4, 5, 7, , , , by employing relations among G(q), Ramanujan-Weber class invariants, and some parameters for eta function. In [8], the values of for n = 1, 4, 9, and for n = 1, 4, 9 were evaluated by using some modular equations of degrees 3 and 9. Moreover, Paek and Yi [5] evaluated the values of for n = 36, 81, 144, 324, , , and for n = 36, , by employing some modular equations of degrees 3 and 9.

In this paper, we further show how to evaluate explicit values of and for n = 4m, 4−m, 2 · 4m, and 2−1 · 4−m for some nonnegative integer m by using modular equations of degree 9.

Since a modular equation is crucial for evaluating such cubic continued fraction, we now give a definition of a modular equation. For |ab| < 1, define Ramanujan’s general theta function f by

Moreover, define the theta functions φ and ψ by, for |q| < 1,

and

where

Let a, b, and c be arbitrary complex numbers except that c cannot be a nonpositive integer. Then, for |z| < 1, the Gaussian or ordinary hypergeometric function 2F1(a, b; c; z) is defined by

where (a)0 = 1 and (a)n = a(a + 1)(a + 2). . .(a + n − 1) for each positive integer n.

Now the complete elliptic integral of the first kind K(k) is defined by

where 0 < k < 1, K' = K(k'), and . The number k is called the modulus of K and k' is called the complementary modulus.

Let K, K', L, and L' denote complete elliptic integrals of the first kind associated with the moduli k, k', l, and l', respectively, where 0 < k < 1 and 0 < l < 1. Suppose that

holds for some positive integer n. A relation between k and l induced by (1.2) is called a modular equation of degree n.

If we set

and ,

we see that (1.2) is equivalent to the relation qn = q'. Hence a modular equation can be viewed as an identity involving theta functions at the arguments q and qn. Following Ramanujan, set α = k2 and β = l2, then we say that β has degree n over α. By the relationship between complete elliptic integrals of the first kind and hypergeometric function, we have

Let zn = φ2(qn). Then the multiplier m for degree n is defined by

We recall the definition of the parameterizations and for the theta functions φ and ψ from [7, 9]. For any positive real numbers k and n, define by

where , and define by

where . For convenience, we write and instead of and , respectively, throughout this paper. We end this section by noting that

and

from [8, Theorem 4.1] and [9, Theorem 3.4], which will play crucial roles in evaluating the values of cubic continued fraction.

 

2. PRELIMINARY RESULTS

In this section, we introduce fundamental theta function identities that will play key roles in deriving a modular equation of degree 9. Let k be the modulus as in (1.1). Set x = k2 and also set

Then

where

Lemma 2.1 ([1, Theorems 5.4.1 and 5.4.2]). If x, q, and z are related by (2.1), (2.2), and (2.3), then

(i) , (ii) , (iii) .

Lemma 2.2 ([2, Entry 3, Chapter 20]). Let γ be the ninth degree and , then

(i) (ii)

The following results exhibit formulas for evaluating in terms of and in terms of .

Lemma 2.3 ([9, Theorem 6.2]). For any positive real number n, we have

(i) (ii)

We end this section by stating the following identity:

Lemma 2.4 ([6, Lemma 6.3.6]). We have

= −

for any positive real number n.

 

3. MODULAR EQUATIONS

In this section, we first derive a modular equation of degree 9 to establish some explicit relation for and for any positive real number n.

Theorem 3.1. If and , then

Proof. By Lemma 2.1,

and ,

where γ has degree 9 over α. Thus

By Lemma 2.2,

and

Combining the last two identities in terms of P and Q, we deduce that

This then completes the proof. □

Corollary 3.2. For any positive real number n, we have

Proof. Let in (1.3). Then P and Q in Theorem 3.1 can be written as and . Rewrite (3.1) in terms of and to complete the proof. □

We next recall a modular equation of degree 9 given in [8] to employ an explicit relation for and for any positive real number n.

Theorem 3.3 ([8, Theorem 3.15]). If and , then

Corollary 3.4 ([8, Corollary 3.16]). For any positive real number n, we have

 

4. EVALUATIONS OF , , AND

In this section, we show how to evaluate the values of , , and for every positive integer m by employing the relations (3.2) and (3.4). We first need the following:

Lemma 4.1. For any nonnegative integer m,

Proof. Let for brevity. Since am > 0 for any nonnegative integer m from the definition of , we have . Hence it is enough to show that for any nonnegative integer m. We prove by induction on m. For m = 0, since from [8, Theorem 4.1], which is approximately equal to 0.44, it follows that . Now assume that for some nonnegative integer k. Then, by (3.2),

Solving the last equality for ak+1 and using the fact that and ak+1 > 0, we have

Since

or equivalently,

it follows that

Dividing both sides of the last inequality by , we conclude that

which completes the proof. □

The following result exhibits an algorithm for evaluating the values of for all positive integers m.

Theorem 4.2. We have

for any nonnegative integer m.

Proof. It is an immediate consequence of Corollary 3.2 and Lemma 4.1. □

We are now ready to show how to evaluate the values of for every positive integer m. We only exhibit the cases when m = 1, 2, and 3.

Corollary 4.3. We have

(i) , (ii) , (iii) ,

where

Proof. For (i), letting m = 0 in (4.2) and putting the value of

from [8, Theorem 4.1], we complete the proof.

For (ii), letting m = 1 in (4.2) and putting the value of from the previous result of (i), we complete the proof.

Part (iii) is clear from Theorem 4.2. □

The following results show a method for evaluating the values of for all positive integers m. We only exhibit the cases when m = 1, 2, and 3.

Theorem 4.4. We have

(i) (ii) (iii)

where

Proof. For (i), letting n = 1 in (3.2), putting the value of

from [8, Theorem 4.1], solving for , and using the fact that has a positive value less than 1, we complete the proof.

For (ii), letting n = in (3.2), putting the value of from the previous result of (i), solving for , and using the fact that has a positive value less than 1, we complete the proof.

For (iii), repeat the same argument as in the proof of (ii). □

We next show how to evaluate the values of for every positive integer m. We only exhibit the cases when m = 1, 2, and 3.

Theorem 4.5. We have

(i) (ii) (iii)

where

Proof. For (i), letting n = 2 in (3.4) and putting the value of

from [9, Theorem 3.4], solving for , and using the fact that has a positive value greater than 1, we complete the proof.

For (ii), letting n = 8 in (3.4), putting the value of from the previous result of (i), solving for , and using the fact that has a value greater than 1, we complete the proof.

For (iii), repeat the same argument as in the proof of (ii). □

The following results show a method for evaluating the values of for every positive integer m. We only exhibit the cases when m = 1, 2, 3, and 4.

Theorem 4.6. We have

(i) (ii) (iii) (iv)

where

Proof. For (i), letting in (3.4), putting the value of

from [7, Theorem 4.16], solving for , and using the fact that has a positive value, we complete the proof.

For (ii), letting in (3.4), putting the value of from the previous result of (i), solving for , and using the fact that has a positive value, we complete the proof.

For (iii) and (iv), repeat the same argument as in the proof of (ii). □

 

5. EVALUATIONS OF G(q)

We turn to evaluations of and for n = 4m, 4−m, 2 · 4m, and 2−1 · 4−m for some positive integer m. We first find G(e−2mπ) for m = 2, 3, and 4 and G(−e−2mπ) for m = 2 and 3.

Theorem 5.1. We have

(i) , (ii) (iii)

where

Proof. For (i), letting n = 4 in Lemma 2.3(i) and putting the value of from Corollary 4.3(i), we complete the proof.

For (ii) and (iii), repeat the same argument as in the proof of (i). □

See [6, Theorem 6.3.7(iii)] for an alternative proof for Theorem 5.1(i), where G(e−4π) was given by

Corollary 5.2. We have

(i) (ii)

where

Proof. Parts (i) and (ii) follow directly from Lemma 2.4 and Theorem 5.1. □

We next find G(e−π/2m) for m = 0, 1, and 2 and G(−e−π/2m) for m = 1 and 2.

Theorem 5.3. We have

(i) , (ii) , (iii)

where

Proof. Part (i) follows directly from Lemma 2.3(i) and Theorem 4.4(i). The proofs of Parts (ii) and (iii) are similar to that of Part (i). □

See [4] for a different proof for Theorem 5.3(i), where G(e−π) was given by

See also [6, Theorem 6.3.3(vii)] for an alternative proof for Theorem 5.3(ii), where G(e−π/2) was given by

Corollary 5.4. We have

(i) (ii)

where

Proof. The results follow directly from Lemma 2.4 and Theorem 5.3. □

See also [6, Theorem 6.3.5(vii)] for an alternative proof for Theorem 5.4(i), where G(−e−π/2) was given by

We now find for m = 1, 2, and 3 and for m = 1 and 2.

Theorem 5.5. We have

(i) , (ii) (iii)

where

Proof. For (i), letting n = 8 in Lemma 2.3(ii) and putting the value of from Corollary 4.5(i), we complete the proof.

For (ii) and (iii), repeat the same argument as in the proof of (i). □

See also [6, Theorem 6.3.7(i)] for an alternative proof for Theorem 5.5(i), where was given by

Corollary 5.6. We have

(i) (ii)

where

Proof. Parts (i) and (ii) follow directly from Lemma 2.4 and Theorem 5.5. □

We end this section by evaluating for m = 0, 1, 2 and 3 and for m = 1, 2, and 3.

Theorem 5.7. We have

(i) (ii) (iii)

where

Proof. Part (i) follows directly from Lemma 2.3 and Theorem 4.6(i). The proofs of Parts (ii), (iii), and (iv) are similar to that of Part (i). □

Corollary 5.8. We have

(i) (ii) (iii)

where

Proof. The results follow directly from Lemma 2.4 and Theorem 5.7. □