Two theta-function identities for the Ramanujan–Selberg continued fraction and applications

Journal of Number Theory 151 (2015) 46–53

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Two theta-function identities for the Ramanujan–Selberg continued fraction and applications Nipen Saikia Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh-791112, Arunachal Pradesh, India

a r t i c l e

i n f o

Article history: Received 29 May 2014 Received in revised form 10 October 2014 Accepted 15 December 2014 Available online 7 February 2015 Communicated by David Goss

a b s t r a c t We prove two theta-function identities for the Ramanujan– Selberg continued fraction which are analogous to those of the Rogers–Ramanujan continued fraction. These identities are then used to prove reciprocity formulas and general theorems for the explicit evaluations of the Ramanujan–Selberg continued fraction. © 2015 Elsevier Inc. All rights reserved.

MSC: primary 33D90 secondary 11F20 Keywords: Ramanujan–Selberg continued fraction Theta-function identities Reciprocity formulas Explicit evaluations

1. Introduction The Ramanujan–Selberg continued fraction S(q) is defined by S(q) :=

q 1/8 q 1/8 ψ(q) q q2 q3 = 2 φ(q) 1 + 1 + q + 1 + q + 1 + q 3 +···

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jnt.2014.12.020 0022-314X/© 2015 Elsevier Inc. All rights reserved.

|q| < 1,

(1.1)

N. Saikia / Journal of Number Theory 151 (2015) 46–53

47

where, for q := e2πiz and Im(z) > 0, φ(q) :=

∞ 

2

qn =

n=−∞

ψ(q) :=

∞ 

(−q; q 2 )∞ (q 2 ; q 2 )∞ , (q; q 2 )∞ (−q 2 ; q 2 )∞

q n(n+1)/2 =

n=0

(1.2)

(q 2 ; q 2 )∞ , (q; q 2 )∞

(1.3)

f (−q) := (q; q)∞ ,

(1.4)

and (a; q)∞ :=

∞ 

(1 − aq k ).

k=0

The continued fraction S(q) was recorded by Ramanujan at the beginning of Chapter 19 of his second notebook [2, p. 221]. The equality in (1.1) was proved by Ramanathan [5]. Zhang [12] established general formulas for explicit evaluations of S(q) in terms of Ramanujan’s singular moduli. Baruah and Saikia [1] established some modular identities for S(q) and also proved some general theorems for the explicit evaluations of S(q). In this paper, we study further properties of the Ramanujan–Selberg continued fraction S(q). We prove two theta-function identities connecting S(q) and f (−q) in Theorems 3.1 and 3.2. As applications to these identities, we prove new general theorems for the explicit evaluations of S(q) in Theorems 4.2 and 4.4 and reciprocity formulas in Theorems 5.1 and 5.3 which are analogous to those of famous Rogers–Ramanujan continued fraction R(q) defined by R(q) :=

q 1/5 q q 2 q 3 . 1 + 1 + 1 + 1 +···

(1.5)

2. Preliminaries For q = e−π

√ n

, Ramanujan’s class invariants Gn and gn are defined by

Gn = 2−1/4 q −1/24

f (q) f (−q 2 )

and

gn = 2−1/4 q −1/24

f (−q) . f (−q 2 )

(2.1)

In his paper [6] and notebooks [7], Ramanujan recorded a total of 116 class invariants. The table at the end of Weber’s book [10, pp. 721–726] contains the values of 107 class invariants. An account of Ramanujan’s class invariants and applications can also be found in Berndt’s book [3]. √ −π n Again, for q = e and any positive real number n, define the parameter Jn by Jn = √

f (−q) . 2q 1/8 f (−q 4 )

(2.2)

48

N. Saikia / Journal of Number Theory 151 (2015) 46–53

The parameter Jn is the particular case k = 4 of the general parameter rk,n defined by Berndt [4, p. 9, (4.6)] as rk,n :=

f (−q) , 1/4 (k−1)/24 k q f (−q k )

q = e−2π



n/k

,

(2.3)

where n and k are positive real numbers. Lemma 2.1. (See [2, p. 43, Entry 27(iii)].) If α and β are such that the modulus of each exponential argument is less than 1 and αβ = π 2 , then  √ e−α/12 4 αf (−e−2α ) = e−β/12 4 βf (−e−2β ). Lemma 2.2. (See [11, p. 21, Theorem 3.2.2].) If P =  4

then

(P Q) +

2 PQ

f (−q) q 1/24 f (−q 2 )

4

 =

Q P

(2.4) and Q =

f (−q 2 ) q 1/12 f (−q 4 )

12 .

For a different proof of Lemma 2.2, see [8, p. 200, Lemma 2.12]. Lemma 2.3. We have ψ(q) =

f 2 (−q 2 ) f (−q)

and

φ(q) =

f 5 (−q 2 ) f 2 (−q)f 2 (−q 4 )

.

Proof. Proof follows from [2, pp. 39–40, Entries 24 & 25]. 2 Lemma 2.4. (See [1, p. 9, Theorem 6.1].) If Jn is as defined in (2.2), then J1/n =

1 . Jn

3. Theta-function identities for S(q) In this section, we prove two new theta-function identities for the Ramanujan–Selberg continued fraction S(q). Theorem 3.1. We have 1 S 4 (q)

− 16S 4 (q) =

f 12 (−q) q 1/2 f 12 (−q 2 )

.

Proof. Employing Lemma 2.3 in (1.1), we note that S(q) =

P q 1/8 f (−q)f 2 (−q 4 ) = 2, f (−q 2 )f 2 (−q 2 ) Q

(3.1)

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where P and Q are as defined in Lemma 2.2. Rewriting the identity in Lemma 2.2 as  P

12

+ 16

P Q2

4

 =

Q2 P

4 ,

(3.2)

employing (3.1) and simplifying, we arrive at the desired result. 2 Theorem 3.2. We have f 8 (−q) 1 − 16 = . S 8 (q) qf 8 (−q 4 ) Proof. Rewriting the identity in Lemma 2.2 as  8

8

P Q + 16 =

Q2 P

8 ,

(3.3)

employing (3.1) and simplifying, we complete the proof. 2 4. General theorems for the explicit evaluations of S(q) This section is devoted to prove new general theorems for the explicit evaluations of S(q) and give examples. Theorem 4.1. We have (i) g2/n =

1 g2n

and

(ii) g2 = 1.

Proof. (i) follows from the definition of gn and Lemma 2.1. (ii) follows from part (i) with n = 1. 2 One can also prove Theorem 4.1(i) by replacing n by 2n in the result g4/n = 1/gn which is established by Ramanujan [6]. Theorem 4.2. We have (i) (ii) √

1

√ S 4 (e−π 2n )

1 S 4 (e−π



− 16S 4 (e−π

2/n )

√ 2n

− 16S 4 (e−π



12 ) = 8g2n ;

2/n

)=

8 12 . g2n

Proof. We set q = e−π 2n in Theorem 3.1 and employ the definition of gn to arrive at (i). To prove (ii), we replace n by 1/n in Theorem 4.2(i) and employ Theorem 4.1(i). 2

N. Saikia / Journal of Number Theory 151 (2015) 46–53

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Remark 4.3. From Theorem 4.2(i) and (ii), it isclear that if we know explicit values of √ −π 2n g2n , then we can evaluate S(e ) and S(e−π 2/n ), respectively. Ramanujan recorded several values of g2n in his notebooks. An account of list of values of g2n can be found in [3, pp. 200–204]. Saikia [8,9] also evaluated several new values of g2n . For example, from [3, p. 200] we note that  g22 =

1+

√ 2.

(4.1)

Setting n = 11 in Theorem 4.2(i) and employing the value of g22 , we find that 1

√ S 4 (e−π 22 )

− 16S 4 (e−π

√ 22

 √ 6 )=8 1+ 2 .

(4.2)

Solving (4.2) and choosing the positive real root, we evaluate S(e−π

√ 22

)=

1 2



  √ √ 1/4 −396 − 280 2 + 12 2178 + 1540 2 .

Similarly, setting n = 11 in Theorem 4.2(ii), employing the value of g22 , solving the resulting equation, and choosing positive real root, we evaluate S(e−π

 2/11

)=

1 2



 1/4  √ √ −1 + 3 2178 + 1540 2 . 396 − 280 2

Theorem 4.4. We have (i) (ii)

1

√ S 8 (e−π n )

− 16 = 16Jn8 ;

1 16 √ − 16 = 8 . Jn S 8 (e−π/ n )

√

Proof. Setting q = e−π n in Theorem 3.2 and employ the definition of Jn to arrive at (i). To prove (ii), we replace n by 1/n in part (i) and employ Lemma 2.4. 2 √

Remark 4.5.√From Theorem 4.4(i) and (ii), it is obvious that explicit values of S(e−π n ) and S(e−π/ n ) can easily be evaluated if we know explicit values of the parameter Jn . Baruah and Saikia [1] evaluated several explicit values of the parameter Jn . For example from [1, p. 9, Theorem 6.1(v)], we note that √ J7 = (8 + 3 7 )1/8 .

(4.3)

Setting n = 7 in Theorem 4.4(i), employing the value of J7 from (4.3), solving the resulting equation, and choosing positive real root, we obtain

N. Saikia / Journal of Number Theory 151 (2015) 46–53

S(e−π

√ 7

)=

51

√ 1/8 1 . 8−3 7 2

(4.4)

Similarly, setting n = 7 in Theorem 4.4(ii) and using the value of J7 from (4.3), we evaluate S(e−π/

√ 7

)=

√ 1/8 1 . 8+3 7 2

(4.5)

5. Reciprocity formulas In this section, we prove two reciprocity formulas for the continued fraction S(q). Theorem 5.1. If a and b are positive real numbers with ab = 2, then



1 − 16S 4 (e−πa ) 4 S (e−πa )



1 − 16S 4 (e−πb ) 4 S (e−πb )

= 64.

Proof. From Theorem 3.1, we deduce that



 1 1 4 −πa 4 −πb − 16S − 16S (e ) (e ) S 4 (e−πa ) S 4 (e−πb ) 12  f (−eπa )f (−e−πb ) . = e−π(a+b)/24 f (−e−2πa )f (−e−2πb )

Setting α = πb and β =

(5.1)

π in Lemma 2.1 and noting ab = 2, we deduce that b f (−e−πa ) e−πb/24 f (−e−2πb )

√ = e−π(b−a)/24 b.

(5.2)

Similarly, by interchanging the role of a and b in (5.2), we deduce that √ f (−e−πb ) = e−π(a−b)/24 a. −πa/24 −2πa e f (−e )

(5.3)

Employing (5.2) and (5.3) in (5.1) and noting ab = 2, we complete the proof. 2 Remark 5.2. Theorem 5.1 implies that if we know explicit values of S(e−πa ), then explicit √ values of S(e−2π/a ) can be√determined. For example, setting a = 7 in Theorem 5.1, employing the value S(e−π 7 ) from (4.4), solving the resulting equation, and choosing the positive real root, we obtain S(e

√ −2π/ 7

    √ √ 1/4 √ 1/4 1  3 71 − 16 7 − 8 8 − 3 7 )= √ 8−3 7 . 2

(5.4)

N. Saikia / Journal of Number Theory 151 (2015) 46–53

52

Theorem 5.3. If a and b are positive real numbers with ab = 1, then



 1 1 − 16 − 16 = 256. S 8 (e−πa ) S 8 (e−πb ) Proof. From Theorem 3.2, we deduce that

1 − 16 S 8 (e−πa )



Setting α = 2πb and β =

1 − 16 S 8 (e−πb )



 =

f (−e−πa )f (−e−πb ) e−π(a+b)/8 f (−e−4πa )f (−e−4πb )

8 . (5.5)

π in Lemma 2.1 and noting ab = 1, we deduce that 2b f (−e−πa ) e−πb/8 f (−e−4πb )

√ = e−π(b−a)/24 2b.

(5.6)

Similarly, by interchanging the role of a and b in (5.6), we deduce that √ f (−e−πb ) = e−π(a−b)/24 2a. −πb/8 −4πa e f (−e )

(5.7)

Employing (5.6) and (5.7) in (5.5) and noting ab = 1, we complete the proof. 2 Remark 5.4. Theorem 5.3 can be used to find explicit values of S(e−π ). Setting a = 1 in Theorem 5.3 and solving the resulting equation, and choosing positive real root, we obtain S(e−π ) = 2−5/8 .

(5.8)

Further, from Theorem 5.3 we note that if we know explicit values of S(e−πa ), then 2 explicit values of S(e−π/a ) can be determined. For example, setting a = √ in Theo7 √ rem 5.3, employing the value of S(e−2π/ 7 ), solving the resulting equation, and choosing positive real root, we evaluate S(e

√ −π 7/2

 )=

   √ √ √ 1/8 −16192 + 6120 7 − 144 6328 − 2387 7 + 381 904 − 341 7 . (5.9)

References [1] N.D. Baruah, N. Saikia, Modular equations and explicit values of Ramanujan–Selberg continued fraction, Int. J. Math. Math. Sci. 2006 (2006), Article ID 54901, pp. 1–15. [2] B.C. Berndt, Ramanujan’s Notebooks, part III, Springer-Verlag, New York, 1991. [3] B.C. Berndt, Ramanujan’s Notebooks, part V, Springer-Verlag, New York, 1998. [4] B.C. Berndt, Flowers which we cannot yet see growing in Ramanujan’s garden of hypergeometric series, elliptic functions, and q’s, in: J. Bustoz, M.E.H. Ismail, S.K. Suslov (Eds.), Special Functions 2000: Current Perspective and Future Directions, Kluwer, Dordrecht, 2001, pp. 61–85.

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[5] K.G. Ramanathan, Hypergeometric series and continued fraction, Proc. Indian Acad. Sci. Math. Sci. 97 (1987) 227–296. [6] S. Ramanujan, Modular equations and approximations to π, Quart. J. Math. 45 (1914) 350–372. [7] S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957. [8] N. Saikia, Ramanujan’s modular equations and Weber–Ramanujan’s class invariants Gn and gn , Bull. Math. Sci. 2 (2012) 205–223. [9] N. Saikia, Ramanujan’s Schlafli-type modular equations and class invariants gn , Funct. Approx. Comment. Math. 49 (2) (2013) 201–409. [10] H. Weber, Lehrburg der Algebra II, Chelsea, New York, 1961. [11] J. Yi, Construction and application of modular equation, PhD thesis, University of Illinois, 2001. [12] L.-C. Zhang, Explicit evaluation of Ramanujan–Selberg continued fraction, Proc. Amer. Math. Soc. 130 (1) (2002) 9–14.