Scholarly article on topic 'Some New Explicit Values of Quotients of Ramanujan’s Theta Functions and Continued Fractions'

Some New Explicit Values of Quotients of Ramanujan’s Theta Functions and Continued Fractions Academic research paper on "Mathematics"

CC BY
0
0
Share paper
Keywords
{""}

Academic research paper on topic "Some New Explicit Values of Quotients of Ramanujan’s Theta Functions and Continued Fractions"

Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences Volume 2014, Article ID 534376, 7 pages http://dx.doi.org/10.1155/2014/534376

Research Article

Some New Explicit Values of Quotients of Ramanujan's Theta Functions and Continued Fractions

Nipen Saikia

Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh, Arunachal Pradesh 791112, India Correspondence should be addressed to Nipen Saikia; nipennak@yahoo.com Received 27 December 2013; Accepted 12 May 2014; Published 25 May 2014 Academic Editor: Seppo Hassi

Copyright © 2014 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 evaluate some new explicit values of quotients of Ramanuj an's theta functions and use them to find explicit values of Ramanuj an's continued fractions.

1. Introduction

Ramanujan's general theta function f(a, b) is defined by

f(a,b):= ^ ak(k+1)/2bk(k-1)/2, \ab\ < 1. (1)

Two important special cases of f(a, b) are the theta functions <p(q) and f(q) [1, page 36, Entry 22] defined by, for \q\ < 1,

0 (q) := f ii' q) = Z q =

f(q) ••= f(1' q3) =

n(n+1)/2 ( iW) c

(q;q2)

where (a;q)m = n™o(l - aqn).

In his notebooks [2], Ramanujan recorded many explicit values of theta functions <p(q) and f(q). All these values were proved by Berndt [3, page 325] and Berndt and Chan [4]. Yi [5] introduced the parameter hkn for positive real numbers k and n defined by

. ___-n^Jk

nk'n k1/40(qk)'

and used the particular case hnn to find explicit values <p(q). Baruah and Saikia [6] defined the parameter gkn for positive real numbers k and n as

= y{-q) = -

0k'n kWft-Wy (-<£)' q £

and used the particular case gn n to find explicit values f(q). Saikia [7] also established some explicit values f(q).

In this paper, we consider the particular cases h3 n and g3 n of the parameters hk n and gk n, respectively. By using theta function identities, we find some new explicit values of the parameters h3 n and g3 n. Particularly, we evaluate h3 n and g3A for n = 3/2, 2/3, 6,1/6, 5/2, 2/5,10, and 1/10. Previously, Y^ [5] evaluated h3^n for n = 1,3, 1/ 3, 9, 1/ 9, 5, 1/5, 25, and 1/25. Saikia [8] evaluated h3 n for n = 2, 1/2, 4, 1/4, 7, 1/7, 49, and 1/49. Baruah and Saikia [6] evaluated g3 n for n = 1, 3, 1/3, 9, 1/9, 5, 1/5, 25, 1/25, 7, 1/7, 13, 1/13, 49, and 1/49. As an application to our new values, we evaluate some old and new explicit values of Ramanujan's cubic continued fraction G(q) and a continued fraction of order twelve H(q) which are, respectively, defined by, for \ \ < 1,

G(q) := H(q)

q1/3 q + q2 q2 + q4 q3 + q6

qf(-q'- q11 ) : f(-q5'-q7)

q(l-q) q3(l- q2)(l - q4) q3(1- q8)(l- q10)

1- q3 + (1- q3)(1 + q6) + (1-q3)(1+q12) _

The continued fraction G(q) was recorded by Ramanujan on page 366 of lost notebook [9]. We refer to [10-14] for explicit evaluations of G(q). The continued fraction H(q) was

introduced by Naika et al. [15]. We refer to [8,15] for explicit evaluations of H(q).

The presentation of the paper is as follows. In Section 2 we record some preliminary results for ready references in this paper. Section 3 is devoted to explicit evaluations of the parameters h3n and g3n. In Section 4 we use new explicit values of h3n and g3n to evaluate some explicit values of the continued fractions G(q) and H(q).

We end this introduction by noting the following remarks regarding g3n and h3 n from [6, page 1764, Theorem 4.1] and [5, page 385, Remark2.3], respectively.

Remark 1. The parameter g3n has positive real value with g31 = 1 and g3n increases as n increases.

Remark 2. The parameter h3 n has positive real value with h31 = 1 and the values of h3 n decrease as n increases.

Lemma7 (see [5,page385, Theorem2.2]). For all positive real numbers k and n, one has

hk,1/n

Lemma 8 (see [6, page 1764, Theorem 4.1]). For all positive real numbers k and n, one has

gk,1/n =

Lemma9 (see [5,page 393, Theorem 4.9(i)]). For any positive real number n, one has

2. Preliminary Results

Lemma 3 (see [16, Theorem 3.2]). IfP := *Jqy(-q)y(-q9)/ f2(-q3) and Q := qy(-q2)y(-q18)/f2(-q6), then

P2 Q2 (P2 Q2sl

+ -T + ( —T +

Q2 P2 Q2 P2

3PQ-±)(3PQ-± + 3P + 3Q + PL + Q3

PQ)\ PQ Q P Q3 P3

^(h3»h39n + JTT

3,nh9,n

Lemma 10 (see [5, page 394, Theorem 4.12(i)]). For any positive real number n, one has

{h3,nh3,25nf +

3,25n)

Lemma 4 (see [16, Theorem 3.1]). If L := ifi(q)ifi(q9)/ifi2(q) and M := ^qf(-q)f(-q9)/f2(-q3), then

l3,25n

t2 M2 +1 L2 =

1 - 3M2'

Lemma 5 (see [16, Theorem 3.2]). If P := f(-q3)f(-q5)/ qy(-q)f(-q15) and Q := f(-q6)f(-q10)/q2f(-q2)f(-q30), then

Lemma 11 (see [6, page 1769, Theorem 5.1(i) and (iii)]). For any positive real number n, one has

4 + Q ) + (P + Q)-(PQ+^

Q2 P2 \Q P J V PQ

JPQ-^.

vm)[\Q3 + \P3 + 1

Lemma 6 (see [16, Theorem 3.9]). If L := $(q3)$(q5)/ 15) and M := q3)y(-q5)/qy(-q)y(-q15), then

1 + L 1 - L

(i) (1+^3g3

,ng3,9n) = 1 + 3g3,9

(ii) 3{(g3 ,ng3,25n) + 1/(g3,ng3,25n) } + 5{(g3,25n/g3,n) + (g3,n/g3,25n)} = {(g3,25n/g3 5{(g3,25n/g3

3. Explicit Evaluations of the Parameters gk>n and hk n

In this section, we prove some general theorems for the explicit evaluations of the parameters gk n and hk n and evaluate some new explicit values therefrom.

Theorem 12. One has

/ \ 2 / \ 2 I 03,n03,36n \ +l 03,4n03,9n

V 03,4n03,9n ) V 03,n03,36n

+ ( ( 03.n03,36n ) + ( 03,4n03,9n

V 03,4n03,9n ) \ 03,n03,36n 03,n03,4n \ I 03,9n03,36n

03,9n03,36n

03,n03,4n

3 , 03,n03,4n ) - ( 03,9n03,36n ) + 03,n03,36n

03,9n03,36n ) V 03,n03,4n ) V 03,4n03,9n 03,n03,4n \ I 03,9n03,36n

03,9n03,36n ) \ 03,n03,4n

3 ( 03,4n03,9n ) + ( 03,n03,36n ) + ( 03,4n03,9n

V n03,36n ) V ,4n03,9n ) V #3,n#3,36n

Proof. We set q := e in Lemma 3 and use the definition

of 03,n.

Corollary 13. One has

03,3/2 =

03,6 =

03,2/3 =

^6(1 +V2)

3^2 + 2V2- ^6 + 6V2 V2

^3^2 + 2V2- ^6 + 6V2

#3,6 =

^3^2 + 2V2- ^6 + 6V2

V6(1 + V2)

Proof. Setting n = 1/6 in Theorem 12 and simplifying using Lemma 8, we obtain

(.03,3/203,6

(.03,3/203,6

(03,3/203,6

- (03,3/203,6) + 8) + 8 = 0.

Solving (16) for g33/2g3fi and noting the fact in Remark 1, we find that

03,3/203,6 = V3(1 + V2)-

Again, setting n = 1/6 in Lemma 11(i) and simplifying using Lemma 8, we deduce that

1 + )) =1 + 3^,3/,

Eliminating g3 6 from (18) using (17) and simplifying, we obtain

3/2 + 3(^1 + V2 - (1 + V2)3/2) #32,3/2 + (3 + 3V2) = 0.

Solving (19) and noting the fact in Remark 1, we obtain

03,3/2

^3^2 + 2V2- ^6 + 6V2

Employing (20) in (17) and simplifying, we obtain

V6(1+V2)

#3,6 =

3^2 + 2V2- ^6 + 6V2

Now the values of ^32/3 and ^31/6 follow from the values of ^3 3/2 and g36, respectively, and Lemma 8. □

Theorem 14. One has

03,n 03,9n

03,n 03,9n

-1 = 0. (22)

Proof. We set q := e

= ^ Ti^n/3 in Lemma 4 and use the definitions

of h3,n and 03,n.

Corollary 15. One has

hX3/2 = 2-5/8(3V2 - 4)3/4(6 + 6V2 + V6)1/2, h3,6 = 21/831/2(3V2 + 4)1/4(6 + 6V2 + V6)-1/2, h3 2/3 = 25/8(3V2 - 4)-3/4(6 + 6V2 + V6)-1/2, h3,1/6 = 2-1/83-1/2(3V2 + 4)-1/4(6 + 6V2 + V6)1/2

Proof. Setting n = 1/6 in Theorem 14 and simplifying using Lemma 7, we obtain

(h3,6h

3,3/2/

(03,603

(03,603

-1 = 0.

Employing (17) in (24), solving the resulting equation, and noting the fact in Remark 2, we deduce that

h , h = 21/4-h3,3/2h3,6 = 2

4 + 3V2

Again setting n = 1/6 in Lemma 9 and simplifying using Lemma 7, we obtain

^ilP + ) = (W«3* )2 + 3. (26)

V n3,6 n3,3/2 /

Employing (25) in (26), solving the resulting equation, and noting the fact in Remark 2, we deduce that

Ks/2 _ 6 + 6/2 + a/6 h3,6 /3(4 + 3/2)'

Multiplying (25) and (27), we evaluate the value of h33/2, and dividing (25) by (27) and simplifying, we evaluate the value of h3 6. Now the values of h3 2/3 and h31/6 easily follow from the values of h3 3/2 and h3 6, respectively, and Lemma 7. □

Theorem 16. One has

22 I 03,4n03,25n \ + ( 03,n03,1OQn ) + / 03,4n03,25n

V 03,n03,1OOn ) V 03,4n03,25n ) V 03,n03,1OOn ,

I 03,n03,1OOn\ ( 03,25n03,1OOn\ ( 03,n03,4n

V 03,4n03,25n ) \ 03,n03,4n 1/2

03,25n 03,1QQn 03,n03,4n

( 03,4n03,25n V 03,n03,

03,n03,4n

03,25n03,1OOn 3/2

03,25n 03,1OOn 1/2

( 03,4n03,25n V 03,n03,1OOn

( 03,n03,1OOn V 03,4n03,25n

I 03,n03,1O

\ 03,4n03,25n

■ _ 0'

Proof. We set q := e in Lemma 5 and use the definition

of 03,n.

Corollary 17. One has

X5/2 _ (l + /3 + ^2+ /3) (a + /5b)1/4c

, 1-s 1/2 /

31O _ (1 + /3 + /2 + /3) (fl+/5fc)-14c

—1/2

32/5 _ (l + /3 + ^2+ /3) (fl + /5fc)—1/4c1/4

—1/2

g31/10 _ (l + /3 + ^2+ /3) (fl + /5b)1/4c—1/4

fl _ 63 + 35/3 + 30^2 + /3 + 20^6 + 3/3,

fc _ 2574 + 1485/3 + 1330^2 + /3 + 770^6 + 3/3, (30)

c _ 25+ 15/3 + 14^2 + /3 + 7^6 + 3/3.

Proof. Setting n = 1/10 in Theorem 16 and simplifying using Lemma 8, we obtain

(03,5/203,1O) +

(03,5/203,1O)

4 ( 03,5/203,1

03,5/2 03,10

-4 _ 0.

Solving (31) and noting the fact in Remark 2, we find that

03,5/203,10

_ 1 + /3+ V2 + /3.

Again, setting n = 1/10 in Lemma 11 (ii) and simplifying using Lemma 8, we obtain

03,5/2 \ + I 03,5/2

03,10 ) \ 03,10

+ 5 ( (03,5/203,10) +

(03,5/2 03,1

( (03,5/203,1O)

(03,5/203,1'

5 l 03,5/203,1

03,5/203,10

Employing (32) in (33), solving the resulting equation, and noting the fact in Remark 1, we obtain

g3Ao _ (fl+ /5b\

03,5/2

where a = 63 + 35/3 + 30^2 + /3+20^6 + 3/3, fc = 2574 +

1485/3+ 1330^2+ /3+770^6+ 3/3,andc = 25+15/3 +

14/2+/3 + 7/6 + 3/3.

Combining (32) and (34), we calculate the values of g3 5/2 and ^310. Then, the values of g3 2/5 and g3y1/10 follow from the values of g3 5/2 and g310, respectively, and Lemma 8. □

Theorem 18. One has

03,25n 03,n

[ - 1 - ( ) _ 0' (35)

Proof. We set q := e in Lemma 6 and use the definitions

of h3,n and 03,n. □

Corollary 19. One has

h3,5/2 = (V3+V2+V3) (2+V3+V2+V3

x(/-4V5fc) d

1/4 ,-1/4

h3,10 = (V3 + V2+V3) (2+V3 + V2 + V3

x(/-4V5fc)-1/4d

h3,2/5 = (V3+V2+V3) (2 + V3+V2+V3

x(/-4V5fc)-1/4d

h3,1/10 = (V3+V2+V3) (2+V3+V2+V3

x(/-4V5fc)1/4 d

wfoere fc is given in Corollary 17, f = 1143 + 655V3 + 582^2+ V3 + 346^6+ 3V3, and d = 1117 + 645V3 + 578^2 + V3 + 334^6+ 3V3.

Proof. Setting n = 1/10 in Theorem 18 and simplifying using Lemma 7, we get

#3,5/2^3,10 (1 - ^3,5/2^3,10) - 1 - ^3,5/2^3,10 = 0 (37)

Employing (32) and solving the resulting equation, we obtain

V3+ ^2+ V3

h3,5/2h3,10 =

2+ V3+ V2+ V3

Again setting n = 1/10 in Lemma 10 and simplifying using Lemma 7, we find that

^3,5/2 \ / h3,10 + ,

- ((^3,5/2^3,10) -

(^3,5/2^3,10)

- 5 ( (^3,5/2^3,10) +

- 5 ( hirnhim -

(h3,5/2h3, 1

3,5/2h3,10 h h

h3,5/2h3,10

Employing (38) in (39), solving the resulting equation, and noting the fact in Remark 2, we obtain

/-4V5fc

where fc is given in Corollary 17, / = 1143 + 655V3 +

582^2+ a/3 + 346^6+ 3V3, and d = 1117 + 645V3 +

578^2 + a/3 + 334^6+ 3V3.

Combining (38) and (40), we evaluate the values of fo35/2 and fo310. Then the values of fo32/5 and ^3,1/10 follow from fo3,5/2 and ^310, respectively, and Lemma 7. □

4. Explicit Evaluations of Continued Fractions

This section is devoted to finding some explicit values of the continued fractions G(^) and H(^) by using new values of n and evaluated in Section 3.

Lemma 20 (see [6, page 1788, Theorem 9.1(i)]). One has

G3 =_-1

1 + 3#4

Theorem 21. One has

G3 f-e-"/V2

9^36 + 42V2 - (38 + 36V2)

G3 f-e-^

(3^2 + 272- ^6 + 6V2)

6^(2 + 2V2) (6 + 6V2) - (58 + 40V2))

G3 f-e-"V2/3

6(^(2 + 2V2) (6 + 6V2) - (6 + 4V2))

G3 f-e-"/3V2

2(3 + 2V2)

^36 + 24V2-(l0 + 8V2)

G3 (-e-^

G3 /^VW3) =

G3 (-e-*/V3ü

c + 3 (a + V5fc) (l + V3 + ^2 + a/3) -(a + V5fc)

(A+V5fc) + 3c(i + V3+ V2+v3) -(«+ V5fc) (l + V3+ ^2+v3)

(a + V5fc) (l + V3 + ^2 + V3) + 3c

-c(l +V3+ ^2 + V3) c(l +V3+^2+V3) +3(fl+V5fc)

where a, fc, and c are ^¿ven ¿n Corollary 17.

Proof. We set n = 3/2, 6, 2/3, 1/6, 5/2, 10, 2/5, and 1/10 in Lemma 20 and use the corresponding values of from Corollaries 13 and 17 to complete the proof. □

TV10/3

= (31/4(3+V2+ Vs) d

The explicit values of G3(-e-7l/3V2), G3(-e-"V576), G3(-e-.Vio73), G3(-e--V27l5), and G3(-e-"/V3°) are new.

Lemma 22 (see [8, page 144, Theorem 5.1]). One foas

H(e-"V"/3) =

31/4^3,w -1 31/4^3,K +1'

-(2 + V3 + V2+V3^2(/- 4Vsfc}1/4)

(31/4(3+V27vS^2d1/4 1/2

+ ^2+V3+ V2+ V3) (/-4v5fc)1/4

TV2/15

By setting n = 3/2, 6, 2/3, 1/6, 5/2, 10, 2/5, and 1/10 in Lemma 22 and employing the corresponding values of from Corollaries 15 and 19, we calculate the following new explicit values of the continued fraction H(^).

Theorem 23. One has

= -2 + 23/831/4 (3V2-4f4 Vl + 6V2+V6, 2 + 23/831/4(3V2-4)3/4^1 + 6V2+ V6 '

21/833/4(4 + 3V2)1/4 - Vl + 6V2+V6 21/8 33/4(4 + 3V2)1/4 + ^1 + 6V2+ VS'

25/831/4 -(3V2-4)3/4Vl + 6V2+V6

= (s1/4(2+V3+V2+V3^2d1/4

-(3+^2+ V3) (/ - 4v5fc) x(s1/4(2+v3+ V2+ V3^2d1/'

+ (3+V2+V3^2(/-4V5fc)1/4

-rc/V30

( -.V2/^ 23 -(3V2-4) r + 6Y2+v6 f - 1/2

(e ) = 25/831/4 + (3V2-4)3/4^1 + 6V2+V6' (/-4^

,(e-„/V^) = 27/8V1 + 6V2+v6-2(12 + 9v2) 27/8^1 + 6V2+ V6 + 2(l2 + 9V2)

1/4 1/4'

-(3 + V2 +V3^2d1/4

31/4(2 + V3+ V2+ VS) (/-4Vsfc)1/4

/ 1/2 = (s1/4(s+v2+v3) (/-4Vsfc)1/4

+ (3+V2+V3) d

-(2+V3+ V2+ V3) d

31/4(s + V2+VS) (/-4Vsfc)1/4

+ I2+V3+ V2+ V3) d

wfoere b is given in Corollary 17 and d and f are given in Corollary 19.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author is thankful to the University Grants Commission,

New Delhi, India, for partially supporting the research work under Grant F. no. 41-1394/2012(SR).

References

[1] B. C. Berndt, Ramanujan's Notebooks, Springer, New York, NY, USA, 1991.

[2] S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay, India, 1957.

[3] B. C. Berndt, Ramanujans Notebooks, Springer, New York, NY, USA, 1998.

[4] B. C. Berndt and H. H. Chan, "Ramanujans explicit values for the classical theta-functions," Mathematika, vol. 42, no. 2, pp. 278-294,1995.

[5] J. Yi, "Theta-function identities and the explicity formulas for theta-function and their applications," Journal of Mathematical Analysis and Applications, vol. 292, no. 2, pp. 381-400, 2004.

[6] N. D. Baruah and N. Saikia, "Two parameters for Ramanujan's theta-functions and their explicit values," Rocky Mountain Journal of Mathematics, vol. 37, no. 6, pp. 1747-1790, 2007.

[7] N. Saikia, "A new parameter for Ramanujan's theta-functions and explicit values," Arab Journal of Mathematical Sciences, vol. 18, no. 2, pp. 105-119, 2012.

[8] N. Saikia, "Modular identities and explicit values of a continued fraction of order twelve," JP Journal ofAlgebra, Number Theory and Applications, vol. 22, no. 2, pp. 127-154, 2011.

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

[10] C. Adiga, M. S. M. Naika, and K. R. Vasuki, "Some new explicit evaluations for Ramanujan's cubic continued fraction," New Zealand Journal of Mathematics, vol. 31, no. 2, pp. 109-114,2002.

[11] C. Adiga, T. Kim, M. S. M. Naika, and H. S. Madhusudhan, "On Ramanujan's cubic continued fraction and explicit evaluations of theta-functions," Indian Journal of Pure and Applied Mathematics, vol. 35, no. 9, pp. 1047-1062, 2004.

[12] N. D. Baruah and N. Saikia, "Some general theorems on the explicit evaluations of Ramanujan's cubic continued fraction," Journal of Computational and Applied Mathematics, vol. 160, no. 1-2, pp. 37-51, 2003.

[13] B. C. Berndt, H. H. Chan, and L.-C. Zhang, "Ramanujan's class invariants and cubic continued fraction," Acta Arithmetica, vol. 73, pp. 67-85, 1995.

[14] H. H. Chan, "On Ramanujan's cubic continued fraction," Acta Arithmetica, vol. 73, pp. 343-355, 1995.

[15] M. S. M. Naika, B. N. Dharmendra, and K. Shivashankara, "A continued fraction of order twelve," Central European Journal of Mathematics, vol. 6, no. 3, pp. 393-404, 2008.

[16] M. S. M. Naika, S. Chandankumar, and M. Harish, "On some new mixed modular equations involving Ramanujan's theta-functions," Matematicki Vesnik, vol. 66, no. 3, pp. 283-293,2014.

Copyright of International Journal of Mathematics & Mathematical Sciences is the property of Hindawi Publishing Corporation and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.