On Continued Fraction of Order Twelve

Transcription

1 Pue Mathematical Sciences, Vol. 1, 2012, no. 4, On Continued Faction of Ode Twelve B. N. Dhamenda*, M. R. Rajesh Kanna* and R. Jagadeesh** *Post Gaduate Depatment of Mathematics Mahaani s Science College fo Women J. L. B. Road,Mysoe , India bndhama@gmail.com, m.ajeshkanna@gmail.com **Depatment of Mathematics, Govenment Science College N. T. Road, Bangaloe , India Jagadeesh1978@gmail.com Abstact In this pape, we establish some new modula elations between a continued faction V (q) of ode 12 (established by M. S. Mahadeva Naika, B. N. Dhamenda and K. Shivashankaa and ecently studied this continued faction by K. R. Vasuki, Abdulawf A. A. Kathtan, G. Shaath, and C. Sathish Kuma) and V (q n ) fo n=6,10,14,18. Mathematics Subject Classification: Pimay 33D20, 11F55, 11S23 Keywods: Continued faction, Modula equation, Theta function 1. Intoduction In Chapte 16 of his second notebook [1], [5], Ramanujan develops the theoy of theta-function and is defined by (1.1) f(a, b) := a n(n+1) 2 b n(n 1) 2, ab < 1, n= = ( a; ab) ( b; ab) (ab; ab) whee (a; q) 0 = 1 and (a; q) = (1 a)(1 aq)(1 aq 2 ). Following Ramanujan, we defined (1.2) ϕ(q) := f(q, q) = (1.3) ψ(q) := f(q, q 3 ) = n= n=0 q n2 = ( q; q) (q; q), q n(n+1) 2 = (q2 ; q 2 ) (q; q 2 ),

2 198 B. N. Dhamenda, M. R. Rajesh Kanna, and R. Jagadeesh (1.4) f( q) := f( q, q 2 ) = and n= (1.5) χ(q) := ( q; q 2 ). ( 1) n q n(3n 1) 2 = (q; q) The odinay hypegeometic seies 2 F 1 (a, b; c; x) is defined by (a) n (b) n 2F 1 (a, b; c; x) = (c) n n! xn, whee Let and (a) 0 = 1, (a) n = a(a + 1)(a + 2)...(a + n 1), fo n 1, x < 1. n=0 ( 1 z() := z(; x) := 2 F 1, 1 ) ; 1; x ( q := q (x) := exp πcsc ( π ) 2 F 1 ( 1, 1 2F 1 ( 1, 1 whee = 2, 3, 4, 6 and 0 < x < 1. Let n denote a fixed natual numbe, and assume that ( (1.6) n 2 F 1 1, 1 ; 1; 1 α) ( 1 2F 1, 1; 1; α) = 2 F 1 ; 1; 1 x) ; 1; x) ( 1, 1 ; 1; 1 β) ( 1 2F 1,, 1 ; 1; β) whee = 2, 3, 4 o 6. Then a modula equation of degee n in the theoy of elliptic functions of signatue is a elation between α and β induced by (1.6). We often say that β is of degee n ove α and m() := z(:α) is called the z(:β) multiplie. We also use the notations z 1 := z 1 () = z( : α) and z n := z n () = z( : β) to indicate that β has degee n ove α. When the context is clea, we omit the agument in q, z() and m(). The class invaiant G n is defined by (1.7) G n := q 1 24 χ(q) = (4α(1 α)) The celebated Roges-Ramanujan continued faction is defined as (1.8) R(q) := q1/5 f( q, q 4 ) f( q 2, q 3 ) = q1/5 q q 2 ). q 3, q < 1, On page 365 of his Lost Notebook [17], Ramanujan ecoded five identities showing the elationships between R(q) and five continued factions R( q), R(q 2 ), R(q 3 ), R(q 4 ), and R(q 5 ). He also ecoded these identities at the scatteed places of his Notebooks [16]. L. J. Roges [18] established the modula equations elating R(q) and R(q n ) fo n=2,3,5, and 11. The last of these equations cannot be found in Ramanujan s woks. Recently K. R. Vasuki and S. R. Swamy [23] found the modula equation elating R(q) with R(q 7 ).

3 On continued faction of ode twelve 199 The Ramanujan s cubic continued faction G(q) is defined as (1.9) G(q) := q1/3 f( q, q 5 ) f( q 3, q 3 ) = q1/3 q + q 2 q 2 + q 4, q < 1, The continued faction (1.9) was fist intoduced by Ramanujan in his second lette to G. H. Hady [12]. He also ecoded the continued faction (1.9) on page 365 of his Lost Notebook [17] and claimed that thee ae many esults fo G(q) simila the esults obtained fo the famous Roges-Ramanujan continued faction (1.8). Motivated by Ramanujan s claim, H. H. Chan [8], N. D. Bauah [4], Vasuki and B. R. Sivatsa Kuma [21] established the modula elations between G(q) and G(q n ) fo n=2,3,5,7,11 and 13. The Ramanjuan Gllinita-Godon continued faction [12, p. 44], [10], [17] is defined as follows: (1.10) H(q) := q1/2 f( q 3, q 5 ) f( q, q 7 ) = q1/2 q 2 q 3 + q 4, q < 1, q 5 + Chan and S. S. Hang [8],M. S. Mahadeva Naika, S. Chandan Kuma and M. Manjunatha [13] have established some new modula elations fo Ramanujan- Göllnitz-Godon continued faction H(q) with H(q n ) fo n = 6, 10, 14 and 16 and also established thei explicit evaluations, Mahadeva Naika, B. N. Dhamenda and S. Chandan Kuma [14] have also established some new modula elations fo Ramanujan-Göllnitz-Godon continued faction H(q) with H(q n ) fo n = 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 17, 19, 23, 25, 29 and 55 and Vasuki and Sivatsa Kuma [21] established the modula elations between H(q) and H(q n ) fo n=3,4,5and 11.Recently, B. Cho, J. K. Koo, and Y. K. Pak [9] extended the esult cited above fo the continued faction (1.10) to all odd pime p by computing the affine models of modula cuves X(Γ) with Γ = Γ 1 (8) Γ 0 (16p). Motivated by the cited woks on the continued factions (1.8)-(1.10), in this pape, we established the modula elation between continued faction V (q)andv (q n ) fo n = 6, 10, 14, 18. (1.11) V (q) := qf( q, q11 ) f( q 5, q 7 ) = q(1 q) (1 q 3 )+ q 3 (1 q 2 )(1 q 4 ) (1 q 3 )( q 6 ) +... The continued faction (1.11) was fist established by Mahadeva Naika, Dhamenda and K. Shivashankaa [15]as a special case of a fascinating continued faction identity ecoded by Ramanujan in his Second Notebook [16, p. 74], they have also established a modula elation between the continued faction V (q) and V (q n ) fo n=3, 5 and ecently Vasuki, Abdulawf A. A. Kathtan, Shaath, and C. Sathish Kuma [24] V (q) and V (q n ) fo n=7,9,11, Peliminay esults In this section, we collect the necessay esults equied to pove ou main esults.

4 200 B. N. Dhamenda, M. R. Rajesh Kanna, and R. Jagadeesh Lemma 2.1. [24] If x := V (q) and y := V (q 2 ), then (2.1) x 2 y + 2xy x 2 y + y 2 = 0. Lemma 2.2. [24] If x := V (q) and y := V (q 3 ), then (2.2) x 3 y 3 + y 2 y + 3xy 3x 2 y 2 + x 3 y 2 x 3 y = 0. Lemma 2.3. [24] If x := V (q) and y := V (q 5 ), then (2.3) xy 6 5x 6 y 5 + 5x 5 y 5 5x 4 y 5 5xy x 3 y 4 + 5xy 4 10x 4 y 3 10x 2 y x 3 y 2 + 5x 5 y 2 5x 5 y 5x 2 y + 5xy y + x 5 = 0. Lemma 2.4. [24] If x := V (q) and y := V (q 7 ), then (2.4) 35x 4 y 3 + y + 28x 2 y 3 + 7x 3 y 2 14xy 6 xy 8 + x 8 y 7 28x 5 y x 5 y 35x 3 y x 5 y x 4 y 5 7x 3 y 7xy 28x 2 y x 2 y x 7 + 7x 7 y 7x 3 y 3 7x 6 y + 28x 6 y 2 14x 7 y 2 + 7x 5 y 3 7x 6 y 3 + 7x 7 y 3 35x 5 y 4 7x 2 y 5 + 7x 3 y 5 7x 7 y 7 7x 7 y x 2 y 6 28x 3 y 6 + 7x 5 y 6 28x 6 y 6 + 7x 7 y 6 + 7xy 7 7x 2 y 7 + 7x 3 y 7 7x 5 y x 6 y 7 + 7xy 2 7xy 3 = 0. Lemma 2.5. [24] If x := V (q) and y := V (q 9 ), then 153x 4 y 3 + y + 297x 2 y x 3 y xy 6 + 9xy 8 45x 8 y 7 99x 5 y 2 + 9x 5 y 135xy x 5 y x 4 y 5 561x 6 y xy 4 30x 3 y 9xy 153x 2 y x 2 y + 10y 3 4y 2 16y 6 4y x 7 y 981x 3 y 3 27x 6 y + 165x 6 y 2 135x 7 y x 5 y 3 333x 6 y x 7 y 3 369x 5 y x 2 y x 7 y 5 243x 2 y x 3 y 6 153x 5 y x 6 y 6 297x 7 y 6 54xy x 2 y 7 165x 3 y x 5 y 7 171x 6 y x 7 y xy 2 90xy 3 x 9 y 8 9x 5 y x 6 y 8 99x 8 y 3 18x 2 y 8 + 4x 9 y 7 + 9x 8 y 8 9x 4 y 8 10x 9 y x 8 y 6 (2.5) 369x 7 y 4 54x 4 y 2 27x 7 y 8 + 9x 4 y + 135x 8 y x 3 y x 6 y 4 x 9 9x 8 y + 54x 8 y 2 10x 9 y 2 3x 9 y 4 + 4x 9 y 378x 2 y x 9 y 5 126x 8 y 5 249x 4 y x 4 y 7 16y y y 7 + y 9 198x 4 y 6 = 0.

5 On continued faction of ode twelve Relation Between H(q) and H(q n ) Theoem 3.1. If u := H(q) and v := H(q 6 ), then (3.1) ( v 3 v 2 + v 5 )u 6 + (6v + 6v 4 6v 5 6v 2 )u 5 + (9v v 3 27v 4 9v 2 )u 4 + ( 2v + 16v 4 2v 5 36v v 2 )u 3 + (27v 3 27v 2 9v 4 + 9v)u 2 + (6v 2 6v + 6v 5 6v 4 )u v 3 + v 4 v 6 + v = 0. Poof. Using the equations (2.1), then we have to expess y in tems of x. (3.2) y = 1/2 x + 1/2x 2 1/2 1 4x + 2x 2 4x 3 + x 4. Replace q to q 2 in the equation (2.2) and using the above equation (3.2), we aive at the equation (3.1). Theoem 3.2. If u := H(q) and v := H(q 10 ), then ( 50v v 8 + 5v 1 5v v 7 30v v 9 )u 12 + ( 50v 11 10v v 6 550v 9 350v v v 10 )u 1 ( 190v v 1 580v 7 + 4v v v 10 50v 2 125v v + 180v v 9 )u 10 + (680v 4 790v 7 410v v 8 850v 5 280v v v v v 10 )u 9 + (230v v v v v v v v v 6 20v v 5 )u 8 + (1120v v 7 140v v v v v v v 9 640v v 2 )u 7 + (2300v v v v v v 4 20v 3 (3.3) + 190v v v 9 40v)u 6 + (3040v 6 320v 1 20v v v v v 4 40v v v 9 + 8v)u 5 + (105v 3 740v v v v v v v 9 130v v v)u 4 + (220v 2 230v 5 190v v v v 6 40v v v 4 50v 480v 9 )u 3 + (15v v 6 v v 190v 2 185v 4 190v v v v v 5 70v 10 )u 2 + (60v 6 110v 3 70v 5 10v 2v v v 2 )u 6v 2 20v 4 + v v 3 10v 6 + v + 15v 5 = 0. Poof. Using the equations (3.2) and eplace q to q 2 in the equation (2.3), we aive at the equation (3.3).

6 202 B. N. Dhamenda, M. R. Rajesh Kanna, and R. Jagadeesh Theoem 3.3. If u := H(q) and v := H(q 14 ), then (3.4) (3528v v v v v v v v 9 126v v v v v v v 3 )u 1 (24199v 9 14v 20020v v v 11 63v v v v v v v v v v 14 )u 10 + (77v v v v v 2 323v v v 5 959v v v 3 357v v v v)u 14 + (86v 8 560v v v v v v 9 14v 15 2v 7 )u 15 + (6104v v v v v 3 56v 3192v v v 9 210v v 6 280v v 1 588v v 12 )u 13 + (2177v v 11 84v v v v v v v v v v v v v 9 )u 12 + v + 28v 3 15v 2 56v v 5 49v v 7 + v 9 8v 8 + (53552v v v v v v v v v v v v v v v 2 )u 9 + (62258v v v v v v v v v v v v v 38003v v 5 )u 8 + (306v 42658v v v 9 616v v v v v v v v v v v 2 )u 7 + (4683v v v v v v v 4 21v v v v v 11 63v v 9 896v 14 )u 6 + (10066v v v v v 8 182v 5236v v v v v v v v 9 98v 15 )u 5 + ( 1106v v v v v v v v v v v v 7 679v v v 4 )u 4 + (112v v 13 28v v v v v v v 4 616v v v v v v 2 )u 3 + ( 805v v v 3 v v v 7 98v v + 736v 9 43v 8 588v v 1 742v v v v 15 )u 2 + (44v v 2 14v + 420v 4 252v v 6 2v 9 196v 3 560v 5 )u + ( 49v v v 1 v 15 8v 14 + v v 13 56v 10 15v 8 )u 16 = 0.

7 On continued faction of ode twelve 203 Poof. Using the equations (2.1) and (2.4), we aive at the equation (3.4). Theoem 3.4. If u := H(q) and v := H(q 18 ), then (2v 15 3v 3193v v 14 2v 2 + v v v v v v 4 3v v v v v v 12 )u 18 + ( 81v v v v 7 945v v v v v 3 378v v v v v v 13 36v v v 8 )u 16 + ( 18912v v v 6138v v v v v v v v v v 6 18v 17 )u 17 + v + 2v 3 + 3v v v v v v v v v v 12 11v 15 v v v 13 2v 16 3v 17 + ( v v v v v v v v v v v v v 8 156v v v v 3 )u 15 + (41886v v v v v v v v v v v v v 91761v v v v 14 )u 14 + ( v v v v v v v v v v v v v v v v v 3 )u 13 + ( v v v v v v v v v v v v v v v 2 675v v 10 )u 12 + ( v v v v v v v v v v v v v v v v v 3 )u 11 + ( 1935v v v v v v v v v v v v v v v v v 8 )u 10 + ( v v v v v v v v v v v v v v v v v 5 )u 9 + (993363v v v v v v v v

8 204 B. N. Dhamenda, M. R. Rajesh Kanna, and R. Jagadeesh (3.5) v v v v v v v v v 6 )u 8 + (5382v v v v v v v v v v v v v v v v v 8 )u 7 + ( 18180v v v v v v v v v v v v v v v v v 14 )u 6 + (20124v v v v v v v v v v v v v v v v v 15 )u 5 + (239229v v v v v v v v v v v v v v v v v 5 )u 4 + (32628v v v v v v v 9 546v 34164v v v v v v v v v 16 )u 3 + (135v v v v v 3 378v v v v v 4 945v v 10 81v v v v v 12 )u 2 + (3132v v 11 36v v 17 18v 20652v v v v v v v v v v 8 )u = 0. Poof. Using the equations (2.1) and (2.5), we aive at the equation (3.5). Refeences [1] C. Adiga, B. C. Bendt, S. Bhagava and G. N. Watson, Chapte 16 of Ramanujan s second notebook: Theta-function and q-seies, Mem. Ame. Math. Soc., 53, No.315, Ame. Math. Soc., Povidence, (1985). [2] Bendt, B.C.: Ramanujan s Notebooks, Pat III. New Yok. Spinge-Velag [3] N. D. Bauah,Modula equations fo Ramanujan s cubic continued factions,268, (2002). [4] N. D. Bauah and R. Baman, Cetain Theta-function identites and Ramanujan s modula equations of degee 3, Indian J. Math, 48, No. 1, (2002). [5] B. C. Bendt, Ramanujan s Notebooks, Pat III, Spinge-Velag, New Yok (1991). [6] B. C. Bendt, Ramanujan s Notebooks, Pat V, Spinge-Velag, New Yok (1998). [7] H.H. Chan, On Ramanujan cubic continued faction, Acta Aithm., 73, No, (1995). [8] H.H. Chan, S.-S. Haung, On the Ramanujan-Göllnitz-Godon continued faction. Ramanujan J. 1, (1997). [9] B. Cho, J. K. Koo, and Y. K. Pak, Aithmetic of the Ramanujan-Göllnitz-Godon continued faction, J. Numbe Theoy, 4, No. 129, (2009).

9 On continued faction of ode twelve 205 [10] H. Göllnitz, Patition mit Diffenzebendingungen, J. Reine Angew. Math., 25, (1967). [11] B. Godon, Some continued factions of the Roges-Ramanujan type, Duke Math. J., 32, (1965). [12] G. H. Hady, Ramanujan, Chelsea, New Yok (1978). [13] M. S. Mahadeva Naika, S. Chandan Kuma and M. Manjunatha, On some new modula equations and thei applications to continued factions, (communicated). [14] M. S. Mahadeva Naika and B. N. Dhamenda and S. Chandankuma, Some identities fo Ramanujan Göllnitz-Godon continued faction, Aust. J. Math. Anal. Appl. (to appea). [15] M. S. Mahadeva Naika, B. N. Dhamenda and K. Shivashankaa, A continued faction of ode twelve. Cent. Eu. J. Math. 6(3), (2008). [16] S. Ramanujan, Notebooks (2 volumes). Bombay. Tata Institute of Fundamental Reseach (1957). [17] S. Ramanujan, The lost notebook and othe unpublished papes. New Delhi. Naosa (1988). [18] L. J. Roges, On a type of modula elation. Poc. Lond. Math. Soc., 19, (1921). [19] K. R. Vasuki and P. S. Guupasad, On cetain new modula elations fo the Roges- Ramanujan type functions of ode twelve, Poc. Jangjeon Math. Soc. (to appea). [20] K. R. Vasuki, G. Shaath and K. R. Rajanna, Two modula equations fo squaes of the cubic functions with applications, Note Math. (to appea). [21] K. R. Vasuki and B. R. Sivatsa Kuma, Cetain identities fo Ramanujan-Göllnitz- Godon continued faction. J. Compu. App. Math. 187, (2006). [22] K. R. Vasuki and B. R. Sivatsa Kuma, Two Identities fo Ramanujan s Cubic Continued Faction, Pepint. [23] K. R. Vasuki and S. R. Swamy, A new idenitity fo the Roges-Ramanujan continued faction. J. Appl. Math. Anal. Appl., 2, No. 1, (2006). [24] K. R. Vasuki, Abdulawf A. A. Kathtan, Shaath, and C. Sathish Kuma, On a continued faction of ode 12. Ukainian Mathematical Jounal, Vol. 62, No. 12 (2011). Received: June, 2012