ON A CONTINUED FRACTION IDENTITY FROM RAMANUJAN S NOTEBOOK

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1 Asian Journal of Current Engineering and Maths 3: (04) Contents lists available at ASIAN JOURNAL OF CURRENT ENGINEERING AND MATHS Journal homepage: ON A CONTINUED FRACTION IDENTITY FROM RAMANUJAN S NOTEBOOK S.N. Fathima, Yudhisthira Jamudulia Department of Mathematics Ramanujan School of Mathematical Sciences Pondicherry University, Puducherry-60504, INDIA ARTICLE INFO Corresponding Author Yudhisthira Jamudulia, Department of Mathematics Ramanujan School of Mathematical Sciences Pondicherry University, Puducherry-60504, INDIA ABSTRACT In this paper, we study a continued fraction of Ramanujan A (q).we establish an integral representation of A (q) and prove its modular identities. We also compute explicit evaluation of this continued fraction. 000 Mathematics Subject Classification: A55. Key Words: Continued Fractions, Modular Equations. 04, AJCEM, All Right Reserved. Introduction Ramanujan recorded about 00 results on continued fractions in his notebooks [] and lost notebook [] without proof. The only result on continued fraction that he published [9], [0, pp.4-5], is related to the now celebrated Roger Ramanujan continued fraction defined by i. R (q) = + qq qq qq , qq <, S (q) =-R (-q), which was first introduced by L.J. Roger [3] and independently rediscovered by Ramanujan. in his first two letters to G.H.Hardy [0,pp. xxvii-xxviii] [8, pp.-30,53-6], Ramanujan communicated several results concerning R (q). In particularly, he asserted that R (ee ππ qq) = 5+ 5 S (ee ππ qq) = , which were first proved by G.N. Watson [5]. In his lost notebook [, pp.6], Ramanujan claims that R (q) = 5 exp ( /5) ( tt)5 tt 5 dddd, (.) qq ( tt 5 )( tt 0 ) tt 39

2 = exp (/5) ( tt)5 tt 5 dddd, (.) qq tt 5 tt 5 tt 4 5 where 0<q<.. The equality (.) was first proved by G.E. Andrews [3] and equality (.) was proved by S.H.Son [4]. On page 365 of his lost notebook [], Ramanujan recorded five modular equations relating R (q) with R (-q), R (qq ), R (qq 3 ),R (qq 4 ) and R (qq 5 ). Motivated by these works in this paper we study the Ramanujan continued fraction i. R (q) = qq +qq qq 4 +qq 4 qq 6 +qq 6, qq <, qq + qq 6 + qq 0 + qq 4 + = qq8 ;qq 8 ( qq 4 ;qq 8 ). (.3) A similar continued fraction is been previously studied by C.Adiga and N.Anitha []. In Section we obtain an interesting q-identity related to A (q) using Ramanujan s ψ summation formula [5, Ch.6, Entry 7] (aa) nn (bb) nn nn= zz nn = (aaaa) (qq/aaaa) (qq) (bb/aa), bb < zz <. (.4) (zz) (bb/aaaa) (bb) (qq/aa) aa In Section 3 we obtain a product representation for A (q). In Section 4 we deduce several identities satisfied by A(q) with theta function φ(q) and ψ(q).in Section 5 we obtain an integral representation of A(q).Section 6 contains relationship of A(q) with ordinary hyper geometric series. We discuss modular equations of degree n with some illustration in Section 7. Finally, in Section 8 we present formula for explicit evaluation of A (q). We conclude this introduction with few customary definition we make use in the sequel. For a and q complex number with q < (aa) (aa; qq) = ( aaqq nn ) nn=0 nn (aa) nn (aa; qq) nn = ( aaqq kk ) = kk=0 (aa) (aaqq nn ), nn aaaaaa iiiiiiiiiiiiii, f (a, b) = nn= aa nn (nn+)/ bb nn(nn )/ (.5) =( aa; aaaa) ( bb; aaaa) (aaaa; aaaa), aaaa <. (.6) Identity (.6) is the Jacobi s triple product identity in Ramanujan s notation [5, Ch. 6, Entry 9]. It follows from (.5) and (.6) that [5, Ch.6, Entry ], and φ (q):=f (q, q)= qq nn = ( qq; qq) nn=, (.7) (qq; qq) 39

3 ψ (q):= f (q,qq 3 ) = qq nn(nn+)/ = qq ;qq nn=. (.8). q-identity Related to A (q) Theorem. (qq;qq ) ii. A (q) = qq4 ;qq 4 (qq 8 ;qq 8 4 ) qq 4nn +qq 4nn + nn=0. (.) Proof: Changing q to qq, then setting a= -qq 4, b=-qq and z= qq 4 in Ramanujan ψ summation formula (.4), we complete the proof of Theorem.. 3. Product Representation for A (q) Theorem 3.. Let A (q) be defined by (.3). Then Proof: From [5, Ch. 6, Entry ], for q < A (q) = ψψ (qq 8 ) ψψ (qq 4 ). (3.) ( aa) (bb) (aa) ( bb) = aa bb (aa bbbb )(aaaa bb) qq(aa bbqq )(aaqq bb) ( aa) (bb) +(aa) ( bb) qq + qq 3 + qq 5 + (3.) Rationalizing left hand side of (3.) and then changing q to qq, a to q and b to q in the resulting identity, we obtain qq;qq qq;qq ( qq;qq ) 4 (qq;;qq ) 4 = qq qq +qq qq 4 +qq 4. (3.3) qq + qq 6 + qq 0 + Multiplying numerator and denominator of left hand side of (3.3) by (qq ; qq ) and using (.6), we obtain {ff(qq,qq) ff( qq, qq)} = qq qq +qq qq 4 +qq 4. (3.4) ff (qq,qq) ff ( qq, qq) qq + qq 6 + qq 0 + Employing [5, Ch. 6, Entry 30 (iii) and (vi)] in (3.4) we obtain qqff (,qq 8 ) = qq qq +qq qq 4 +qq 4 ff(,qq 4 )ψψ(qq 4 ) qq + qq 6 + qq 0 + (3.5) Finally applying [5, Ch. 6, Entry 8 (ii)] and (.8), we complete the proof of (3. ). 4. Some Identities Involving A (q) We obtain several relations of A (q) in terms of theta functions φ (q) and ψ (q). Theorem 4. A (q) = ψψ (qq 4 ) φφ (qq 4 ), (4.) A (q) = ψψ(qq8 ) φφ (qq 4 ), (4.) ψψ 4 ( qq ) A (q) =, (4.3) φφ ( qq )φφ (qq 4 ) 393

4 AA nn + (qq)+aa nn + ( qq) AA(qq)+AA( qq) A (q) = ψψ (qq 4 )φφ( qq 4 ) φφ ( qq 8 )φφ(qq 4 ), (4.4) =AA nn (qq), (4.5) A(q)A(q )A(q 3 )A(q 4 ) = ψψ (qq 4 ), (4.6) AA (q)a(q )A(q 3 )A(q 4 ) = ψψ (qq 8 ) ψψ 4 (qq 4 ). (4.7) Proof:Replacing q by qq in [5, Ch. 6, Entry 3 (v)] and then squaring both sides, we obtain (4.). On using [5, Ch. 6, Entry 5 (iv)] in (4.) we obtain (4.). Setting a = b = qq and a = b = qq 4 in [5, Ch.6, Entry 30(i)] and substituting in (3.) we have A (q) = ψψ(qq4 ) ψψ(qq ) 4 φφ(qq ) φφ(qq 4 ). (4.8) Using [5, Ch. 6, Entry 5 (iii)] twice in (4.8), we obtain (4.3). Using [5, Ch. 6, Entry 5(iii)] and [5, Ch. 6, Entry 5(iv)] in (3.), we have Employing [5, Ch. 6, Entry 5 (ii)] in (4.9), we obtain A (q) = φφ( qq4 ) φφ ( qq 8 ) φφ(qq) φφ( qq) 4qq A (q) = φφ( qq4 )ψψ(qq 8 ). (4.9) φφ ( qq 8 ). (4.0) From [5, Ch. Entry 5(i)] and [5, Ch.6, Entry 5 (v)], we have φ(q)-φ(-q)= φφ (qq)φφ ( qq) φφ(qq)+φφ( qq) = 4q ψψ (qq 4 ) φφ (qq 4 ) (4.) Substituting (4.) in (4.0), we obtain (4.4). Proofs of (4.5), (4.6) and (4.7) follow directly from (3.). Theorem 4.If u= A (q), v=aa (qq) and w=aa 4(qq), then Proof: From (3.) we have which completes the proof of (4.). Also from (3.), we have ww 6 =uv, (4.) uu + vv + ww vv ww uu ww 3= v+w. (4.3) uv=a (q) AA (qq)= ψψ (qq8 ) ψψ (qq 4 ) 3, 394

5 uu + vv + ww (qq 8 ) vv ww uu ww 3=ψψ ( qq 4 ) ψψ(qq 4 ) + ψψ (qq8 ) ψψ (qq 4 ) ψψ(qq 8 ) ψψ (qq 4 ) ψψ + ψψ (qq 8 )ψψ (qq 4 ) (qq 8 ) ψψ (qq 4 )ψψ (qq 8 ) = ψψ(qq8 ) + ψψ ψψ(qq 4 ) (qq 8 ) ψψ (qq 4 ) which completes the proof of (4.3). 5. Integral Representation of A (q) Theorem 5. For 0< q <, where φ (q) is defined in (.7)., ψψ (qq 4 ) ψψ (qq 8 ) A (q) = exp φφ 4 qq 4 dddd, (5.) qq Proof:Taking log on both sides of (3.), we have log A (q) = log ψ (qq 8 )- log ψ (qq 4 ). (5.) Employing [5, Ch. 6, Entry 3(ii)] on right hand side of (5.), we obtain ( ) nn qq 4nn nn(+qq 4nn ) log A (q) = nn=. (5.3) Differentiating (5.3) and simplifying, we have dd ( ) nn qq 4nn log AA dddd (qq)=8 qq nn=. (5.4) (+qq 4nn ) Using Jacobi s identity [5, Ch. 6, Identity 33.5, pp. 54] and integrating both sides and finally exponentiating both sides of identity (5.4), we complete the proof of Theorem Relation Between A (q) and Hyper geometric Function In this section we deduce relations between A (q) and hyper geometric function F (a, b; c; x), where (aa) kk (bb) kk (cc) kk kk! F (a, b; c; x) = kk=0 xx kk, x < (6.) Theorem 6. and then x =kk = - φφ 4 ( qq) φφ 4 (qq), q = exp(-π F (/, /:, -k )/ F (/, /:, k ), (i) A(q) = qq, z = F (, ; ; kk ), 395

6 (ii) A (q) + = 4 { + ( xx) ( xx) qq qqqq Proof of (i):from [5, Ch. 7, Entry (i), pp. 3], we have 4 ( xx) 3 }. ψψ(qq) = z xx qq /8. (6.) Then employing (6.) in (3.) we obtain (i). Proof of (ii): From [5, Ch. 7, Entry (iv), (v), pp. 3] and (3.), we have A (q) = qq 4 xx xx. (6.3) Rationalizing (6.3) and simple manipulation completes the proof of (ii). 7. Modular Equations of Degree n In this section we obtain new modular equations of B (q), where B (q) = q A (q). We say modulus ββ has degree n over the modulus α when n F (/,/;;-α)/ F (/,/;:α)= F (/,/;;-β)/ F (/,/;:β). (7.) where F (a, b; c; x) is defined as in (6.). A modular equation of degree n is an equation relating α and β induced by (7.). Theorem 7.If then q= exp(-π F (/, /; ; -α)/ F (/, /; ; α)), α =- BB(qq) +BB(qq) 4. (7.3) Proof:On employing [5, Ch. 6, Entry 5 (ii)] and [5, Ch. 6, Entry 5(v)] in (3.), we have Thus A (q) = φφ ( qq ) qq φφ (qq ) φφ ( qq ) + φφ (qq ). B (q) = φφ ( qq ) φφ (qq ). (7.4) φφ ( qq ) + φφ (qq ) Also from [5, Ch. 7, Entry 5, pp. 00] and (7.) it is implied that α= - φφ 4 ( qq) φφ 4 (qq). (7.5) Using (7.5) in (7.4), we complete the proof of (7.3). Let q and α is related by (7.). If β has degree n over α then from Theorem 5., we obtain 396

7 β =- BB(qqnn ) +BB(qq nn ) 4. (7.6) Corollary 7. Let l=b (q), m=b (q 3 ), n= B (q 4 ), then (i) l 4-4l 3 m 3 + 6l m -4lm +m 4 =0. (ii) l 4 + l 4 n 4 + 4l 4 n 3 +6 l 4 n + 4 l 4 n -8n 3-8n =0. Proof of (i): From [5, Ch. 9, Entry 5 (ii) pp. 30], we have (αααα) 4 + {( αα)( ββ)} 4 = (7.7) On using (7.6) with n=3 and (7.3) in (7.7) and simplifying we complete the proof of Corollary 7.. (i). Proof of (ii): When β has degree 4 over α then we have from [5, Ch. 8, Eq. (4.) pp.5] On using (7.6) with n=4 and (7.3) in (7.8), we obtain nn +nn 4 = ll +(αα) 4 ββ = (αα) 4. (7.8) +ll + ll +ll Squaring both sides of the above identity and simplifying we complete the proof of Corollary 7.. (ii). 8. Explicit Formula For The evaluation of A (q) Let q n =ee ππ nn and αα nn denote the corresponding values of α in (7.).From Theorem 7.., we have. Hence B (ee ππ nn ) = ( αα nn ) 4. +( αα nn ) 4 A (ee ππ nn ) = eeππ nn ( αα nn ) 4. (8.) +( αα nn ) 4 From [5, Ch. 7, pp. 97], we haveαα =,αα =,αα 4 = 4.Thus from (8.), we have A (ee ππ ) = 4 eeππ 4. + A (ee ππ ) = eeππ + 4, A (ee ππ ) = eeππ The Ramanujan-Weber class invariant is defined by GG nn = 4qq nn 4 ( qq nn ; qq nn ), 397

8 gg nn = 4qq nn 4 (qq nn ; qq nn ), where q n = ee ππ nn. Chan and Huang [7] has derived few explicit formula for evaluation of S (ee ππ nn/ ) in terms of Ramanujan Weber class. Similar works are also carried out by Adiga et. al. []. Analogous to these works we obtain explicit formula for the evaluation of A(ee ππ nn ). Theorem 8.For Ramanujan Weber class invariant as defied in (8.) and let p=gg nn and pp = gg nn, then A (ee ππ nn ) = eeππ nn pp(pp+ pp ) /4 pp(pp+ pp /4 pp(pp+ pp ) /4 + pp(pp+ pp /4, (8.3) /4 A (ee ππ nn ) = pp pp pp + eeππ nn /4. (8.4) + pp pp pp + Proof:From [7, Eq. 4.7, pp. 85], we have Hence, GG nn = {4αα nn ( αα nn )} /4. αα nn = pp(pp+)+ pp(pp ). (8.5) Using (8.5) in (8.) we obtain (8.3). Also from [7, Eq. 4.9, pp. 85], we have Hence gg nn = αα nn αα nn. αα nn = (pp + ) pp. Using (8.6) in (8.) we obtain (8.4). Examples: Let n=. Since G =, from Theorem 8. we have A (ee ππ ) = eeππ /4 /4 +. Let n=. Since gg =, from Theorem 8. we have Let n=3. Since GG 3 =, then Theorem 8. we have A (ee ππ/ ) = eeππ ( )/4 +( ) /4. 398

9 A (ee ππ/ 3 ) = eeππ 3 (8+4 3)/4 (7+4 3) /4 (8+4 3) /4 +(7+4 3) /4. Acknowledgement The first author is thankful to UGC, New Delhi for awarding research project [No. F4-39/0/ (SR)] under which this work has been done. References [] C. Adiga and N. Anitha, A Note on a Continued Fraction of Ramanujan, Bull.Austral.Math.Soc, 70(004), [] C. Adiga, T. Kim, M.S. Mahadeva Naika and H.S. Madhusudhan, On Ramanujan s Cubic Continued Fraction and Explicit Evaluations of Theta Functions, Indian J. Pure and App. Math., No. 55, 9 (004), [3]G.E. Andrews, Ramanujan s lost`notebook III. The Rogers-Ramanujan continued fraction, Adv. Math., 4(98), [4] G.E.Andrews, The Lost Notebook Part I, Springer Verlang, New York, 005. [5] B.C.Berndt, Ramanujan s Notebooks, Part III, Springer-Verlag, New York, 99. [6] B.C. Berndt, Ramanujan s Notebooks, Part V, Springer-Verlag, New York, 998. [7] H.H.Chan and S.S. Huang,On the Ramanujan Gollnitz Gordon Continued Fraction, Ramanujan J., (997), [8] B.C. Berndt and R.A. Rankin, Ramanujan: Letters and Commentary, Amer. Math.Soc., Providence, 995. [9] S. Ramanujan, Proof of certain identities in combinatory analysis, Proc. Cambridge Philos.Soc., 9(99), 4-6. [0] S. Ramanujan, Collected Papers, Chelsea, New York, 96. [] S. Ramanujan, Notebooks, volumes, Tata Institute of Fundamental Research, Bombay, 957. [] S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 988. [3] L.J. Rogers, Second memoir on the expansion of certain infinite products, Proc.London Math.Soc., 5(894), [4] S.H.Son, Some integrals of theta functions in Ramanujan s lost`notebook, CRM Proceedings and Lecture Notes, 9(999), [5] G.N. Watson, Theorems stated by Ramanujan (VII): Theorems on continued fractions, J.London Math. Soc., 4(99),