a new parameter for ramanujan s theta functions and explicit values

Arab Journal of Mathematical Sciences (2012) 18, 105–119 King Saud University Arab Journal of Mathematical Sciences www.ksu.edu.sa www.sciencedirect.com ORIGINAL ARTICLE A new parameter for Ramanujan’s theta-functions and explicit values Nipen Saikia Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh 791 112, Arunachal Pradesh, India Received January 2012; revised 12 January 2012; accepted 12 January 2012 Available online 31 January 2012 KEYWORDS Theta-functions; Parameters; Explicit values Abstract We define a new parameter Ak,n involving Ramanujan’s thetafunctions /(q) and w(q) for any positive real numbers k and n and study its several properties We also prove some general theorems for the explicit evaluations of the parameter Ak,n and find many explicit values Finally, we establish an explicit formula for values of w(eÀ2np) for any positive real number n in terms of Ak,n and give examples ª 2012 King Saud University Production and hosting by Elsevier B.V All rights reserved Introduction For q :¼ e2piz, Im(z) > 0, define Ramanujan’s theta-functions /(q), w(q), and f(Àq) as E-mail address: nipennak@yahoo.com 1319-5166 ª 2012 King Saud University Production and hosting by Elsevier B.V All rights reserved Peer review under responsibility of King Saud University doi:10.1016/j.ajmsc.2012.01.004 Production and hosting by Elsevier 106 N Saikia /qị :ẳ X qn ẳ #3 0; 2zị nẳ1 wqị :ẳ X qnnỵ1ị=2 ẳ 21 q1=8 #2 0; zị; nẳ0 and fqị :ẳ q; qị1 ẳ qÀ1=24 gðzÞ; where #2, #3 are classical theta-functions [7, p 464], g denotes the Dedekind eta-function and Y ð1 aqk ị: a; qị1 :ẳ kẳ0 In his rst notebook [6, vol I, p 248] S Ramanujan recorded many elementary and non elementary values of /(q) and w (q) All these values were proved by Berndt [4, p 325] and Berndt and Chan [5] They also found new explicit values /(q) Recently, Yi [8,9] evaluated many new values of /(q) and f(q) using modular identities, transformation formulae for theta-functions and the parameters rk;n ; r0k;n ; hk;n , and h0k;n , dened, respectively, by p fqị rk;n :ẳ 1=4 ; q ẳ e2p n=k ; 1:1ị k qk1ị=24 fqk Þ pffiffiffiffiffi fðqÞ ; q ¼ eÀp n=k ; r0k;n :ẳ 1=4 1:2ị k qk1ị=24 fqk ị p /qị ; q ẳ ep n=k ; hk;n :ẳ 1=4 1:3ị k /qk ị and h0k;n :ẳ /qị k1=4 /qk ị ; p q ẳ e2p n=k : 1:4ị Baruah and Saikia [2] also obtained several new explicit values of the theta-function w(q) by finding explicit values of the parameters gk,n and g0k;n which are dened, respectively, by p wqị q ẳ ep n=k ; gk;n :ẳ 1=4 1:5ị k qk1ị=8 wqk Þ and g0k;n :¼ wðqÞ k1=4 qðkÀ1Þ=8 wðqk Þ ; p q ẳ ep n=k : 1:6ị In sequel to the above work, we now define a new parameter Ak,n for any positive real numbers k and n and involving theta-functions /(q) and w(q) as A new parameter for Ramanujan’s theta-functions Ak;n ẳ /qị 2k1=4 qk=4 wq2k ị ; p q ẳ ep n=k : 107 1:7ị In this paper, we study several properties of Ak,n which are analogous to those of hk,n and gk,n and also establish its connections with the parameter rk,n and Ramanujan’s class invariants We prove some general theorems for the explicit evaluations of Ak,n by using some new theta-function identities and find many explicit values of Ak,n We also offer a general formula for explicit evaluations of w(q) in terms of Ak,n and find some particular values In Section 2, we record some transformation formulae for theta-functions and list the explicit values of rk,n and hk,n from [1] and [8] for ready reference in the later sections In Section 3, we prove some new theta-function identities which will be used in the subsequent sections In Section 4, we study several properties of Ak,n and establish relations connecting Ak,n with rk,n and Ramanujan’s class invariants In Section 5, we prove some general theorems for the explicit evaluations of Ak,n and find many explicit values of Ak,n by using the results in Sections 2–4 Finally, in Section 6, we offer a general formula for the evaluations of w(q) in terms of Ak,n and find some particular values To end this introduction, we define Ramanujan’s modular equation Let K, K0 , L, and L0 denote the complete elliptic integrals of the first kind associated with the moduli k, k0 , l, and l0 , respectively Suppose that the equality K0 L0 1:8ị n ẳ K L holds for some positive integer n Then a modular equation of degree n is a relation between the moduli k and l which is implied by (1.8) Ramanujan recorded his modular equations in terms of a and b, where a = k2 and b = l2 We say that b has degree n over a By denoting zr = /2(qr), where q = exp(ÀpK0 /K), ŒqŒ < 1, the multiplier m connecting a and b is defined by m = z1/zn Preliminary results Lemma 2.1 [3, p 43, Entry 27(ii)] If a and b are such that the modulus of each exponential argument is less than and ab = p, then p p 2 awe2a ị ẳ bea =4 /ðÀeÀb Þ: Lemma 2.2 [3, p 122, Entry 10(i), (ii), (iii)] We have p /qị ẳ z1 ; p /qị ẳ z1 aị1=4 ; p /q2 Þ ¼ z1 ð1 À aÞ1=8 : ð2:1Þ ð2:2Þ ð2:3Þ ð2:4Þ 108 N Saikia Lemma 2.3 ([3, p 123, Entry 11(iii), (iv), (v)]) We have pffiffiffiffi 1=4 z1 a wðq Þ ¼ ; 2q1=4 pffiffiffiffiffiffiffiffiffiffiffi 1=2 pffiffiffiffi z1 f1 À ag p wq ị ẳ ; 2q1=2 pffiffiffiffi z1 f1 À ð1 À aÞ1=4 g wðq8 Þ ¼ : 4q ð2:5Þ ð2:6Þ ð2:7Þ We also note that if we replace q by qn in the Lemmas 2.2 and 2.3, then z1 and a will be replaced by zn and b, respectively, where b has degree n over a Lemma 2.4 ([3, p 233, (5.2), (5.5)]) If m = z1/z3 and b has degree over a, then 1aẳ m ỵ 1ị3 mị m 1ị ỵ mị and b ẳ : 16m 16m ð2:8Þ Lemma 2.5 ([3, p 231, Entry 5(x)]) If b has degree over a, then mð1 aị 1=2 1=2 ỵ bị 1=2 1=2 bị aị m ẳ 2f1 aị1 bịg1=8 : ẳ 2:9ị Lemma 2.6 ([1,8]) If rk,n is as defined in (1.1), then pffiffiffi p p p r2;3 ẳ ỵ 2ị1=4 ; r2;12 ẳ 25=24 21 ỵ ỵ 6ịị1=8 ; r4;4 ẳ 25=16 ỵ 2ị1=4 ; s p p 1=2 p 5ỵ 3=8 1=4 3=4 1=8 ; r4;16 ẳ ỵ 2ị 16 ỵ 15 ỵ 12 ỵ ị ; r5;5 ẳ s p p p p p p 5ỵ 1ỵ ỵ 5ị5=8 ỵ ỵ 5ị1=8 pffiffiffi ; r2;20 ¼ ; r2;5 ¼ ; r5;20 ¼ 1=4 À1 pffiffiffi pffiffiffi 1=8 pffiffiffi pffiffiffi p f21 ỵ 35 28 3ịg ; r2;8 ẳ 23=16 ỵ 2ị1=4 ; r2;9 ẳ ỵ 3ị1=3 ; r2;36 ẳ p p 2=3 2ị p 1=4 p p 3=16 1=4 r2;32 ẳ ỵ 2ị 16 ỵ 15 ỵ 12 ỵ 23=4 ị1=8 ; r2;3=2 ẳ 27=24 ỵ 3ị1=4 ; p p1=4 p 5ỵ1ỵ p r2;6 ẳ 21=24 ỵ 3ị1=4 ; r2;5=2 ¼ ; 21=4  qffiffiffiffiffiffiffiffiffiffiffiffiffiffi   qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1=4 p 1=4 p p p p 1 5ỵ1ỵ ỵ ỵ 2ị r2;10 ẳ ỵ 5ị ; r2;10 ẳ ỵ 5ị ; 2 p p 51=4 ỵ 25=8 ; r4;2 ẳ 21=8 ỵ 2ị1=8 ; r4;3 ẳ ỵ 3ị1=4 ; r2;25=2 ¼ 5=8 ; r2;50 ¼ 1=4 À1 109 A new parameter for Ramanujan’s theta-functions   pffiffiffi p q p 1=2 r4;5 ẳ p ỵ ỵ 21 ỵ 5ị ; r4;7 ẳ ỵ 7Þ1=4 ; pffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=8 pffiffiffi 3=8 pffiffiffi 31=4 1=4 ; ỵ ỵ 10 ; r4;9 ẳ ỵ p ỵ r4;8 ẳ ỵ 2ị 2 r q r q!2 ffiffiffiffiffi p ffiffi ffi p p ffiffi ffi p ffiffi ffi p ffiffi ffi p ffiffi ffi pffiffiffi 1 4 r4;25 ẳ ỵ ỵ ỵ 53 ; r4;49 ẳ ỵ ỵ 21 ỵ ỵ ỵ 21 ỵ ; pffiffiffi pffiffiffi pffiffiffi pffiffiffi 3=8  pffiffiffi pffiffiffi 1=8 ỵ 5ị9=8 ỵ ỵ 5ị1=8 ; 21 ỵ ỵ 6ị ; r4;10 ẳ r4;6 ẳ ỵ 2ị p p p p p 1=8 pffiffiffi pffiffiffi 3=4 1=3 1=3 1=8 À11=24 r4;18 ¼ ỵ 2ị1 ỵ 35 28 3ị ; r2;18 ẳ ỵ 3ị ỵ ỵ 23 ị ; p p ffi p ffiffi ffi p ffiffi ffi p ffiffi ffi p 5=12 1=3 r2;72 ẳ 213=48 1ị ỵ 3ị ỵ þ 33=4 ð þ 1ÞÞ1=3 : Baruah and Saikia [1] corrected Yi’s incorrect value of r2,72 From [8, p 12, Theorem 2.1.2(i)–(iii)], we note that rk;1 ¼ 1; rk;n ¼ rn;k and rk;1=n ¼ 1=rk;n : ð2:10Þ Lemma 2.7 ([8]) If hk,n is as defined in (1.3), then pffiffiffi p 21=3 ỵ 41=3 2 1=4 p h3;3 ẳ 3ị ; h3;9 ẳ ; h3;4 ¼ pffiffiffi ; 3À1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 5À1 : h3;5 ¼ Theta-function identities In this section, we prove some new theta-function identities which will be used in the subsequent sections Theorem 3.1 If P ẳ /qị and Q ẳ /q2 ị qwq8 ị q1=2 wq4 ị   32 4P Q2 ỵ ẳ 0: then P ỵ ỵ P Q Q2 Proof Transcribing P by using (2.3) and (2.6) and simplifying, we obtain p 1aẳ P2 : ỵ P2 3:1ị Similarly, transcribing Q by using (2.4) and (2.7) and simplifying, we arrive at p p Q4 ỵ aị2 16 ỵ 2Q2 ị2 a ẳ 0: ð3:2Þ 110 N Saikia Employing (3.1) in (3.2) and simplifying, we obtain 32P2 ỵ 4P4 ỵ 8P2 Q2 ỵ P4 Q2 Q4 ẳ 0: 3:3ị Dividing (3.3) by P2Q2 and rearranging the terms, we complete the proof Theorem 3.2 If P ẳ  then h /qị q1=2 wq4 ị and Q ẳ /q ị q3=2 wq12 ị      P2 Q2 32 P Q þ À þ PQ þ þ ¼ 0: PQ Q P Q2 P2 Proof Transcribing P and Q using (2.3) and (2.6) and simplifying, we obtain x :¼ pffiffiffiffiffiffiffiffiffiffiffi 1aẳ P2 ỵ P2 and y :ẳ p 1bẳ Q2 ; ỵ Q2 3:4ị where b has degree over a Employing (3.4) in Lemma 2.5, we obtain mx ỵ y ẳ 2xyị1=4 3:5ị and 3y x ¼ 2ðxyÞ1=4 : m Eliminating m between (3.5) and (3.6) and then simplifying, we arrive at pffiffiffiffiffiffi pffiffiffiffiffiffi xyð xy 1ị ẳ x yịxyị1=4 : 3:7ị Squaring (3.7) and simplifying, we obtain p xyxy ỵ 1ị ẳ x2 ỵ y2 ỵ 6xy: 3:8ị 3:6ị Squaring (3.8) and then factorizing by employing (3.4), we deduce that ðP4 À 32PQ À 6P3 Q À 6PQ3 À P3 Q3 Q4 ị P4 ỵ 32PQ ỵ 6P3 Q ỵ 6PQ3 ỵ P3 Q3 Q4 ị ẳ 0: 3:9ị It is examined that there exists a neighborhood about origin, where the first factor of (3.9) is not zero Then the second factor is zero in this neighborhood By the identity theorem the second factor is identically zero Thus, we conclude that P4 ỵ 32PQ ỵ 6P3 Q ỵ 6PQ3 ỵ P3 Q3 Q4 ẳ 0: 2 3:10ị Dividing (3.10) by P Q and rearranging the terms, we complete the proof Theorem 3.3 If P ẳ /qị q3=4 wq6 ị and Q ẳ /qị /q3 ị h 111 A new parameter for Ramanujan’s theta-functions then P4 ¼ 432 288Q4 ỵ 128Q6 16Q8 : Q8 6Q4 þ 8Q2 À Proof Transcribing P by using (2.3) and (2.5) and Q by (2.2), we obtain   pffiffiffiffi À a 1=4 P¼2 m b and Q ¼ pffiffiffiffi m; where b has degree over a and m = z1/z3 From (3.11), we deduce that   1a P ẳ 16Q : b 3:11ị ð3:12Þ Employing Lemma 2.4 in (3.12) and simplifying, we find that P4 m 1ị3 ỵ mịm2 16Q4 m ỵ 1ị3 mị3 ẳ 0: 3:13ị Substituting for m from (3.11) in (3.13) and then simplifying, we complete the proof h Properties of Ak,n Theorem 4.1 For all positive real numbers k and n, we have (i) Ak,1 = 1, (ii) Ak,1/n = 1/Ak,n Proof Using the definition of Ak,n and Lemma 2.1, we easily arrive at (i) Replacing n by 1/n in Ak,n and using Lemma 2.1, we find that Ak,nAk,1/n = which completes the proof of (ii) h Remarks 4.2 By using the definitions of /(q), w(q) and Ak,n, it can be seen that Ak,n has positive real value and that the values of Ak,n increase as n increases when k > Thus, by Theorem 4.1(i), Ak,n > for all n > if k > Theorem 4.3 For all positive real numbers k, m, and n, we have Ak;n=m ¼ Amk;n : Ank;m Proof Using the definition of Ak,n, we obtain 112 N Saikia pffiffiffiffiffiffiffiffi Amk;n n1=4 /ðÀeÀ2p n=mk Þ p : ẳ Ank;m m1=4 /e2p m=nk ị 4:1ị Using Lemma 2.1 in the denominator of right hand side of (4.1) and simplifying, we complete the proof Corollary 4.4 For all positive real numbers k and n, we have Ak2 ;n ¼ Ank;n Ak;n=k : Proof Setting k = n in Theorem 4.3 and simplifying using Theorem 4.1(ii), we obtain Ak2 ;m ẳ Amk;k Ak;m=k : 4:2ị Replacing m by n, we complete the proof h Theorem 4.5 Let k, a, b, c, and d be positive real numbers such that ab = cd Then Aa;b Akc;kd ¼ Aka;kb Ac;d : Proof From the definition of Ak,n, we deduce that, for positive real numbers k, a, b, c, and d, pffiffiffiffi pffiffiffiffi epðkÀ1Þ ab=4 wðeÀ2p ab Þ À1 pffiffiffiffi ð4:3Þ Aka;kb Aa;b ẳ k1=4 we2kp ab ị and Akc;kd A1 c;d ẳ p we2p cd ị p : k1=4 we2kp cd Þ epðkÀ1Þ pffiffiffi cd=4 Now the result follows readily from (4.3), (4.4) and the hypothesis ab = cd ð4:4Þ h Corollary 4.6 For any positive real numbers n and p, we have Anp;np ¼ Anp2 ;n Ap;p : Proof The result follows immediately from Theorem 4.5 with a = p2, b = 1,c = d = p and k = n h Now, we give some relations connecting the parameter Ak,n with rk,n and Ramanujan’s class invariants A new parameter for Ramanujan’s theta-functions 113 Theorem 4.7 Let k and n be any positive real numbers Then r24k;n , (i) Ak;n ¼ rk;n r24n;k (ii) An;k ¼ Ak;n r4k;n pffiffiffiffiffi f2 ðÀqÞ f2 ðÀq2 Þ Proof Let q ¼ eÀp n=k Using the results /ðÀqÞ ¼ and wqị ẳ fqị from [3, p 39] in the denition of Ak,n, we nd that fq ị Ak;n ẳ fðÀq2k Þf2 ðÀqÞ 2k1=4 qk=4 fðÀq2 Þf2 ðÀq4k Þ : ð4:5Þ Employing the definition of rk,n from (1.1) in (4.5), we complete the proof of (i) Proof of (ii) follows easily from (i) h Corollary 4.8 For all positive real numbers n and p, we have (i) A1;n ¼ r24;n , r24n;n (ii) An;n ¼ , rn;n (iii) Anp;np ¼ An;np2 Ap;p rÀ2 4;p2 Proof To prove (i), we set k = in Theorem 4.7(i) and use the result rk,1 = from (2.10) Proof of (ii) follows from Theorem 4.7(i) with k = n To prove (iii), we set a = 1,b = p2, c = d = p, and k = n in Theorem 4.5 and use part (i) Theorem 4.9 For all positive real number n, we have (i) An;2 ¼ g28n , g2n (ii) An=2;2 ¼ 21=2 gn G2n Proof (i) From [8, p 18, Theorem 2.3.3(i)], we note that gn ¼ r2;n=2 ; where the class invariant gn is given by gn ¼ 2À1=2 qÀ1=24 vðÀqÞ; pffiffi where q :¼ eÀp n ; n is a positive real number, and v(q) = (Àq;q2)1 Setting n = and replacing k by n, we get 4:6ị 114 N Saikia An;2 ẳ r22;4n ; r2;n ð4:7Þ where we used the result rk,n = rn,k from (2.10) Using the result (4.6) in (4.7), we complete the proof To prove (ii), we replace n by n/2 in part (i) and use the result g4n = 21/4gnGn from [4, p 187, Entry 2.1], where the class invariant Gn is given by Gn = 2À1/2qÀ1/24v(q) h Explicit evaluations of Ak,n In this section, we prove some general theorems on Ak,n and then use these theorems to find explicit values of Ak,n Theorem 5.1 We have !  2  2 2A2;n A2;4n 5=2 ỵ A2;n ỵ ỵ ẳ 0: A2;4n A2;n A2;4n Proof We use the definition of Ak,n in Theorem 3.1 to prove the theorem h Theorem 5.2 We have p p (i) A2;2 ẳ ỵ 2, r  p p p (ii) A2;4 ẳ 2 ỵ ỵ ỵ , p p p q (iii) A2;1=2 ẳ 1= ỵ ẳ 22 2, ffi r rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi p p p (iv) A2;1=4 ẳ 1= 2 ỵ ỵ ỵ ẳ 1ị ỵ =2 Proof Setting n = 1/2 in Theorem 5.1 and using Theorem 4.1(ii), we obtain ! 27=2 5:1ị ỵ A42;2 ỵ ¼ 0: A42;2 A22;2 Solving (5.1) and using the fact in Remark 4.2, we complete the proof of (i) Again setting n = in Theorem 5.1 and using Theorem 4.1(i), we obtain ! ! 5=2 ỵ 5:2ị 1ỵ A2;4 ỵ ẳ 0: A2;4 A22;4 115 A new parameter for Ramanujan’s theta-functions Solving (5.2) and using the fact in Remark 4.2, we prove (ii) Proofs of (iii) and (iv) easily follow from parts (i) and (ii), respectively and the Theorem 4.1(ii) h Theorem 5.3 We have  A2;n A2;9n 2  A2;9n A2;n 2 5=2 ỵ2     A2;n A2;9n A2;n A2;9n ỵ ỵ ỵ6 ẳ 0: A2;n A2;9n A2;9n A2;n Proof Proof follows from Theorem 3.2 and the definition of Ak,n h Theorem 5.4 We have ffi q p p p ỵ 2, (i) A2;3 ẳ ỵ ỵq p p (ii) A2;9 ẳ ỵ 2 ỵ 29 ỵ 2ị, p p p p 2 ị ỵ ỵ 2, (iii) A2;1=3 ẳ q p p (iv) A2;1=9 ẳ 29 ỵ 2ị ỵ 2ị Proof Setting n = 1/3 in Theorem 5.3 and using Theorem 4.1(ii), we obtain ! ! 1 A42;3 ỵ ỵ A22;3 ỵ 27=2 ẳ 0: 5:3ị A42;3 A2;3 Solving (5.3) and using the fact in Remark 4.2, we arrive at (i) Again setting n = in Theorem 5.3 and using Theorem 4.1(i), we obtain !   1 5=2 A2;9 ỵ ỵ ị ỵ A2;9 ẳ 0: 5:4ị A2;9 A22;9 Solving (5.4) and using the fact in Remark 4.2, we complete the proof of (ii) Noting A2,1/3 = 1/A2,3 and A2,1/9 = 1/A2,9 from Theorem 4.1(ii), we prove (iii) and (iv), respectively h Theorem 5.5 We have p p p 1=4 ỵ 2ị1 ỵ ỵ 6ị , (i) A3;2 ẳ  1=4 p p (ii) A4;4 ẳ 21=4 ỵ 2ị3=4 16 ỵ 1521=4 ỵ 12 ỵ 923=4 , p ffiffi ffi p ffiffi ffi pffiffiffi (iii) A5;5 ¼ 21=2 ỵ 5ị1=2 51=4 1ị2 ẳ 27=2 þ 5Þ1=2 ð51=4 þ 1Þ2 ð þ 1Þ2 , pffiffiffi pffiffiffi 1=4 pffiffiffi 3=4 (iv) A5;2 ¼ 2À1=2 ð1 þ 5Þ ð2 þ þ 5Þ , pffiffiffi p p p (v) A9;2 ẳ 21=4 ỵ 2ị1 ỵ 35 28 3ị1=4 , p 1=4 p 1=4 (vi) A8;2 ẳ 23=16 ỵ 2ị 16 þ 1521=4 þ 12 þ 23=4 Þ , p 1=4 (vii) A3=2;2 ẳ 23=4 ỵ 3ị , 116 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffi (viii) A5=2;2 ¼ 2À1=4 ỵ 5ị1=2 ỵ ỵ 2ị1=4 , p (ix) A25=2;2 ẳ 217=8 ỵ 1ị2 51=4 þ 1Þ N Saikia Proof The proof of the theorem follows from Theorem 4.7(i) and the corresponding values of rk,n from Lemma 2.6 h One can easily find values of A3,1/2, A4,1/4, A5,1/5, A5,1/2, A9,1/2, A8,1/2, A3/2,1/2, A5/2,1/2, and A25/2,1/2 by using Theorems 5.5 and 4.1(ii) Theorem 5.6 We have (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (xii) (xiii) (xiv) A1,1 = 1, À pffiffiffiÁ1=4 A1;2 ẳ 21=4 ỵ , p1=2 A1;3 ẳ ỵ , p1=2 A1;4 ẳ  25=8 ỵ 2q , 0 p p 2, A1;5 ẳ ỵ ỵ 21 ỵ 5ị p3=4 p p 21 ỵ ỵ 6ị 1=4 A1;6 ẳ ỵ p1=2 A1;7 ẳ ỵ ,  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiÁ3=4 pffiffiffi À A1;8 ¼ 21=2 ỵ ỵ ỵ 10 1=4, p!2 31=4 ỵ p ỵ , A1;9 ẳ 2 p p p29=4 ỵ ỵ 1=4=4, A1;10 ẳ ỵ p p A1;16 ẳ 23=4 ỵ 16 ỵ 15 21=4 ỵ 12 ỵ Á 23=4 1=4, pffiffiffi pffiffiffiÁ Àpffiffiffi pffiffiffiÁ2 À A1;18 ¼ 21=4 ỵ ỵ 35 28 1=4,  ffiffiffiffiffi2 pffiffiffi pffiffiffi p A1;25 ¼ ỵ ỵ ỵ 53 =4, q qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi A1;49 ẳ ỵ ỵ 21 ỵ ỵ ỵ 21 ỵ 16 Proof The proof of the theorem follows from Corollary 4.8(i) and the corresponding values of r4,n from Lemma 2.6 h The values of A1,1/n for n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 16, 18, 25, and 49 can easily be calculated by applying Theorems 5.6 and 4.1(ii) Theorem 5.7 We have A43;n p 18h43;n ỵ 3h63;n À 3h83;n pffiffiffi : ¼ 9h3;n À 18h43;n ỵ 3h23;n 117 A new parameter for Ramanujan’s theta-functions Proof A proof of the theorem follows directly from Theorem 3.3 and the definitions of hk,n and Ak,n from (1.3) and (1.7), respectively h Theorem 5.8 We have pffiffiffiÁ3=8  pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiÀ1=4 À 30 À 18 ỵ ỵ , (i) A3;3 ẳ 31=8 ỵ p p p p p1=4 p p1=4 105 90 ỵ 126 80 , (ii) A3;4 ẳ 84 ỵ 69 ỵ 36 31 p p1=4 À pffiffiffi pffiffiffiffiffiÁÀ1=4 pffiffiffi pffiffiffi À (iii) A3;5 ¼ À57 32 ỵ 27 ỵ 16 15 ỵ 15 , pffiffiffiÁÀ1=4 pffiffiffiÁ1=4 À pffiffiffi À 1=3 1=2 1=3 2=3 2=3 39 ỵ 182 ỵ 92 ỵ 22 (iv) A3;9 ẳ ỵ ị þ Proof The proof of the theorem follows from Theorem 5.7 and the corresponding values of h3,n from Lemma 2.7 h The values of A3,1/n for n = 3, 4, 5, and can easily be found by applying Theorems 5.8 and 4.1(ii) Theorem 5.9 We have pffiffiffi 5=6  p1=3 p p2=3  p3=8 1ỵ 3 þ A6;6 ¼ 2À1=12 31=8 À 2ỵ 2=3  1=3  p p p p p ỵ ỵ 233=4 þ þ 33=4 ð þ 1Þ  q1=4 p p 30 18 ỵ þ : Proof Setting n = and p = in Corollary 4.6, we get A6;6 ¼ A18;2 A3;3 : ð5:5Þ Again setting k = 18 and n = in Theorem 4.7(i), we find that À1 A18;2 ¼ r272;2 rÀ1 18;2 ¼ r2;72 r2;18 ; ð5:6Þ where we used the result rk,n = rn,k from (2.10) Employing the values of r2,72 and r2,18 from Lemma 2.6 in (5.6), we obtain pffiffiffi À5=6  À1=3 pffiffiffiÀ1=3 pffiffiffi pffiffiffi2=3  pffiffiffi pffiffiffi A18;2 ¼ 2À1=12 À 1ỵ ỵ ỵ 233=4 2ỵ  2=3 p p p ỵ ỵ 33=4 ỵ 1ị : 5:7ị Employing the values of A18,2 from (5.7) and A3,3 from Theorem 5.8(i) in (5.5), we complete the proof h 118 N Saikia The value of A6,1/6 easily follows from Theorems 5.9 and 4.1(ii) Explicit evaluations of w(q) In this section, we establish an explicit formula for w(eÀ2np), for positive real number n and give some examples Lemma 6.1 Let a = p1/4/C(3/4) Then /ep ị ẳ a21=4 : For proof, see Entry 1(ii) in Chapter 35 of [4, p 325] Theorem 6.2 For every positive real number n, we have we2np ị ẳ 25=4 aenp=4 : n1=4 An;n Proof Using the definition of An,n and Lemma 6.1, we complete the proof h Theorem 6.3 We have (i) w(eÀ2p) = a25/4ep/4, p 1=2 (ii) we4p ị ẳ a22 ep=2 À ,  pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi1=4 À pffiffiffiÁÀ3=8 (iii) we6p ị ẳ a25=4 33=8 e3p=4 30 18 þ À9 þ À3 þ , pffiffiffiÁÀ3=4 À pffiffiffi À Á À2 p 1=4 3=4 À1=4 8p 16 ỵ 152 ỵ 12 ỵ 92 , (iv) we ị ẳ a2 e ỵ p ffiffi ffi À Á À Á À1=2 (v) wðeÀ10p ị ẳ a27=4 51=4 e5p=4 51=4 ỵ , p ffiffi ffi p ffiffi ffi À Á p p2=3 5=6 1=3 17=12 3=8 21 2ỵ 3 1ỵ (vi) we12p ị ẳ ae3p=2 pffiffiffi pffiffiffiÁÀ3=8 À pffiffiffi pffiffiffi 3=4 Á1=3 À pffiffiffi pffiffiffi 2=3 ỵ ỵ þ 23 À þ þ 33=4 ỵ 1ị  p p p1=4 30 18 ỵ ỵ Proof The proof of the theorem follows from Theorem 6.2 and the corresponding values of An,n from Theorem 5.6(i), Theorem 5.2(i), Theorem 5.8(i), Theorem 5.5(ii) and (iii), and Theorem 5.9 h The values of w(eÀnp) for n = 2, 4, and can also be found in [4, p 325] References [1] N.D Baruah, N Saikia, Modular equations and explicit values of Ramanujan–Selberg continued fraction, Int J Math Math Sci 2006 (2006) 1–15 Article ID 54901 A new parameter for Ramanujan’s 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