MODULAR EQUATIONS FOR THE RATIOS OF RAMANUJAN S THETA FUNCTION ψ AND EVALUATIONS

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1 NEW ZEALAND JOURNAL OF MATHEMATICS Volume , -48 MODULAR EUATIONS FOR THE RATIOS OF RAMANUJAN S THETA FUNCTION ψ AND EVALUATIONS M. S. MAHADEVA NAIKA, S. CHANDANKUMAR AND K. SUSHAN BAIR Received August 010 Abstract. In this paper, we establish several new modular equations of degree 9 using Ramanujan s mixed modular equations. We also establish several general formulas for explicit evaluations of ratios of Ramanujan s theta function. 1. Introduction In Chapter 16, of his Second notebook 16] Ramanujan has defined his theta function as fa, b := a nn1/ b nn1/, ab < Following Ramanujan, we define where n= ϕq := fq, q = ψq := fq, q = fq := a; q := n= n= n=0 q n = q; q q; q, 1. q nn1/ = q ; q q; q, 1. q nn1/ = q; q, 1.4 χq := q; q, 1. 1 aq n, q < 1. n=0 In ] and 1], the authors have defined two parameters l k,n and l k,n as follows: and ψe πn/k l k,n := 1.6 k 1/4 e k1π 8 n/k ψe π nk, l k,n ψe πn/k := 1. k 1/4 e k1π 8 n/k ψe π nk. They have established several properties and some explicit evaluations of l k,n and l k,n for different positive rational values of n and k. They also listed some applications of 010 Mathematics Subject Classification B6, D10, Secondary F. Key words and phrases: Modular equation, Theta-function. Research supported by DST grant SR/S4/MS:09/0, Govt. of India.

2 4 M. S. MAHADEVA NAIKA, S. CHANDANKUMAR AND K. SUSHAN BAIR these parameters for the Rogers-Ramanujan continued fraction and Ramanujan s cubic continued fraction. In ], M. S. Mahadeva Naika and S. Chandankumar have established several new modular equations of degree and also general formulas for the explicit evaluation of ratios of Ramanujan s theta function ϕ. In 1], Mahadeva Naika, K. S. Bairy and M. Manjunatha have established several new modular equations of degree 4 and also general formulas for the explicit evaluations of ratios of Ramanujan s theta function ϕ. For more details see 1], ], 8], 9], 1], 14]. Now we define a modular equation in brief. The ordinary hypergeometric series F 1 a, b; c; x is defined by F 1 a, b; c; x := n=0 a n b n x n, c n n! where a 0 = 1, a n = aa 1a a n 1 for any positive integer n, and x < 1. Let z := zx := F 1 1, 1 ; 1; x 1.8 and q := qx := exp π F 1 1, 1 ; 1; 1 x F 1 1, 1, 1.9 ; 1; x where 0 < x < 1. Let r denote a fixed natural number and assume that the following relation holds: r F 1 1, 1 ; 1; 1 α 1 F 1, 1 ; 1; α = F 1 1, 1 ; 1; 1 β 1 F 1, ; 1; β Then a modular equation of degree r in the classical theory is a relation between α and β induced by We often say that β is of degree r over α and m := zα is called the zβ multiplier. We also use the notations z 1 := zα and z r := zβ to indicate that β has degree r over α. In Section, we collect the identities which are useful in proving our main results. In Section, we establish some new modular equations of degree 9 which are analogous to Ramanujan s modular equations. In Section 4, we establish general formulas for explicit evaluations of l 9,n and also establish relations among l 9,n and l 9,n.. reliminary Results In this section, we collect some identities which are useful in proving our main results. Lemma.1. 10, Theorem.] We have f q f q 9 f q f q 18 = ψq ψq 9 ψq qψq 9 ψq qψq 9,.1 = ψ q ψq qψq 9 ψ q 9 ψq qψq 9..

3 RATIOS OF RAMANUJAN S THETA FUNCTION ψ AND EVALUATIONS Lemma.. 6, Ch. 1, Entry 10i and Entry ii, pp. 1 1] If β, γ and δ are of degrees r, s and rs over α respectively, then ϕq = z 1,. ϕq r = z r,.4 ϕq s = z s,. ϕq rs = z rs,.6 q 1/8 ψq = z 1 {α1 α} 1/8,. q r/8 ψq r = z r {β1 β} 1/8,.8 q s/8 ψq s = z s {γ1 γ} 1/8,.9 q rs/8 ψq rs = z rs {δ1 δ} 1/8..10 Lemma.. 6, Ch.0, Entry 1ii, p. 4] We have 1 q ψq9 ψq = 1 9q ψ4 q ψ 4 q 1/.. Lemma.4. 6, Ch. 0, Entry viii, ix, p. 84] Let α, β, γ and δ be of the first, third, fifth and fifteenth degrees respectively. Let m denote the multiplier connecting α, β and m be the multiplier relating γ, δ, then 1/8 1/8 1/8 αδ 1 α1 δ αδ1 α1 δ m = βγ 1 β1 γ βγ1 β1 γ m,.1 1/8 1/8 1/8 βγ 1 β1 γ βγ1 β1 γ m = αδ 1 α1 δ αδ1 α1 δ m..1 Lemma.. 6, Ch. 0, Entry 1 i and ii, p. 401] Let α, β, γ and δ be of the first, third, seventh and twenty first degrees respectively. Let m denote the multiplier connecting α, β and m be the multiplier relating γ, δ, then 1/4 1/4 1/4 βγ 1 β1 γ βγ1 β1 γ αδ 1 α1 δ αδ1 α1 δ 1/6.14 βγ1 β1 γ 4 = m 1/4 αδ βγ αδ1 α1 δ m, 1/4 1 α1 δ αδ1 α1 δ 1 β1 γ βγ1 β1 γ 1/6 αδ1 α1 δ = m βγ1 β1 γ m. 4 1/4.1 Lemma.6. 6, Ch. 0, Entry 14 i and ii, p. 408] Let α, β, γ and δ be of the first, third, eleventh and thirty third degrees respectively. Let m denote the multiplier connecting α, β and m be the multiplier relating γ, δ, then 1/8 1 β1 δ βδ1 β1 δ βδ αγ 1/8 1/8 1 α1 γ αγ1 α1 γ 1/1 βδ1 β1 δ = mm αγ1 α1 γ,.16

4 6 M. S. MAHADEVA NAIKA, S. CHANDANKUMAR AND K. SUSHAN BAIR 1/8 αγ βδ 1/8 1 α1 γ 1 β1 δ 1/1 αγ1 α1 γ 4 = βδ1 β1 δ 1/8 αγ1 α1 γ βδ1 β1 δ.1. mm Lemma.. 6, Ch. 0, Entry 19 iv, p. 46] If β, γ and δ are of degrees, 1 and 9 respectively over α, then 1/8 1/8 1/8 αδ 1 α1 δ αδ1 α1 δ βγ 1 β1 γ βγ1 β1 γ 1/1.18 αδ1 α1 δ m = βγ1 β1 γ m, 1/8 1/8 βγ 1 β1 γ βγ1 β1 γ αδ 1 α1 δ αδ1 α1 δ 1/1 βγ1 β1 γ m = αδ1 α1 δ m. Lemma.8., Theorem 4.1ii] and 1, Theorem.ii] We have Lemma.9., Ch., Entry 6, p. 10] If M = then Lemma.10., Theorem.1] If =. Modular Equations of Degree 9 1/8.19 l k,n l k,1/n = 1..0 fq q 1/ fq 9 and N = fq q / fq 18, M N = M N MN..1 ψq q 1/4 ψq and = ϕq ϕq, then = In this section, we establish some new modular equations of degree 9 for the ratios of Ramanujan s theta function. Theorem.1. If = ψq qψq 9 and = ψq qψq 9, then = 4..1 roof. Using the equation. by changing q to q and the equation.1 in the equation.1, we deduce that M = and N = The equation.1 can be rewritten as a a A = 0,.

5 RATIOS OF RAMANUJAN S THETA FUNCTION ψ AND EVALUATIONS where a = MN and A = M N. Solving the equation. for a and cubing both sides, we find that 8M N = m,.4 where m = ± 9 4A. Eliminating m from the above equation.4, we deduce that = 0.. By examining the behavior of the above factors near q = 0, we can find a neighborhood about the origin, where the third factor is zero; whereas other factors are not zero in this neighborhood. By the Identity Theorem third factor vanishes identically. This completes the proof. Theorem.. If = ψq qψq 9 and = ψq q ψq 18, then =..6 roof. Replacing q with q in the equation.1 and equating with the equation., we find that = On factorizing the above equation, we deduce that = 0..8 By examining the behavior of the above factors near q = 0, we can find a neighborhood about the origin, where the second factor is zero; whereas the first factor is not zero in this neighborhood. By the Identity Theorem second factor vanishes identically. This completes the proof. Theorem.. If = ψq qψq 9 and = ψq q ψq 18, then = 9 ] 1. roof. Using the equations.1 and.6, we arrive at the equation.9..9

6 8 M. S. MAHADEVA NAIKA, S. CHANDANKUMAR AND K. SUSHAN BAIR Theorem.4. If = ψqψq q 6 ψq 9 ψq 4 and = ψqψq4 q 4 ψq 9 ψq, then 1 = ] ] roof. Using the equations..10 and the equations.1 and.1 with r = and s =, we deduce that p a b = q 1 b a,. where p = ϕq ϕq, q 1 = ϕq ϕq 1, a = ψq q 1/4 ψq, b = ψq q /4 ψq 1. Using the equations. and., we deduce that 9b a 9ba a b 6a 6 b a b a b a b b a 6b 6 a b a a 8 b 8 = 0..1 By squaring the above equation.1, we deduce that 10a 4 b 4b a 180a 4b 4 180a 4 b 4 0a 4b 4b a 180b a 4 a b 4 14a 4b 4 a b 4b a 4 a 4a b b 4 81b 4 b a 81a 4 a b 16a 4 b 4 90a 4b a 1b 4b a 1a 4a b a 4 4 b a 4b 4 0a 4b 4 0a 4b 4 a 4b 4 14a 4 b 4 10a b 4b a 4 10a 4a b b 4 90b 4b a 10a 4b a b 4 = 0,.1 where a 4 = a 4, b 4 = b 4, a = a and b = b. Isolating the terms containing a b on one side of the equation.1, squaring both sides

7 RATIOS OF RAMANUJAN S THETA FUNCTION ψ AND EVALUATIONS 9 and by using the equation., we deduce that 40A B 1BA 4BA 4 90BA 1BA 81BA 10B A 0B A 4 90B A 90B A 1B A 4B A 4 90B A 180B A 1B A B 4 A 0B 4 A 4 0B 4 A 4B 4 A 4B 4 A B A B A 4 10B A 1B A 1B A B 6 A A 6 B A B A B 9064A 6 B 6 810A 9 B 6 810A 6 B A 9 B 9 40A 9 B 40A B A 6 B 1 810A 9 B 1 A 1 B 1 180A B 1 19A 10 B 1849A 1 B 6 810A 1 B 9 180A 1 B 9841BA BA B A B A 890B A 8686B A B A 4166B A B A 160B A 6 090B A 91484B A B A 48969B A 496B 4 A B 4 A 0444B 4 A B 4 A 14841B 4 A B 4 A 91484B 4 A 9610B 6 A 8 694B 6 A 8090B 6 A 0444B 6 A 4 160B 6 A 9900B A B A 8090B A 6 160B A B A 4 890B A 090B A B 8 A B 8 A 9610B 8 A B 8 A 496B 8 A B 8 A 4166B 8 A 9841B 8 A B A B A 694B A B A B A B A 91009B A 9049B A 4040B 10 A 8 918B 10 A B 10 A B 10 A 10B 10 A B 10 A 9660B 10 A 80B 10 A 19B 9 A 8 48B 9 A 9800B 9 A B 9 A B 9 A 6610B 9 A 60A 1 B 4 440A B 4 10A 10 B A 9 B A B A 10 B 6 814A 1 B 04A B 19490A 10 B 9800A 9 B 98A 1 B 16A B 918A 10 B 48A 9 B 1A B A 10 B 9 90A 1 B 8 100A B A 10 B 8 19A 9 B 8 18A 1 B 69A B 410A B 1A 9 B 100A 8 B 16A B 1A 1 B 10 19A B A 10 B A 9 B A 10 B 1A 10 B 1 90A 8 B 1 98A B A 6 B 18A B 1 410A B 9660B A B A 9 80BA 10

8 40 M. S. MAHADEVA NAIKA, S. CHANDANKUMAR AND K. SUSHAN BAIR 814A B 1 60A 4 B 1 04A B 440A 4 B 9A 1 9B 1 44A 1 B 19A 1 B 64A B 44B 1 A 19B 1 A 109B A 64B A 109BA 6610BA 9 = By observing the above factors of the equation.14 near q = 0, it can be seen that there is a neighborhood about the origin where the first factor is zero, whereas the second factor is not zero. Hence by the Identity Theorem the first factor vanishes identically and then by setting = AB and = A, we arrive at the equation.10. B Theorem.. If = ψqψq q 8 ψq 9 ψq 6 and = ψqψq6 q 6 ψq 9 ψq, then = ] ] ] ] 1. ] 1.1 The proof of.1 is similar to the proof of the equation.10, except that in place of results.1 and.1, the results.14 and.1 with r = and s = are used. Theorem.6. If = ψqψq q 1 ψq 9 ψq 99 and = ψqψq 99 q 10 ψq 9 ψq, then ] ] ] 8 1

9 RATIOS OF RAMANUJAN S THETA FUNCTION ψ AND EVALUATIONS ] ] ] = ] The proof of.16 is similar to the proof of the equation.10, except that in place of results.1 and.1, the results.16 and.1 with r = and s = are used. Theorem.. If = ψqψq1 q 14 ψq 9 ψq and = ψqψq q 1 ψq 9 ψq 1, then 1 { ] ] ] 1 ] 91 ] ] 1 ] ] 4 1 ] 419 ] 1 6 ]] 1 ] ] ]] 1 ] ] 0 1 ] ] ]] ] 1 ] ] ] ] ] ]] ] ] ] ] ]

10 4 M. S. MAHADEVA NAIKA, S. CHANDANKUMAR AND K. SUSHAN BAIR 91 ] ]]} = ]] ] 9 ] 40 1 ] 6 ] { ]] ] 1 ]] 81 ] ] 04 1 ] 1 ] ]] ] ] 6 1 ] ] ] 1 ]]}. ] 1.1 The proof of.1 is similar to the proof of the equation.10, except that in place of results.1 and.1, the results.18 and.19 with r = and s = 1 are used. 4. General Formulas for Explicit Evaluations of l 9,n In this section, we establish some general formulas for explicit evaluations of l 9,n. We also establish relations among l 9,n and l 9,n. Theorem 4.1. If = l 9,n and = l 9,4n, then = 1 ] roof. Using the equation.6 along with the equation 1.6, we arrive at the equation 4.1. Corollary 4.1. We have 1 l 9, =, 4. 1 l 9,1/ =, 4. l 9,4 =, 4.4 l 9,1/4 =, 4.

11 RATIOS OF RAMANUJAN S THETA FUNCTION ψ AND EVALUATIONS 4 l 9,8 = l 9,1/8 = , roof of 4.. utting n = 1/ in the equation 4.1 and using the fact that l 9, l 9,1/ = 1, we deduce that l 9, l 9, l 9, l 9, 1 = We observe that the first factor of the equation 4.8 vanishes for the specific value of q = e π/9, but the other two factors does not vanish. Since l 9, > 1, we arrive at the equation 4.. roof of 4.. By using the fact that l 9, l 9,1/ = 1, we arrive at the equation 4.. roofs of 4.4 and 4.. utting n = 1 in the equation 4.1 and by using the fact that l 9,1 = 1, we deduce that l 4 9,4 l 9,4 l 9,4 l 9,4 l 9,4 1 = The above equation 4.9 can be rewritten as where x = l 9,4 1. l 9,4 Solving the above equation for x, we deduce that Since x > 0, we deduce that x = x x = ,. 4. l 9,4 1 =. 4.1 l 9,4 On solving the above equation 4.1, we arrive at the equations 4.4 and 4.. roofs of 4.6 and 4.. Using the equation 4. in the equation 4.1, we obtain the equations 4.6 and 4.. Theorem 4.. If = l 9,n and = l 9,n, then 1 1 ] = roof. Using the equation.1 along with the equations 1.6 and 1. with k = 9, we arrive at the equation 4.1 which completes the proof. Corollary 4.. We have l 9, = , 4.14 l 9, =

12 44 M. S. MAHADEVA NAIKA, S. CHANDANKUMAR AND K. SUSHAN BAIR roof. Using the values of l 9,, 1] and l 9, in the above equation 4.1, we arrive at the equations 4.14 and 4.1 respectively. Theorem 4.. If = l 9,n and = l 9,4n, then = roof. Changing q to q in the equation.6 and by using the equations 1.6 and 1. with k = 9, we arrive at the equation Corollary 4.. We have l 9,4 = 1, 4.1 l 1/9,4 = 1, 4.18 l 9,8 =, 4.19 l 9,1 = 1 1] ], 4.0 l 9,8 = roof. utting n = 1,,, in Theorem 4. and using the values of l 9,1, l 9,, l 9, and l 9,, we arrive at the equations 4.1, 4.19, 4.0 and 4.1 respectively. This completes the proof. Theorem 4.4. If = l 9,n l 9,n and = l 9,n, then l 9,n 1 = ] ] roof. Employing the equation.10 along with the equation 1.6 with k = 9, we obtain the equation 4.. Corollary 4.4. We have where a = l 9, = 1, 4. l 9,1/ = 1, 4.4 l 9, = 4 1 a, 4. l 9,1/ = 4 1 a, 4.6

13 RATIOS OF RAMANUJAN S THETA FUNCTION ψ AND EVALUATIONS 4 roof of 4.. utting n = 1/ in the equation 4. and then using the equation.0, we find that l 9, l 9, l 9, l 9, 1 l 9, l 9, l 9, 1 = 0. Since l 9, > 1, we find that 4. l 9, l 9, = Solving the above equation 4.8, we arrive at 4.. roof of 4.4. Using the equations.0 with n =, k = 9 and 4., we obtain 4.4. roofs of 4. and 4.6. utting n = 1 in the equation 4. and using the fact that l 9,1 = 1, we deduce that l9, 1 l9, 6 0 l 9, 1 6 = l 9, The above equation 4.9 reduces to where a = l 9, 1 l 9,. On solving the above equation and a > 1, we deduce that a 6 0 a 4 = 0, 4.0 l 9, 1 l 9, = Again by solving the above equation, we obtain 4. and 4.6. Theorem 4.. If = l 9,n l 9,49n and = l 9,n, then l 9,49n = ] 1 { ] ] ]}. 1 1 ] 4. roof. Employing the equation.1 along with the equation 1.6 with k = 9, we obtain the equation 4..

14 46 M. S. MAHADEVA NAIKA, S. CHANDANKUMAR AND K. SUSHAN BAIR Corollary 4.. We have l 9, = , 4. l 9,1/ = roofs of 4. and 4.4. utting n = 1/ in the equation 4. and then using the equation.0, we find that l 8 9, l 9, 1l 6 9, 1 l 9, 48l 4 9, 1 l 9, 1l 9, l 9, 1 l 4 9, 4 l 9, 14l 9, 4 l 9, 1l 4 9, l 9, l 9, l 9, 1 = We observe that the first and the last factors are not zero for specific value of q = e π /9. Hence the second factor is zero, x 4 x 16 = 0, 4.6 where x = l 9, 1 l 9,. On solving the above equation 4.6 for x and x > 1, we deduce that l 9, 1 l 9, =. 4. On solving the above equation, we arrive at the equations 4. and 4.4 respectively. Theorem 4.6. If = l 9,n l 9,169n and = l 9,n, then l 9,169n 1 { ] ] ] ] ] 91 1 ] ] ] ] ] ]] 1 ] ] ] ]] 1 ] 0 1 ] ] ] 1 ]] 1 ] 89 1 ] ] ] 44 1 ] ]] 9 ] ] ] ] 46 1 ] ] 4 1 ]]} = ] 9 6

15 RATIOS OF RAMANUJAN S THETA FUNCTION ψ AND EVALUATIONS 4 { ] ]] 4 1 ] ] 6 1 ]] 1 ] ] 10 1 ] 1 ]] 1 ] ] 04 1 ] 1 ] ]] 4 1 ] ] 6 1 ] ] ] 4 1 ]]}. 4.8 roof. Employing the equation.1 along with the equation 1.6 with k = 9, we obtain the equation 4.. Corollary 4.6. We have l 9,1 =, l 9,1/1 = roof. utting n = 1/1 in the equation 4.8 and using the equation.0, we find that l 9,1 4 1l 9,1 6 l 9,1 6l 9,1 16 l 9,1 1l 9,1 6 l 9,1 1 l 9,1 4 1l 9,1 6 l 9,1 6l 9,1 16 l 9,1 1l 9,1 6 l 9,1 1 l 9,1 l 9,1 l 9,1 1 l 9,1 l 9,1 l 9,1 1 l 9,1 1 l 9,1 l 9,1 l 9,1 1 l 9,1 l 9,1 l 9,1 1 l 9,1 1 = We observe that the first factor is zero for the specific value of q = e π 1/9, where as the other factors are not zero. Hence we deduce that x 1 6 x 8 16 = 0, 4.4 where x = l 9,1 1. Solving the above equation for x and x > 0, we deduce that l 9,1 l 9,1 1 l 9,1 = On solving the above quadratic equation, we arrive at the equations 4.9 and Remark 1. By using the values of l 9,n and l 9,n established in the earlier section, one can compute the explicit evaluations of Ramanujan s cubic continued fractions. For details see ], 4], 1]. Acknowledgement The authors are thankful to Dr. S. Cooper and the referee for their valuable suggestions which considerably improved the quality of the paper.

16 48 M. S. MAHADEVA NAIKA, S. CHANDANKUMAR AND K. SUSHAN BAIR References 1] C. Adiga, M. S. Mahadeva Naika and K. Shivashankara, On some - eta-function identities of Ramanujan, Indian J. Math., 44 00, 6. ] C. Adiga, Taekyun Kim and M. S. Mahadeva Naika, Modular equations in the theory of signature and - identities, Adv. Stud. Contemp. Math., 1 00, 40. ] C. Adiga, Taekyun Kim, M. S. Mahadeva Naika and H. S. Madhusudhan, On Ramnujan s cubic continued fraction and explicit evaluations of theta-functions, Indian J. pure appl. math., 9 004, ] C. Adiga, K. R. Vasuki, and M. S. Mahadeva Naika, Some new explicit evaluations of Ramanujan s cubic continued fraction, The New Zealand J. Math., 1 00, 1 6. ] N. D. Baruah and Nipen Saikia, Two parameters for Ramanujan s theta-functions and their explicit values, Rocky Mountain J. Math., 6 00, ] B. C. Berndt, Ramanujan s Notebooks, art III, Springer-Verlag, New ork, ] B. C. Berndt, Ramanujan s Notebooks, art IV, Springer-Verlag, New ork, ] S. Bhargava, C. Adiga and M. S. Mahadeva Naika, A new class of modular equations akin to Ramanujan s - eta-function identities and some evaluations there from, Adv. Stud. Contemp. Math., 1 00, 48. 9] M. S. Mahadeva Naika, - eta-function identities and computation of Ramanujan- Weber class invariants, J. Indian Math. Soc., , ] M. S. Mahadeva Naika, Some theorems on Ramanujan s cubic continued fraction and related identities, Tamsui Oxf. J. Math. Sci , 4 6. ] M. S. Mahadeva Naika and S. Chandankumar, Some new modular equations and their applications, Communicated. 1] M. S. Mahadeva Naika, S. Chandankumar and M. Manjunatha, On some new modular equations of degree 9 and their applications, Communicated. 1] M. S. Mahadeva Naika, M. C. Maheshkumar and K. Sushan Bairy, General formulas for explicit evaluations of Ramanujan s cubic continued fraction, Kyungpook Math. J., , ] M. S. Mahadeva Naika, K. Sushan Bairy and S. Chandankumar, On some explicit evaluation of the ratios of Ramanujan s theta-function, Communicated. 1] M. S. Mahadeva Naika, K. Sushan Bairy and M. Manjunatha, Some new modular equations of degree four and their explicit evaluations, Eur. J. ure Appl. Math., 6 010, ] S. Ramanujan, Notebooks volumes, Tata Institute of Fundamental Research, Bombay, 19. 1] J. i, ang Lee and Dae Hyun aek, The explicit formulas and evaluations of Ramanujan s theta-function ψ, J. Math. Anal. Appl., 1 006, M. S. Mahadeva Naika Department of Mathematics Bangalore University Central College Campus Bengaluru INDIA msmnaika@rediffmail.com S. Chandankumar Department of Mathematics Bangalore University Central College Campus Bengaluru INDIA chandan.s1@gmail.com K. Sushan Bairy Department of Mathematics Bangalore University Central College Campus Bengaluru INDIA ksbairy@rediffmail.com