Jacobi-Sohncke Type Mixed Modular Equations and their applications

Palestine Journal of Mathematics

Vol. 6(1)(2017)

, 228237

© Palestine Polytechnic University-PPU 2017

Jacobi-Sohncke Type Mixed Modular Equations and their applications to overpartitions M. S. Mahadeva Naika, K. Sushan Bairy, D. S. Gireesh and N. P. Suman

Communicated by P. K. Banerji MSC 2010 Classications: 33D10, 11A55, 11F27 Keywords and phrases: Modular equation, theta-function, continued fraction. This research is supported by DST, Govt. of India under the project grant SR/S4/MS:739/11. The authors would like to thank Prof. P. K. Banerji for his useful comments.

Abstract. The modular relations between the Ramanujan-Selberg continued fractions H (q ), H (q 5 ), H (q n ), and H (q 5n ) for n = 2, 3, 4, 5, 7, 9, 11, and 13 are established as an application to several new Jacobi-Sohncke type mixed modular equations for composite degrees 1, 5, n, and 5n. We also obtain congruence properties for color overpartition of n with odd parts.

1

Introduction

The theory of modular equations commenced when A. M. Legendre [12], in his paper derived a modular equation of degree 3 in 1825. In fact prior to this, Landen records Landen's transformation in his papers [10], [11] in 1771 and 1775. After Legendre, Jacobi established modular equations of degree 3 and 5 in his famous book [8]. Sohncke [22, 23] established modular equations of degrees 7, 11, 13, 17, and 19. Subsequently many mathematicians have contributed to the theory of modular equations. Some of them are Guetzlaff, Schröter, Schläi, Klein, Cayley, Russell, Weber. Ramanujan's contributions in the area of modular equations are immense. Perhaps, Ramanujan found more modular equations than all of his predecessors discovered together. For more on modular equations one can refer [7, 9, 13, 14, 18]. The complete elliptic integral of rst kind is dened as K (k ) :=

∫

π 2

0

√

dϕ

1 − k2 sin2 ϕ

∞ π∑

=

( 1 )2

2 n 2n k 2

2 n=0 (n!)

=

π

2

( 2 F1

)

1 1 , ; 1; k 2 , 2 2

(1.1)

where 0 < k < 1 and 2 F1 is the ordinary or Gaussian hypergeometric function dened by 2 F1 (a, b; c; z )

:=

∞ ∑ (a)n (b)n n z , 0 ≤ |z| < 1, (c)n n! n=0

(a)0 = 1, (a)n = a(a + 1) · · · (a + n − 1) for n a positive integer

and a, b, c are complex numbers such that c ̸= 0, −1, −2, . . . . The number k is called the √ modulus of K , and k′ := 1 − k2 is called the complementary modulus. Now we dene modular equation in brief. Let K , K ′ , L, and L′ denote the complete elliptic integrals of the rst kind associated with the moduli k, k′ , l, and l′ , respectively. Suppose that the equality n

K′ L′ = K L

(1.2)

229

Jacobi-Sohncke Type Mixed Modular Equations

holds for some positive integer n. Then a modular equation of degree n is a relation between the moduli k and l which is induced by (1.2). Following Ramanujan, set α = k2 and β = l2 . Then we say β is of degree n over α. The multiplier m is dened by m=

However, if we set

K . L

(1.3)

q = exp{−πK ′ /K}

and q ′ = exp{−πL′ /L} (1.4) n ′ we see that (1.2) is equivalent to the relation q = q . Thus a modular equation can be viewed as an identity involving theta-functions at the arguments q and q n . Let K , K ′ , L1 , L′1 , L2 , L′2 , L3 , and L′3 denote complete √ elliptic integrals of the rst kind √ √ √ corresponding, in pairs, to the moduli α, β , γ , and δ , and their complementary moduli, respectively. Let n1 , n2 , and n3 be positive integers such that n3 = n1 n2 . Suppose that the equalities n1

L′ L′ L′ K′ K′ K′ = 1 , n2 = 2 and n3 = 3 K L1 K L2 K L3

(1.5) √

√

√

√

hold. Then a mixed modular equation is a relation between the moduli α, β , γ , and δ that is induced by (1.5). We say that β , γ , and δ are of degrees n1 , n2 , and n3 , respectively over α. The multipliers m and m′ are associated with α, β and γ , δ . The equations involving α1/8 and β 1/8 are referred to them as Jacobi-Sohncke type modular equations. Remark 1.1.

Recently, M. S. Mahadeva Naika, K. Sushan Bairy and C. Shivashankar [17] have obtained Jacobi-Sohncke type mixed modular equations of degrees 1, 3, n and 3n. In [21, eq.(54)], Selberg rst proved the continued fraction identity 1

H (q ) :=

=

q8

q

q2 + q

1 +1+ 1 1 q 8 (−q 2 ; q 2 )∞ , (−q ; q 2 )∞

q3

q4 + q2

+ 1 +

1

+

...

(1.6)

|q| < 1,

which appears as Formula 5 [20, p. 290]. Other proofs have been given by Ramanathan [19] and Andrews, Berndt, Jacobsen and Lamphere [2]. Recently , two integral representations for H (q ), a relation between H (q ) and H (q n ) and some explicit evaluations of H (q ) have been obtained by Adiga, Mahadeva Naika and Ramya Rao [1]. Mahadeva Naika et al. [15, 16], have obtained several integral representations and also Rogers-Ramanujan type functions for H (q ). In section 2, we collect the identities which helps us to prove our main results. In sections 3, we establish several new mixed modular equations. In section 4, we establish modular relations for Ramanujan-Selberg continued fraction with four moduli. In section 5, we establish some congruence properties for overpartitions of n in p−colors with odd parts. 2

Preliminary Results

In this section, we collect the results which are useful in obtaining our main results. Lemma 2.1.

Lemma 2.2.

[3, Entry 17.3.1, p. 385] If β is of degree 2 over α, then √ √ √ (1 − 1 − α)(1 − β ) = 2 β (1 − α).

(2.1)

[4, Entry 5 (ii), p. 230] If β has degree 3 over α, then (αβ )1/4 + {(1 − α)(1 − β )}1/4 = 1.

(2.2)

230

M. S. M. Naika, K. S. Bairy, D. S. Gireesh, N. P. Suman

Lemma 2.3.

Lemma 2.4.

[3, Entry 17.3.2, p. 385] If β has degree 4 over α, then √ √ √ (1 − 4 1 − α)(1 − 4 β ) = 2 4 β (1 − α). [4, Entry 13 (i), p. 280] If β has degree 5 over α, then (αβ )1/2 + {(1 − α)(1 − β )}1/2 + 2{16αβ (1 − α)(1 − β )}1/6 = 1.

Lemma 2.5.

[4, Entry 3 (x), (xi), p. 352] If β has degree 9 over α, then ( ( )1/8 ( ) )1/8 √ 1 − β 1/8 β (1 − β ) β + − = m, α 1−α α(1 − α) ( )1/8 ( ) ( )1/8 α 1 − α 1/8 α(1 − α) 3 + − =√ . β 1−β β (1 − β ) m

Lemma 2.7.

(2.5)

(2.6) (2.7)

[4, Entry 7, p. 363] If β has degree 11 over α, then (αβ )1/4 + {(1 − α)(1 − β )}1/4 + 2{16αβ (1 − α)(1 − β )}1/12 = 1.

Lemma 2.8.

(2.4)

[4, Entry 19 (i), p. 314] If β has degree 7 over α, then (αβ )1/8 + {(1 − α)(1 − β )}1/8 = 1.

Lemma 2.6.

(2.3)

[5, pp. 387388] Let U1 = 1 − V1 = 64

√ √ αβ − (1 − α)(1 − β ),

( ) √ √ √ αβ + (1 − α)(1 − β ) − αβ (1 − α)(1 − β )

(2.8)

(2.9) (2.10)

and

√ W1 = 32 αβ (1 − α)(1 − β ),

(2.11)

then

√ √ U1 (U13 + 8W1 ) = W1 (11U12 + V1 ),

(2.12)

where β is of degree 13 over α. Lemma 2.9.

[4, Entry 13. (xv), p. 382] If M = (αβ )1/4 and N = (β/α)1/8 , then ( N−

Lemma 2.10.

We have

1 N

)3

( ) ( ) 1 1 +8 N − =4 M− . N M

(2.13)

χ(q ) = 21/6 (α(1 − α)/q )−1/24 ,

(2.14)

χ(−q ) = 21/6 (1 − α)1/12 (α/q )−1/24 ,

(2.15)

where α = k 2 , k is called the modulus of K . Proof.

For proofs of (2.14) and (2.15), see [4, Entry 12 (v), (vi), Ch.17, p. 124].

[4, Ch.18, Entry 24(v), p. 216] If we replace α by 1 − β , β by 1 − α, and m by n/m, where n is the degree of the modular equation, we obtain a modular equation of the same degree. Lemma 2.11.

231

Jacobi-Sohncke Type Mixed Modular Equations 3

Mixed Modular Equations

In this section, we derive several Jacobi-Sohncke type mixed modular equations for the degrees 1, 5, n and 5n for n = 2, 3, 4, 5, 7, 9, 11, and 13. Throughout this section, we set R :=

{

αγ βδ

}1/16

(3.1)

and T :=

{

αδ βγ

}1/16

(3.2)

.

We use the following notations (

Rκ := R + κ

and

(3.3)

Rκ

( ) 1 κ T κ := T + κ . T

Lemma 3.1.

α=

( γ=

1 where u = 4 1 4

Proof.

(3.4)

If α, β , γ and δ are of degrees 1, 5, n and 5n respectively, then (

v=

)

1

(( N−

((

N′ −

1 N′

)3

1 N

u+s 2N 2

v+t 2N ′ 2

)2 ,

(3.5)

,

(3.6)

)2

( )) 1 1/8 +8 N − , N = (β/α) , s2 = u2 + 4, N ( ))

)3

+ 8 N′ −

1

1/8

N′

, N ′ = (δ/γ )

, and t2 = v 2 + 4.

The proof of equations (3.5) and (3.6) directly follows from the equation (2.13).

Theorem 3.2.

If α, β , γ , and δ are of degrees 1, 5, 3, and 15 respectively, then R2 + 3T2 − 6 = T4 .

Proof.

From equations (3.1) and (3.2) we see that N =

1 RT

(3.7) and N ′ =

T . Now using equations R

232


(3.5) and (3.6) in the equation (2.2), we deduce R12 T 20 + 10R10 T 18 − 4R9 T 17 s + 8R9 T 13 mt + 15R8 T 16 − 2R12 T 10 m − 10R10 T 12 m − 12R9 T 13 t + 10R8 T 14 m − 20R7 T 15 s + 2R6 T 16 m − 2T 10 m

+ 15R10 T 12 − 15R8 T 14 − 3R6 T 16 + 40R7 T 11 mt − 20R6 T 14 − 10R10 T 8 m − 50R8 T 10 m − 60R7 T 11 t + 50R6 T 12 m − 32R6 T 10 mst + 20R5 T 13 s

+ 10R4 T 14 m + 15R10 T 8 + 8R9 T 7 ms + 75R8 T 10 + 40R7 T 9 ms − 75R6 T 12 + 48R6 T 10 st − 40R5 T 11 ms − 15R4 T 14 − 8R3 T 13 ms − 12R9 T 7 s + T 8 + 60R5 T 11 s + 12R3 T 13 s − 40R5 T 9 mt + 15R4 T 12 + 10R8 T 6 m + 50R6 T 8 m

(3.8)

+ 60R5 T 9 t − 50R4 T 10 m + 4R3 T 11 s − 10R2 T 12 m − 15R8 T 6 − 75R6 T 8 + 75R4 T 10 + 15R2 T 12 − 4R9 T 3 t − 20R7 T 5 t + 20R5 T 7 t + 4R3 T 9 t − 128R2 T 10 m + R12 + 10R10 T 2 + 15R8 T 4 − 20R6 T 6 + 15R4 T 8 + 3T 10

+ 20R2 T 10 + T 12 + 2R6 T 4 m + 10R4 T 6 m + 12R3 T 7 t − 10R2 T 8 m − 3R6 T 4 − 15R4 T 6 + 15R2 T 8 + 3R12 T 10 − 60R7 T 9 s − 8R3 T 7 mt = 0,

where m := (αβ )1/4 . Now eliminating m, s and t from the above eqaution (3.8), we deduce R4 T 4 − R2 T 8 + 3R2 T 6 − 6R2 T 4 + 3R2 T 2 + T 4 − R2 = 0.

(3.9)

By employing equations (3.3) and (3.4) in the above equation (3.9), we arrive at (3.7). If α, β , γ , and δ are of degrees 1, 5, 2, and 10 respectively, then ) ( )( ) ( ) ( 1 1 1 1 Y + − Y 2 + 2 − 2 = 0. 2 X2 + 2 + X2 − 2 X X Y Y

Theorem 3.3.

(3.10)

The proof of Theorem (3.3) is similar to the proof of Theorem (3.2), except that in the T place of equation (2.2); equation (2.1) is used and where we set X := T R and Y := instead R ( ) ( ) 1 1 κ κ of setting Rκ := R + κ and Tκ := T + κ . Proof.

R

T

If α, β , γ , and δ are of degrees 1, 5, 4, and 20 respectively, then ) {( ) ( )} ( ) 1 1 1 1 X4 − 4 Y 3 + 3 + 15 Y + + 4 X2 − 2 X Y Y X {( ) ( )} ( ){ ( ) } 1 1 1 1 3 4 2 Y + 3 −5 Y + −2 X + 4 3 Y + 2 + 10 Y Y X Y ( ) ( ) ( ) 1 1 1 + 32 X 2 + 2 − Y 4 + 4 + 16 Y 2 + 2 − 30 = 0. X Y Y

Theorem 3.4.

(

(3.11)

The proof of Theorem (3.4) is similar to the proof of Theorem (3.3), except that in the place of equation (2.1); equation (2.3) is used. Proof.

Theorem 3.5.

If α, β , γ , and δ are of degrees 1, 5, 5, and 25 respectively, then

R8 + R6 (5T2 + 14) − R4 (T6 + 5T4 + 10T2 + 4) + R2 (T6 − 10T2 − 6) − T6 + 15T2 + 42 = 0.

(3.12)


233

The proof of Theorem (3.5) is similar to that of Theorem (3.2), but rather than (2.2), we use (2.4). Proof.

Theorem 3.6.

If α, β , γ , and δ are of degrees 1, 5, 7, and 35 respectively, then T8 + 28T4 − 42T2 + 126 = R6 − 7R4 (4 − T2 ) + 7R2 (1 + 4T2 ).

(3.13)

The proof of Theorem (3.6) is similar to the proof of the Theorem (3.2), except that in the place of (2.2); equation (2.5) is used. Proof.

Theorem 3.7.


R12 − 8R10 − 8R8 + 18R6 + 693R4 + 432R2 − T8 − 66T6 + 330T4 − 1020T2 − 462R2 T2 + 141R4 T4 − R2 T8 − 54R2 T6 + 15R8 T2 − 12R6 T2

(3.14)

− 471R4 T2 − 9R6 T4 + 285R2 T4 − 9R4 T6 + 1566 = 0.

The proof of Theorem (3.7) is similar to that of Theorem (3.2), but rather than (2.2), we use (2.6) and (2.7). Proof.

Theorem 3.8.


R10 + 88R8 + 11 (T2 − 8) + R6 (22T4 + 264T2 + 737) − R4 (22T6 − 473T4 −1826T2 − 2838) + R2 (11T8 + 330T6 + 1958T4 + 1458T2 + 6522) − T12

(3.15)

− 11T10 − 99T8 − 671T6 − 2409T4 − 5852T2 − 8294 = 0.

The proof of Theorem (3.8) is similar to the proof of Theorem (3.2), except that in the place of (2.2); equation (2.8) is used. Proof.

Theorem 3.9.


T12 − 8T8 + 18T6 + 693T4 + 432T2 + 1566 = R8 (1 + T2 ) + 3R6 (22 + 18T2 + 3T4 ) − 3R4 (110 + 47T4 − 3T6 + 95T2 ) + R2 (1020 + 462T2 + 471T4 + 12T6 − 15T8 ).

(3.16)

The proof of Theorem (3.9) is similar to that of Theorem (3.2), but rather than (2.2), we use (2.12). Proof.

4

Modular identities for Ramanujan-Selberg continued fraction

In this section, we establish modular relations connecting Ramanujan-Selberg continued fractions H (q ), H (q 5 ), H (q n ), and H (q 5n ) for n = 2, 3, 4, 5, 7, 9, 11, and 13. We dene H (q )H (q n ) x := (4.1) 5 5n H (q )H (q )

and y :=

H (q )H (q 5n ) . H (q 5 )H (q n )

(4.2)

Throughout this section, we use the following notations

and

) ( 1 κ Xκ := x + κ x

(4.3)

) ( 1 Yκ := y κ + κ . y

(4.4)

234


Lemma 4.1.

[15, Th.3.2] We have

α1/8 H (q ) = √ ,

2

(4.5)

where q is as dened in (1.4) and α = k 2 , k is called the modulus of K .

Using the above lemma in the corresponding Jacobi-Sohncke type mixed modular equations obtained in the previous section, we deduce the following theorems. If α, β , γ and δ are of degrees 1, 5, 2, and 10 respectively, then √ ) ( ( ) ( ) (√ ) 1 1 x y x y 2 xy + + xy − + − + − 2 = 0. xy xy y x y x

Theorem 4.2.

Theorem 4.3.

(4.6)

If α, β , γ and δ are of degrees 1, 5, 3, and 15 respectively, then X1 + 3Y1 − 6 = Y2 .

If α, β , γ and δ are of degrees 1, 5, 4, and 20 respectively, then   √  √ ) ( ) √ 3 ) (√ ( 3 1 x y 1  x + y  + 15 x2 y 2 − 2 2 + + 32 xy + x y  y3 x3 y x  xy √  √  √ ) ( 2 ( ) (√ ) 1  x3 y3  x y x y2 + 4 xy − + −5 + − + 2 xy  y3 x3 y x  y2 x ( ){ ( ) } ( ) 1 x y x y 2 2 −2 x y + 2 2 3 + + 10 + 16 + − 30 = 0. x y y x y x

(4.7)

Theorem 4.4.

Theorem 4.5.

If α, β , γ and δ are of degrees 1, 5, 5, and 25 respectively, then

X4 + X3 (5Y1 + 14) − X2 (Y3 + 5Y2 + 10Y1 + 4) + X1 (Y3 − 10Y1 − 6) − Y3 + 15Y1 + 42 = 0. Theorem 4.6.

(4.9)

If α, β , γ and δ are of degrees 1, 5, 7, and 35 respectively, then Y4 + 28Y2 − 42Y1 + 126 = X3 − 7X2 (4 − Y1 ) + 7X1 (1 + 4Y1 ).

Theorem 4.7.

(4.8)

(4.10)


X6 − 8X5 − 8X4 + 18X3 + 693X2 + 432X1 − Y4 − 66Y3 + 330Y2 − 1020Y1 − 462X1 Y1 + 141X2 Y2 − X1 Y4 − 54X1 Y3 + 15X4 Y1 − 12X3 Y1 − 471X2 Y1

(4.11)

− 9X3 Y2 + 285X1 Y2 − 9X2 Y3 + 1566 = 0. Theorem 4.8.


X5 + X4 + 11 (Y1 − 8) + X3 (22Y2 + 264Y1 + 737) − X2 (22Y3 − 473Y2 −1826Y1 − 2838) + X1 (11Y4 + 330Y3 + 1958Y2 + 1458Y1 + 6522) − Y6

(4.12)

− 11Y5 − 99Y4 − 671Y3 − 2409Y2 − 5852Y1 − 8294 = 0. Theorem 4.9.


Y6 − 8Y4 + 18Y3 + 693Y2 + 432Y1 + 1566 = X4 (1 + Y1 ) + 3X3 (22 + 18Y1 + 3Y2 ) − 3X2 (110 + 47Y2 − 3Y3 + 95Y1 ) + X1 (1020 + 462Y1 + 471Y2 + 12Y3 − 15Y4 ).

(4.13)

Jacobi-Sohncke Type Mixed Modular Equations 5

235

Color partition identities for overpartition with odd parts

An overpartition of n is a non increasing sequence of natural numbers whose sum is n in which the rst occurrence of a number may be overlined. Let p(n) denote the number of overpartitions of an integer n. For convenience, we set p(0) = 1. For example, there are eight overpartitions of 3 are 3, 3, 2 + 1, 2 + 1, 2 + 1, 2 + 1, 1 + 1 + 1, 1 + 1 + 1. The generating function for p(n) is given by Corteel and Lovejoy [6] as ∞ ∑

p(n)q n =

n=0

(−q ; q )∞ . (q ; q )∞

(5.1)

Similarly, let P(n) be the number of overpartitions of n in which only odd parts are considered. ∞ ∑

P(n)q n =

n=0

(−q ; q 2 )∞ . (q ; q 2 )∞

(5.2)

Now from (2.14) and (2.15), we can write −1/8

(1 − α)

∞

=

∑ χ(q ) = P(n)q n . χ(−q )

(5.3)

n=0

By Lemma 2.11, (3.1) and (3.2) reduces to C := D :=

{

(1 − β )(1 − δ ) (1 − α)(1 − γ )

{

(1 − β )(1 − γ ) (1 − α)(1 − δ )

}1/16 ,

(5.4)

.

(5.5)

}1/16

Let l, m be positive integers with l ̸= m, we dene the following : ∞ ∑

Rpl,m (n)q n =

n=0

χp (q )χp (−q l )χp (q m )χp (−q lm ) , χp (−q )χp (q l )χp (−q m )χp (q lm )

(5.6)

where Rpl,m (n) denotes the number of overpartitions of n in 2p colors in which p colors appears only in odd parts that are not multiples of l, and another p colors appears only in odd parts that are multiples of m but are not multiples of lm. For l, m relatively prime, let Tpl,m (n) denotes the number of overpartitions of n into odd parts with p colors that are not multiples of l or m. The generating function for Tpl,m (n) is dened as, ∞ ∑ n=0 Theorem 5.1.

Proof.

Tpl,m (n)q n =

χp (q )χp (−q l )χp (−q m )χp (q lm ) . χp (−q )χp (q l )χp (q m )χp (−q lm )

(5.7)

We have R5,3 (2n) ≡ T25,3 (2n) (mod 3),

(5.8)

R45,7 (2n) ≡ T35,7 (2n) (mod 7),

(5.9)

R55,11 (2n) ≡ T65,11 (2n) (mod 11).

(5.10)

Using Lemma 2.11 in equation (3.7), we get

( ) ( ) 1 1 C 2 + 2 ≡ D4 + 4 C D

(mod 5).

(5.11)

236


Using (5.3)(5.5), (5.11) can be expressed as χ(q )χ(−q 5 )χ(−q 3 )χ(q 15 ) χ(−q )χ(q 5 )χ(q 3 )χ(−q 15 ) + ( χ(−q )χ(q 5 )χ(q 3 )χ(−q 15 ) χ q )χ(−q 5 )χ(−q 3 )χ(q 15 ) χ2 (q )χ2 (−q 5 )χ2 (q 3 )χ2 (−q 15 ) χ2 (−q )χ2 (q 5 )χ2 (−q 3 )χ2 (q 15 ) ≡ 2 + χ (−q )χ2 (q 5 )χ2 (−q 3 )χ2 (q 15 ) χ2 (q )χ2 (−q 5 )χ2 (q 3 )χ2 (−q 15 )

(5.12) (mod 3).

With the aid of (5.6) and (5.7), we arrive at (5.8). Proofs of (5.9) and (5.10) are similar to that of (5.8), but rather than (3.7), we use (3.13) and (3.15). References

[1] C. Adiga, M. S. Mahadeva Naika and Ramya Rao, Integral representations and some explicit evaluations of a continued fraction of Ramanujan, JP Jour. Algebra, Number Theory & Appl. 2 (1), 520 (2002). [2] G. E. Andrews, B. C. Berndt, L. Jacobsen and R. L. Lamphere, The continued fractions found in the unorganized portions of Ramanujan's notebooks, Memoir No. 477, Amer. Math. Soc. 99, Providence, Rhode Island, (1992). [3] G. E. Andrews and B. C. Berndt, Ramanujan's Lost Notebook, Part I, Springer, New York (2005). [4] B. C. Berndt, Ramanujan's Notebooks, Part III, Springer-Verlag, New York (1991). [5] B. C. Berndt, Ramanujan's Notebooks, Part V, Springer-Verlag, New York (1998). [6] S. Corteel, J. Lovejoy, Overpartitions, Trans. Amer. Math. Soc. 356, 16231635 (2004). [7] H. H. Chan and S. S. Huang, On the Ramanujan-Göllnitz-Gordon continued fraction, Ramanujan J. 1, 7590 (1997). [8] C. G. J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum, Sumptibus Fratrum Bornträger, Regiomonti (1829). [9] L. Jacobsen, Domains of validity of some Ramanujan's continued fraction formulas, J. Math. Anal. Appl. 143, 412437 (1989). [10] J. Landen, A disquisition concerning certain uents, which are assignable by the arcs of the conic sections; where in are investigated some new and useful theorems for computing such uents, Philos. Trans. R. Soc., London 61, 298309 (1771). [11] J. Landen, An investigation of a general theorem for nding the length of any arc of any conic hyperbola, by means of two elliptic arcs, with some other new and useful theorems deduced therefrom, Philos. Trans. R. Soc., London 65, 283289 (1775). [12] A. M. Legendre, Traite´ des fonctions elliptiques, t. 1, Huzard-Coureier, Paris (1825). [13] M. S. Mahadeva Naika, S. Chandankumar and M. Harish, Some new modular relations for Ramanujan's theta function, Indian J. Math. 55 (1), 4155 (2013). [14] M. S. Mahadeva Naika, S. Chandankumar and B. Hemanthkumar, On some new Modular relations for a remarkable product of theta-functions, Tbilisi Mathematical Journal 7 (1), 5568 (2014). [15] M. S. Mahadeva Naika, Remy Y Denis and K. Sushan Bairy, On some RamanujanSelberg continued fraction, Indian J. Math. 51 (3), 585596 (2009). [16] M. S. Mahadeva Naika, K. Sushan Bairy and M. Manjunatha, A continued fraction of order 4 found in Ramanujan's `lost' notebook, South East Asian J. Math. Math. Sci. 9 (3), 4363 (2011). [17] M. S. Mahadeva Naika, K. Sushan Bairy and C. Shivashankar, Jacobi-Sohncke type mixed modular equations and their applications, Note di Matematica 36 (1), 3754 (2016). [18] M. S. Mahadeva Naika, K. Sushan Bairy and N. P. Suman, Modular relations for a remarkable product of theta functions and evaluations, Recent Advances in Mathematics, RMS-Lecture Notes Series 21, 133145 (2015). [19] K. G. Ramanathan, Ramanujan's continued fraction, Indian J. Pure Appl. Math. 16, 695724 (1985). [20] S. Ramanujan, Notebooks, (2 volumes), Tata Institute of Fundamental Research, Bombay (1957).


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[21] A. Selberg, Uber einige arithmetische identitaten, Avh. Norske Vid.-Akad. Oslo I. Mat. -Naturv. Kl. 223 (1936). [22] L. A. Sohncke, Aequationes modulares pro transformatione functionum ellipticarum et undecimi et decimi terti et decimi septimi ordinis, J. Reine Angew. Math. 12, 178 (1834). [23] L. A. Sohncke, Aequationes modulares pro transformatione functionum ellipticarum, J. Reine Angew. Math. 16, 97130 (1837). Author information

M. S. Mahadeva Naika, Department of Mathematics, Bangalore University, Central College Campus, Bengaluru 560 001, INDIA. E-mail: [email protected] K. Sushan Bairy, PG Department of Mathematics, Vijaya College, R. V. Road, Basavanagudi, Bengaluru 560 004, INDIA. E-mail: [email protected] D. S. Gireesh, Department of Mathematics, Bangalore University, Central College Campus, Bengaluru 560 001, INDIA. E-mail: [email protected] N. P. Suman, Department of Mathematics, R. V. College of Engineering, R. V. Vidyaniketan Post, Mysore Road, Bengaluru 560 059, INDIA. E-mail: [email protected] Received:

May 3, 2016.

Accepted:

May 27, 2016.