A New Parameter For Ramanujan’s Function χ(q) Of Degree 3 And Their New Explicit Evaluations.

A New Parameter For Ramanujan’s Function

𝜒(𝑞)

Of Degree 3 And Their New Explicit

Evaluations.

S. Vasanth Kumar and M.R. Rajesh kanna

Abstract: In this article, We will come to know new modular identities of Ramanujan’s Remarkable product of theta-function

of degree 3 and their explicit values

Index Terms: Theta-function, Cubic continued fraction. 2000 Mathematics Subject Classification. Primary 33D10, 40A15,

11A55, 30B70.

————————————————————

1 INTRODUCTION

amanujan defined theta-function as

[2] f(a b) ∑n an(n )bn(n ) |ab|

a ab b ab ab ab (1.1)

Where(a q) ∏n aqn q .

[2] (q) f (q q) ∑ qn ( q q) (q q)

n (1.2)

[2] (q) f (q q ) ∑ qn n (q q )

(q q )

n (1.3)

[2] f( q) f ( q q ) ∑ qn n- _{(q q)}

n - (1.4)

and

[2, Ch.16, Entry 22(iv), p.37] q q q (1.5)

A new parameter I_m,n introduced by Nipen Saikia, which is

defined as

[6] m n (q) q

( m)

(qm₎

q e √n m⁄ _{. (1.6)}

He has listed as many as properties of Im,n. Recently we

noticed in [3, 4] have derived some new parameters for

Ramanujan’s function q of degree 5, degree 9 and their

explicit values.

In this article, we add as many as Modular identities and

explicit evaluation of I_3,n for n = 2, 3, 5, 7.

We now first give brief explanation on a modular equation. The

ordinary hyper-geometric series is defined by

F (a b c x) ∑ a n b n

c _n n xn

n

Where, (a)0=1,(a)n=a(a+1)(a+2)···(a+n−1) for any positive

integer n, and |x|<1.

Let u u(x) F ( x) (1.7)

[2] q q(x) exp( F( x)

F( x)

) (1.8)

Where 0 < x < 1. R

————————————————

 Rajesh Kanna. M. R , Assistant Professor Dept of Mathematics, Sri D D Urs Government First Grade

College, Hunsur, India, PH-09448150508.

E-mail: [email protected]

 Vasanth Kumar. S Research scholar, Dept of Mathematics, Bharathiar University, Coimbatore- 641046, India / Dept of

Mathematics, Mysuru Royal Institute of Technology S.R.

Patna - 571606, PH-09945002174.

Let r to be a fixed natural number and holds good the

following relation:

r F( )

F( )

F ( )

F( )

(1.9)

The relation between and of modular equation of degree r

is induced by (1.9).

In the last section of my article, we have establish the modular

relation between I3,n and I3,k2n, also given explicit evaluations

of I_3,n for n = 2,3,5 and 7.

2PRELIMINARY RESULTS

Lemma 2.1. [1] f P _q ⁄ -q _q and Q

q

( ) then

Q - P Q - . (2.1)

Lemma 2.2. [7] f P (q)

(q) and Q (q ) (q ) then

Q Q P Q Q P Q Q .

(2.2)

Lemma 2.3. f P (q)

(q) and Q (q ) (q ) then

P Q P-Q - P Q . (2.3)

Lemma 2.4. f P (q)

(q) and Q (q ) (q ₎ then

P Q - Q P P Q - Q P Q - Q P - Q

Lemma 2.5. f P (q)

(q) and Q (q ) (q ₎ then

P Q Q P P Q Q Q P

Q - Q P - P Q Q- Q P-Q (2.5)

Lemma 2.6.

[2] q q

f q (2.6)

Lemma 2.7.

[2] f (q) (q) ( q) (2.7)

Lemma 2.8.

f (q ) (q ) ( q ) (2.8)

Lemma 2.9. [6]

Im,1 =1. (2.9)

Lemma 2.10. [6]

Im,nIm,1/n= 1. (2.10)

3MODULAR IDENTITIES AND EXPLICIT EVALUATIONS OF I_M,N

Theorem 3.1. If v q ⁄ q

q and p q

⁄ q

q then

p20v32+(16p16−p28+p4)v28+(71p12−25p24)v24

+(71p8−200p20+p32)v20+(16p4+16p28−550p16)v16

+(71p24−200p12+1)v12+(71p20−25p8)v8

+(p28+16p16−p4)v4+p12=0. (3.1)

Proof. Substitute q by q3 in (2.6), we arrive

q q

f q (3.2)

Cubing and dividing (2.6) by (3.2), we get

(q) (q)

(q) (q){

f (q)

f (q)} (3.3)

Using (2.7) and (2.8) in (3.3), we obtain

(q) (q)

(q) (q ){

( q)

( q)} (3.4)

Raising by four and multiplying by q on both side of (3.4),

we arrive at

q _{(q) (q )}

(q) (q){q

( q )

( q)} (3.5)

By using lemma (2.1), we obtain

P Q

Q (3.6)

Using the equation (3.6) in (3.5), we obtain

v Q Q

Q (3.7)

Where, Q (q)

(q) and v q

_q

_q , the equation (3.7) can

be expressed as

Q ( v) √v v (3.8)

3630

Adopting the equation (3.8) in (2.2), we obtain (3.1).

This completes the proof.

Corollary 3.1. We have

{- √ √ - √ } ⁄

, (3.9)

⁄ { √ √ √ } ⁄

. (3.10)

{ √ √ √ } ⁄

, (3.11)

⁄ { √ √ √ } ⁄

. (3.12)

Proof. Considering the Theorem (3.1), substituting n =1/3 to it

and with the Lemma (2.10), we obtain

- . (3.13)

By solving (3.13), we arrive at (3.9) and (3.10) equations.

Substituting n in (3.1) and using (2.9), we get

{ ⁄ ⁄ ⁄ ⁄ }

{ ⁄ ⁄ }

(3.14)

Since one of the factor vanishes identically where as another

factor does not satisfies the condition I3,4<1, so (3.14) can be

written as

x x- where x - . (3.15)

By solving the (3.15), we arrive at (3.11) and (3.12) equations.

Theorem 3.2. If v q ⁄ q

q and p q ⁄ q

q then

p6v18+(p3−p15)v15 −10p12)v12−20v9p9

+(p18−10p6) v6 +(p15−p3)v3+p12 = 0. (3.16)

Proof. Adopting the equation (3.8) in (2.3), we obtain,

(v15p3−v3p3+p18v6−20v9p9+p6v18−10p6v6−v15p15−

10v12p12+v12+p12+v3p15)(v3p3−v15p3+p18v6+20v9p9

+p6v18+p12−10p6v6+v15p15−10v12p12+v12−v3p15)

(v36p12+p30v30+18p18v30−p6v30+62v24p24+v24+18

p30v18+200p18v18+18p6v18+p36v12+62v12p12−p30v6

+18p18v6 +p6v6+p24) = 0. (3.17)

Since q→ , one of the factors vanishes of the (3.17), but the

remaining factor does not disappear. So we come at the

equation (3.16). Thus it concludes the proof.

Corollary 3.2. We have

{ -√ } ⁄

, (3.18)

⁄ { √ } ⁄

. (3.19)

{ ⁄ } ⁄

, (3.20)

⁄ { ⁄ ⁄ } ⁄

. (3.21)

Proof. Considering Theorem (3.2), Lemma (2.10) and (2.9), we

arrive (3.18) - (3.21).

Theorem 3.3. If v q ⁄ q

q and p q ⁄ q

q then

v6−v5p5−vp−5v3p3+p6=0. (3.22)

Proof. Adopting the equation (3.8) in (2.4), we obtain

(−v5p5−vp+v6−5v3p3+p6)(v5p5+vp+v6+5v3p3+p6)(−5

p4v4−v7p−v11p5+v2p2−vp7+26p6v6+v12+p12+10v3p9−

5p8v8+p10v10+10v9p3−v5p11)(−5p4v4+v7p+v11p5+v2

p2+vp7+26p6v6+v12+p12−10v3p9−5p8v8+p10v10−10v9

p3+v5p11)(−10p4v4+v2p2+25p6v6+v12+p12−10p8v8+

p10v10)(v24−p10v22+p20v20+10p8v20+10p18v18+50p6

v18+75p16v16+10v16p4+230p14v14−v14p2+526v12p12−p2

2v10+230p10v10+10p20v8+75p8v8+50p18v6+10p6v6+10

p16v4+p4v4−p14v2+p24)(68884p36−85p48n+102p60

+3903p48+550383p24−p60n+p72 −183461p12n−

269297n+1615782p12−34442p24n−2602p36n) = 0. (3.23)

Where n √_{b b .}

Since q→ , one of the factors vanishes of the (3.23), but the

remaining factor does not disappear. So we come at the

equation (3.22). Thus it concludes the proof.

Corollary 3.3. We have,

{ √ } ⁄

{ √ } ⁄

(3.25)

⁄ ⁄ _{√ (} ⁄_{) (} ⁄₎

, (3.26)

⁄ ⁄ √ ( ⁄_{) (} ⁄₎

. (3.27)

Proof. Considering Theorem (3.3), Lemma (2.10) and (2.9), we

obtain (3.24 )- (3.27).

Theorem 3.4. If v q ⁄ q

q and p q ⁄ q

q then

v8−v7p7−vp−7v6p2−7p6v2+14v4p4+p8=0. (3.28)

Proof. Adopting the equation (3.8) in (2.5), we obtain,

(v7p7+vp+v8−7v6p2−7p6v2+14v4p4+p8)(−v7p7−vp

+v8−7v6p2−7p6v2+14v4p4+p8)(84p10v6+28v5p5−vp9

+v16+7v9p13+148p8v8+v2p2−v15p7+7v14p2+84p6v10

+35p12v4+7p14v2+p14v14−v9p+7v13p9+28v11p11−v7

p15+p16+35v12p4+7v3p7+7v7p3)(84p10v6−28v5p5+v

p9+v16−7v9p13+148p8v8+v2p2+v15p7+7v14p2+84p6

v10+35p12v4+7p14v2+p14v14+v9p−7v13p9−28v11p11

+v7p15+p16+35v12p4−7v3p7−7v7p3)(210p10v6+v16

+294p8v8+v2p2+14v14p2+210p6v10+77p12v4+14p14v2

+p14v14+p16+77v12p4) −658p26v6−77p26v18−77p18v26

−30492p18v14−p18v2+588p10v10+p28v28+28p28v16−210

p20v24+18676p20v12+119p28v4+2695p8v24−210p8v12−

v18p2+v32+18676p12v20−210p24v20+2695p24v8+p32

−210p12v8−14v30p2−658v26p6−p14v30+36800p16v16+28

p16v28−77p14v6−30492p14v18−14p30v2−p30v14−77v14

p6+28v16p4+119v28p4+v4p4−8148p10v22+588p22v22

+28p16v4−8148p22v10)(128039644p24+3453190p48

+59823937+210156040p12+29964472p36+136p84+7516

p72+p96+216376p60−26269505n−5637p60n−p84n−1123

6677np24−119np72−135235np48−32009911p12n−1726595

p36n) = 0. (3.29)

Where n √b b

Since q→ , one of the factors vanishes of the (3.29), but the

remaining factor does not disappear. So we come at the

equation (3.28). Thus it concludes the proof.

Corollary 3.4.We have,

√ √ √

(3.30)

√ √ √ (3.31)

( ⁄₎ ⁄ _{√ (} ⁄_{) (} ⁄₎

(3.32)

( ⁄₎ ⁄ _{√ (} ⁄_{) (} ⁄₎

(3.33)

Proof. Considering Theorem (3.4), Lemma (2.10) and (2.9), we

obtain (3.30) - (3.33).

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