Parameters for Ramanujan’s function $χ(q)$ of degree five and their explicit evaluation
We study the ratios of parameters for Ramanujan’s function $χ(q)$ and their explicit values.
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2017
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507557105238016 |
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| author | Dharmendra, B. N. Дхармендра, Б. Н. |
| author_facet | Dharmendra, B. N. Дхармендра, Б. Н. |
| author_sort | Dharmendra, B. N. |
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| datestamp_date | 2019-12-05T09:24:35Z |
| description | We study the ratios of parameters for Ramanujan’s function $χ(q)$ and their explicit values. |
| first_indexed | 2026-03-24T02:11:12Z |
| format | Article |
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UDC 512.5
B. N. Dharmendra (Maharani’s Sci. College for Women, India)
PARAMETERS FOR RAMANUJAN’S FUNCTION \bfitchi (\bfitq ) OF DEGREE FIVE
AND THEIR EXPLICIT EVALUATION
ПАРАМЕТРИ ФУНКЦIЇ РАМАНУДЖАНА \bfitchi (\bfitq ) П’ЯТОГО СТУПЕНЯ
ТА ЇХ ЯВНЕ ЗНАХОДЖЕННЯ
We study the ratios of parameters for Ramanujan’s function \chi (q) and their explicit values.
Вивчаються вiдношення параметрiв функцiї Рамануджана \chi (q) та їх явнi значення.
1. Introduction. In Chapter 16 of his second notebook [1], Ramanujan develops the theory of
Theta-function and is defined by
f(a, b) :=
\infty \sum
n= - \infty
a
n(n+1)
2 b
n(n - 1)
2 =
= ( - a; ab)\infty ( - b; ab)\infty (ab; ab)\infty , | ab| < 1, (1.1)
where (a; q)0 = 1 and (a; q)\infty = (1 - a)(1 - aq)(1 - aq2) . . . .
Following Ramanujan, we defined
\varphi (q) := f(q, q) =
\infty \sum
n= - \infty
qn
2
=
( - q; - q)\infty
(q; - q)\infty
,
\psi (q) := f(q, q3) =
\infty \sum
n=0
q
n(n+1)
2 =
(q2; q2)\infty
(q; q2)\infty
,
f( - q) := f( - q, - q2) =
\infty \sum
n= - \infty
( - 1)nq
n(3n - 1)
2 = (q; q)\infty
and
\chi (q) := ( - q; q2)\infty .
Now we define a modular equation in brief. The ordinary hypergeometric series 2F1(a, b; c;x) is
defined by
2F1(a, b; c;x) :=
\infty \sum
n=0
(a)n(b)n
(c)nn!
xn,
where (a)0 = 1, (a)n = a(a+ 1)(a+ 2) . . . (a+ n - 1) for any positive integer n, and | x| < 1.
c\bigcirc B. N. DHARMENDRA, 2017
520 ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 4
PARAMETERS FOR RAMANUJAN’S FUNCTION \chi (q) OF DEGREE FIVE . . . 521
Let
z := z(x) := 2F1
\biggl(
1
2
,
1
2
; 1;x
\biggr)
and
q := q(x) := \mathrm{e}\mathrm{x}\mathrm{p}
\left( - \pi
2F1
\biggl(
1
2
,
1
2
; 1; 1 - x
\biggr)
2F1
\biggl(
1
2
,
1
2
; 1;x
\biggr)
\right) ,
where 0 < x < 1.
Let r denote a fixed natural number and assume that the following relation holds:
r
2F1
\biggl(
1
2
,
1
2
; 1; 1 - \alpha
\biggr)
2F1
\biggl(
1
2
,
1
2
; 1;\alpha
\biggr) =
2F1
\biggl(
1
2
,
1
2
; 1; 1 - \beta
\biggr)
2F1
\biggl(
1
2
,
1
2
; 1;\beta
\biggr) . (1.2)
Then a modular equation of degree r in the classical theory is a relation between \alpha and \beta induced
by (1.2). We often say that \beta is of degree r over \alpha and m :=
z(\alpha )
z(\beta )
is called the multiplier. We also
use the notations z1 := z(\alpha ) and zr := z(\beta ) to indicate that \beta has degree r over \alpha .
The function \chi (q) is intimately connected to Ramanujan’s class invariants Gn an gn which are
defined by
Gn = 2 - 1/4q - 1/24\chi (q), gn = 2 - 1/4q - 1/24\chi ( - q),
where q = e - \pi
\surd
n and n is a positive rational number. Since from [1, p. 56] (Entry 12(v), (vi))
\chi (q) = 21/6
\bigl\{
\alpha (1 - \alpha )q - 1
\bigr\} - 1/24
,
\chi ( - q) = 21/6(1 - \alpha )1/12\alpha - 1/24q - 1/24.
Nipen Saikia [4] introduce the parameter Im,n which is defined as
Im,n :=
\chi (q)
q( - m+1)/24\chi (qm)
, q = e - \pi
\surd
n/m, (1.3)
where m and n are positive real numbers.
In Section 3, we study the modular relation between I5,n and I5,k2n, their explicit evaluations of
I5,n for n = 2, 3, 4, 5, 7 and 11.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 4
522 B. N. DHARMENDRA
2. Preliminary results.
Lemma 2.1 [3]. If P :=
\varphi (q)
\varphi (q5)
and Q :=
\varphi (q2)
\varphi (q10)
, then
\biggl(
P
Q
\biggr) 2
+
\biggl(
Q
P
\biggr) 2
+ (PQ)2 +
\biggl(
5
PQ
\biggr) 2
+ 16
\biggl(
P
Q
- Q
P
\biggr)
=
= 2
\biggl(
P 2 +
5
P 2
\biggr)
+ 2
\biggl(
Q2 +
5
Q2
\biggr)
+ 4. (2.1)
Lemma 2.2 ([2, p. 233], Ch. 25, Entry 66). If P =
\varphi (q)
\varphi (q5)
and Q =
\varphi (q3)
\varphi (q15)
, then
PQ+
5
PQ
= -
\biggl(
P
Q
\biggr) 2
+
\biggl(
Q
P
\biggr) 2
+ 3
\biggl(
P
Q
+
Q
P
\biggr)
. (2.2)
Lemma 2.3 [3]. If P :=
\varphi (q)\varphi (q4)
\varphi (q5)\varphi (q20)
and Q :=
\varphi (q)\varphi (q20)
\varphi (q5)\varphi (q4)
, then
Q4 +
1
Q4
- 112
\biggl(
Q3 +
1
Q3
\biggr)
+ 1440
\biggl(
Q2 +
1
Q2
\biggr)
- 3184
\biggl(
Q+
1
Q
\biggr)
+ 7316 =
= 8
\biggl(
P +
1
P
\biggr) \biggl[
22
\biggl(
Q2 +
1
Q2
\biggr)
- 31
\biggl(
Q+
1
Q
\biggr)
+ 170
\biggr]
- 2
\biggl(
P 2 +
52
P 2
\biggr)
\times
\times
\biggl[
3
\biggl(
Q2 +
1
Q2
\biggr)
+ 24
\biggl(
Q+
1
Q
\biggr)
+ 64
\biggr]
+ 4
\biggl(
P 3 +
53
P 3
\biggr) \biggl[ \biggl(
Q+
1
Q
\biggr)
+ 4
\biggr]
. (2.3)
Lemma 2.4 [3]. If P :=
\varphi (q)
\varphi (q5)
and Q :=
\varphi (q5)
\varphi (q25)
, then
Q3
P 3
- 5Q2
P 2
- 15Q
P
+ 5
\biggl(
PQ+
5
PQ
\biggr)
+ 5
\biggl(
Q2 +
5
P 2
\biggr)
=
= P 2Q2 +
52
P 2Q2
+ 15. (2.4)
Lemma 2.5 [3]. If P :=
\phi (q)\phi (q7)
\phi (q5)\phi (q35)
and Q :=
\phi (q)\phi (q35)
\phi (q5)\phi (q7)
, then
Q4 - 1
Q4
- 14
\biggl[ \biggl(
Q3 +
1
Q3
\biggr)
-
\biggl(
Q2 - 1
Q2
\biggr)
+ 10
\biggl(
Q+
1
Q
\biggr) \biggr]
+ P 3 +
53
P 3
=
= 7
\biggl\{ \biggl(
P 2 +
52
P 2
\biggr) \biggl(
Q+
1
Q
\biggr)
-
\biggl(
P +
5
P
\biggr) \biggl[
2
\biggl(
Q2 +
1
Q2
\biggr)
+ 9
\biggr] \biggr\}
. (2.5)
Lemma 2.6 [3]. If P =
\phi (q)\phi (q11)
\phi (q5)\phi (q55)
and Q =
\phi (q)\phi (q55)
\phi (q5)\phi (q11)
, then
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 4
PARAMETERS FOR RAMANUJAN’S FUNCTION \chi (q) OF DEGREE FIVE . . . 523
Q6 +
1
Q6
+ 33
\biggl(
Q5 +
1
Q5
\biggr)
- 99
\biggl(
Q4 +
1
Q4
\biggr)
+
+1529
\biggl(
Q3 +
1
Q3
\biggr)
- 1683
\biggl(
Q2 +
1
Q2
\biggr)
+ 8800
\biggl(
Q+
1
Q
\biggr)
=
= 6534 +
\biggl(
P 5 +
55
P 5
\biggr)
- 11
\biggl\{ \biggl(
P 4 +
54
P 4
\biggr) \biggl(
Q+
1
Q
\biggr)
-
\biggl(
P 3 +
53
P 3
\biggr) \biggl[
11 + 4
\biggl(
Q2 +
1
Q2
\biggr) \biggr]
-
-
\biggl(
P 2 +
52
P 2
\biggr) \biggl[
18 - 56
\biggl(
Q+
1
Q
\biggr)
+ 3
\biggl(
Q2 +
1
Q2
\biggr)
- 8
\biggl(
Q3 +
1
Q3
\biggr) \biggr]
-
-
\biggl(
P +
5
P
\biggr) \biggl[
324 - 126
\biggl(
Q+
1
Q
\biggr)
+ 160
\biggl(
Q2 +
1
Q2
\biggr)
- 18
\biggl(
Q3 +
1
Q3
\biggr)
+
+9
\biggl(
Q4 +
1
Q4
\biggr) \biggr]
-
\biggl(
P 3 +
53
P 3
\biggr) \biggl[
11 + 4
\biggl(
Q2 +
1
Q2
\biggr) \biggr] \biggr\}
. (2.6)
Lemma 2.7 [7, p. 56; 5].
f6( - q)
qf6( - q5)
=
\varphi 4( - q)
\varphi 4( - q5)
\biggl\{
5\varphi 2( - q5) - \varphi 2( - q)
\varphi 2( - q5) - \varphi 2( - q)
\biggr\}
. (2.7)
Lemma 2.8 ([1, p. 39], Ch. 16, Entry 24(iii)).
\chi (q) =
\varphi (q)
f(q)
. (2.8)
Lemma 2.9 [4]. We have
Im,1 = 1. (2.9)
Lemma 2.10 [4]. We get
Im,nIm,1/n = 1. (2.10)
3. General theorems and explicit evaluations of \bfitI \bfitm ,\bfitn .
Theorem 3.1. If P := q1/3
\chi (q)\chi (q2)
\chi (q5)\chi (q10)
and Q := q - 1/6\chi (q)\chi (q
10)
\chi (q5)\chi (q2)
, then
\biggl\{
Q3 +
1
Q3
\biggr\} \biggl[ \biggl\{
P 5 +
1
P 5
\biggr\}
+ 8
\biggl\{
P 3 +
1
P 3
\biggr\}
+ 19
\biggl\{
P +
1
P
\biggr\} \biggr]
+
\biggl\{
Q6 +
1
Q6
\biggr\}
=
=
\biggl\{
P 6 +
1
P 6
\biggr\}
+ 13
\biggl\{
P 4 +
1
P 4
\biggr\}
+ 52
\biggl\{
P 2 +
1
P 2
\biggr\}
+ 82.
Proof. Replace q by q5 in Lemma 2.8, we obtain
\chi (q5) =
\varphi (q5)
f(q5)
. (3.1)
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 4
524 B. N. DHARMENDRA
Dividing the equations (2.8) by (3.1), we get
\chi (q)
\chi (q5)
=
\varphi (q)
\varphi (q5)
f(q5)
f(q)
. (3.2)
Raising the power six and also multiplying q on both side of the equation (3.2), we have
q
\chi 6(q)
\chi 6(q5)
=
\varphi 6(q)
\varphi 6(q5)
\biggl\{
q
f6(q5)
f6(q)
\biggr\}
. (3.3)
Using the equations (2.7) and (3.3), we obtain
P 4 - P 2(1 - a) - 5a = 0, (3.4)
where P :=
\varphi (q)
\varphi (q5)
, a := q
\chi 6(q)
\chi 6(q5)
. Then the equation (3.4) can be written as
b2 - b(1 - a) - 5a = 0,
where b = P 2. Solve the above equation we get
b =
1 - a+
\surd
1 + 18a+ a2
2
. (3.5)
Using the equations (3.5) and (2.1), we have\bigl(
19x10y4 - 52x10y10 + x10y16 + x16y10 + x14y2 + 8x14y8 - x14y14 + x6+
+x2y14 + 19x4y10 - 13x4y4 + 8x2y8 - x2y2 + y6 - 52x6y6 + 19x6y12+
+19x12y6 - 13x12y12 + 8x8y2 - 82x8y8 + 8x8y14
\bigr) \bigl(
22x14y8 - 3x4y10 -
- 422x10y10 + x28y4 + x4y28 - x20y2 + x20y32 + x28y28 + x30y24 + x24y30+
+x32y20 - x30y12 - x2y20 - y30x12 - 3x10y4 - 25x10y16 - 25x16y10 -
- 8x14y2 - 2523x14y14 - 8x2y14 + x12 + y12 + x4y4 + x2y8 - 11x6y6+
+19x6y12 + 19x12y6 + 1358x12y12 + x8y2 + 101x8y8 + 22x8y14 -
- 3x28y22 + 16x28y10 + 43x28y16 + 3387x16y16 + 43x16y4 + 16x10y28 -
- 398x10y22 - 101x14y26 + 58x14y20 - 25x22y16 - 398x22y10 + 253x20y8+
+19x20y26 + 1358x20y20 + 58x20y14 + 253x8y20 - 19x8y26 - 25x16y22+
+43x16y28 + 16x22y4 - 3x22y28 - 422x22y22 + 43x4y16 + 16x4y22 -
- 101x26y14 - 11x26y26 - 19x26y8 + 19x26y20 - 8x30y18 - 8x18y30 -
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 4
PARAMETERS FOR RAMANUJAN’S FUNCTION \chi (q) OF DEGREE FIVE . . . 525
- 101x6y18 - 101x18y6 + 58x18y12 - 19x24y6 + 253x24y12 + 58y18x12 -
- 19y24x6 + 253y24x12 - 2523x18y18 + 22x18y24 + 22x24y18 + 101x24y24
\bigr)
= 0,
where x = q1/6
\chi (q)
\chi (q5)
and y = q1/3
\chi (q2)
\chi (q10)
. By examining the behavior of the above factors near
q = 0, we can find a neighborhood about the origin, where the first factor is zero, whereas other
factor are not zero in this neighborhood. By the Identity Theorem first factor vanishes identically.
Theorem 3.1 is proved.
Remark 3.1. Here by using the definition of (1.3), then above Theorem 3.1 is also can be written
as P = I5,nI5,4n and Q =
I5,n
I5,4n
.
Theorem 3.2. If P := q1/3
\chi (q)
\chi (q5)
and Q := q - 1/6 \chi (q
3)
\chi (q15)
, then
P 6
Q6
+
Q6
P 6
+ 18 = 9
\biggl\{
P 3
Q3
+
Q3
P 3
\biggr\}
+
\biggl\{
P 3Q3 +
1
P 3Q3
\biggr\}
.
Proof. Employing the equations (3.5) and (2.2), we obtain\bigl(
- 9Q9P 3 - Q3P 3 - Q9P 9 - 9Q3P 9 + 18P 6Q6 +Q12 + P 12
\bigr)
\times
\times
\bigl(
9Q9P 3 +Q3P 3 +Q9P 9 + 9Q3P 9 + 18P 6Q6 +Q12 + P 12
\bigr)
= 0.
By examining the behavior of the above factors near q = 0, we can find a neighborhood about
the origin, where the first factor is zero; whereas other factor are not zero in this neighborhood. By
the Identity Theorem first factor vanishes identically.
Theorem 3.2 is proved.
Remark 3.2. Here by using the definition of (1.3), then above Theorem 3.2 is also can be written
as P = I5,n and Q = I5,9n.
Remark 3.3. Im,n has positive real value less than 1 and that the values of Im,n decrease as n
increases when m > 1. Thus, by Lemma 2.9, Im,n < 1 for all n > 1 if m > 1.
Corollary 3.1. We have
I5,3 =
\Biggl[
7 - 3
\surd
5
2
\Biggr] 1/6
, (3.6)
I5,1/3 =
\Biggl[
7 + 3
\surd
5
2
\Biggr] 1/6
, (3.7)
I5,9 =
\Bigl[
4 -
\surd
15
\Bigr] 1/3
, (3.8)
I5,1/9 =
\Bigl[
4 +
\surd
15
\Bigr] 1/3
. (3.9)
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 4
526 B. N. DHARMENDRA
Proof. Setting n = 1/3 in Theorem 3.2 and using the Lemma 2.10, we obtain
I125,3 + I - 12
5,3 - 9
\bigl(
I65,3 + I - 6
5,3
\bigr)
+ 16 = 0.
Equivalently,
C2 - 9C + 14 = 0, (3.10)
where C = I65,3 + I - 6
5,3 .
Solving (3.10) and using the fact in Remark 3.3, we get
C = 7. (3.11)
Employing (3.10) and (3.11), solving the resulting equation for I5,3 and nothing that I5,3 < 1,
we arrive (3.6).
Again setting n = 1 in Theorem 3.2 and using Lemma 2.9, we obtain
I65,9 + I - 6
5,9 - 10
\bigl(
I35,9 + I - 3
5,9
\bigr)
+ 18 = 0.
Equivalently,
D2 - 10D + 16 = 0, (3.12)
where D = I35,9 + I - 3
5,9 .
Solving (3.12) and using the fact in Remark 3.3, we have
D = 8. (3.13)
Employing (3.12) and (3.13), solving the resulting equation for I5,9 and nothing that I5,9 < 1,
we arrive (3.8).
Theorem 3.3. If P := q5/6
\chi (q)\chi (q4)
\chi (q5)\chi (q20)
and Q := q - 1/2\chi (q)\chi (q
20)
\chi (q5)\chi (q4)
, then
\biggl\{
Q14 +
1
Q14
\biggr\}
- 56
\biggl\{
Q12 +
1
Q12
\biggr\}
+ 861
\biggl\{
Q10 +
1
Q10
\biggr\}
- 5824
\biggl\{
Q8 +
1
Q8
\biggr\}
+
+22524
\biggl\{
Q6 +
1
Q6
\biggr\}
- 59015
\biggl\{
Q4 +
1
Q4
\biggr\}
+ 102884
\biggl\{
Q2 +
1
Q2
\biggr\}
-
- 224
\biggl\{
P 6Q6 +
1
P 6Q6
\biggr\}
+ 1800
\biggl\{
P 3Q3 +
1
P 3Q3
\biggr\}
-
- 224
\biggl\{
P 6
Q6
+
Q6
P 6
\biggr\}
+ 1800
\biggl\{
P 3
Q3
+
Q3
P 3
\biggr\}
=
=
\biggl\{
P 12 +
1
P 12
\biggr\} \biggl\{
Q2 +
1
Q2
\biggr\}
-
\biggl\{
P 9 +
1
P 9
\biggr\} \biggl[ \biggl\{
Q7 +
1
Q7
\biggr\}
-
- 8
\biggl\{
Q5 +
1
Q5
\biggr\}
- 3
\biggl\{
Q3 +
1
Q3
\biggr\}
- 16
\biggl\{
Q+
1
Q
\biggr\} \biggr]
-
\biggl\{
P 6 +
1
P 6
\biggr\}
\times
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 4
PARAMETERS FOR RAMANUJAN’S FUNCTION \chi (q) OF DEGREE FIVE . . . 527
\times
\biggl[
24
\biggl\{
Q8 +
1
Q8
\biggr\}
+ 628
\biggl\{
Q4 +
1
Q4
\biggr\}
- 1357
\biggl\{
Q2 +
1
Q2
\biggr\}
1196
\biggr]
+
+
\biggl\{
P 3 +
1
P 3
\biggr\} \biggl[
16
\biggl\{
Q11 +
1
Q11
\biggr\}
- 109
\biggl\{
Q9 +
1
Q9
\biggr\}
+ 336
\biggl\{
Q7 +
1
Q7
\biggr\}
-
- 1261
\biggl\{
Q5 +
1
Q5
\biggr\}
- 2228
\biggl\{
Q+
1
Q
\biggr\} \biggr]
+ 127634. (3.14)
Proof. Employing the equations (3.5) and (2.3), we obtain (3.14).
Remark 3.4. Here by using the definition of (1.3), then above Theorem 3.3 is also can be written
as P = I5,nI5,16n and Q =
I5,n
I5,16n
.
Corollary 3.2. We have
I5,4 =
(11 + 5
\surd
5)1/4 -
\sqrt{} \sqrt{}
11 + 5
\surd
5 - 4
2
, (3.15)
I5,1/4 =
(11 + 5
\surd
5)1/4 +
\sqrt{} \sqrt{}
11 + 5
\surd
5 - 4
2
. (3.16)
Proof. Employing Theorem 3.3 and Lemma 2.10, solving the resulting equation for I5,4 and
nothing that I5,4 < 1, we arrive (3.15).
Theorem 3.4. If P := q
\chi (q)\chi (q5)
\chi (q5)\chi (q25)
and Q := q - 2/3\chi (q)\chi (q
25)
\chi 2(q5)
, then
\biggl\{
Q3 +
1
Q3
\biggr\} \biggl[ \biggl\{
P 2 +
1
P 2
\biggr\}
+
\biggl\{
P +
1
P
\biggr\}
+ 1
\biggr]
=
=
\biggl\{
P 4 +
1
P 4
\biggr\}
+ 6
\biggl\{
P 3 +
1
P 3
\biggr\}
+ 11
\biggl\{
P 2 +
1
P 2
\biggr\}
+ 16
\biggl\{
P +
1
P
\biggr\}
+ 22. (3.17)
Proof. Employing the equations (3.5) and (2.4), we obtain (3.17).
Remark 3.5. Here by using the definition of (1.3), then above Theorem 3.4 is also can be written
as P = I5,nI5,25n and Q =
I5,n
I5,25n
.
Corollary 3.3. We have
I5,5 =
\Bigl[
9 - 4
\surd
5
\Bigr] 1/3
,
I5,1/5 =
\Bigl[
9 + 4
\surd
5
\Bigr] 1/3
,
I5,25 =
a -
\surd
a2 - 4
2
,
I5,1/25 =
a+
\surd
a2 - 4
2
,
(3.18)
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 4
528 B. N. DHARMENDRA
where
a =
b4 + xb3 + yb2 + z
5b3
, b =
\left\{ 125
\Bigl[
250 + 110
\surd
5 - (5 + 3
\surd
5)
\sqrt{}
10 + 2
\surd
5
\Bigr]
4
\right\}
1/5
,
x = 5(5 + 2
\surd
5), y =
25
\Bigl[
5 +
\surd
5 +
\sqrt{}
10 + 2
\surd
5
\Bigr]
2
,
and
z =
125
\Bigl[
12 + 4
\surd
5 - (
\surd
5 + 1)
\sqrt{}
10 + 2
\surd
5
\Bigr]
4
.
Proof. Employing Theorem 3.4, Lemmas 2.9 and 2.10, solving the resulting equation for I5,5,
I5,25 and nothing that I5,5 < 1 and I5,25 < 1, we arrive (3.18).
Theorem 3.5. If P := q4/3
\chi (q)
\chi (q5)
and Q := q - 1 \chi (q
7)
\chi (q35)
, then
P 4
Q4
+
Q4
P 4
- 7
\biggl\{
P 3
Q3
+
Q3
P 3
\biggr\}
+ 21
\biggl\{
P 2
Q2
+
Q2
P 2
\biggr\}
- 42
\biggl\{
P
Q
+
Q
P
\biggr\}
+ 56 =
= P 3Q3 +
1
P 3Q3
. (3.19)
Proof. Employing the equations (3.5) and (2.5), we obtain (3.19).
Remark 3.6. Here by using the definition of (1.3), then above Theorem 3.5 is also can be written
as P = I5,n and Q = I5,49n.
Corollary 3.4. We have
I5,7 =
a -
\surd
a2 - 4
2
,
I5,1/7 =
a+
\surd
a2 - 4
2
,
(3.20)
where a =
(71 - 3
\surd
105)
\bigl[
16x2 + (x - 1)(71 + 3
\surd
105)
\bigr]
12288
and x = [71 + 3
\surd
105]1/3,
I5,49 =
b -
\surd
b2 - 4
2
,
I5,1/49 =
b+
\surd
b2 - 4
2
,
(3.21)
where b =
(9 - 2
\surd
15)
\bigl[
y2 + (y + 6)(9 + 2
\surd
15)
\bigr]
16
and y = [189 + 42
\surd
15]1/3.
Proof. Employing Theorem 3.5, Lemmas 2.9 and 2.10, solving the resulting equation for I5,7,
I5,49 and nothing that I5,7 < 1 and I5,49 < 1, we arrive (3.20), (3.21).
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 4
PARAMETERS FOR RAMANUJAN’S FUNCTION \chi (q) OF DEGREE FIVE . . . 529
Theorem 3.6. If P := q2
\chi (q)\chi (q11)
\chi (q5)\chi (q55)
and Q := q - 5/3 \chi (q)\chi (q
55)
\chi (q5)\chi (q11)
, then
\biggl\{
P 5 +
1
P 5
\biggr\}
- 11
\biggl\{
P 4 +
1
P 4
\biggr\}
+ 11
\biggl\{
P 3 +
1
P 3
\biggr\}
+
+88
\biggl\{
P 2 +
1
P 2
\biggr\}
+ 33
\biggl\{
P +
1
P
\biggr\}
=
=
\biggl\{
Q6 +
1
Q6
\biggr\}
- 11
\biggl\{
Q3 +
1
Q3
\biggr\} \biggl[ \biggl\{
P 2 +
1
P 2
\biggr\}
+ 2
\biggl\{
P +
1
P
\biggr\}
+ 3
\biggr]
+ 165. (3.22)
Proof. Employing the equations (3.5) and (2.6), we obtain (3.22).
Remark 3.7. Here by using the definition of (1.3), then above Theorem 3.6 is also can be written
as P = I5,nI5,121n and Q =
I5,n
I5,121n
.
Corollary 3.5. We have
I5,11 =
\left[ 9(11 + 5
\surd
5) -
\sqrt{}
110(181 + 81
\surd
5)
4
\right] 1/6
, (3.23)
I5,1/11 =
\left[ 9(11 + 5
\surd
5) +
\sqrt{}
110(181 + 81
\surd
5)
4
\right] 1/6
. (3.24)
Proof. Employing Theorem 3.6 and Lemma 2.10, solving the resulting equation for I5,11 and
nothing that I5,11 < 1, we arrive (3.23).
References
1. Berndt B. C. Ramanujan’s notebooks, Pt III. – New York: Springer-Verlag, 1991.
2. Berndt B. C. Ramanujan’s notebooks, Pt IV. – New York: Springer-Verlag, 1994.
3. Mahadeva Naika M. S., Dharmendra B. N., Chandankumar S. New modular relations for Ramanujan parameter
\mu (q) // IJPAM. – 2012. – 74, № 4. – P. 413 – 435.
4. Nipen Saikia. A parameter for Ramanujan’s function \chi (q): its explicit values and applications // ISRN Comput.
Math. – 2012. – 2012. – Article ID 169050. – 14 p. http://dx.doi.org/10.5402/2012/169050.
5. Kang S.-Y. Some theorems on the Rogers – Ramanujan continued fraction and associated theta function identities in
Ramanujan’s lost notebook // Ramanujan J. – 1999. – 3, №. 1. – P. 91 – 111.
6. Ramanujan S. Notebooks. – Bombay: Tata Inst. Fundam. Res., 1957. – Vols 1, 2.
7. Ramanujan S. The ‘lost’ notebook and other unpublished papers. – New Delhi: Narosa, 1988.
Received 02.02.13,
after revision — 24.02.17
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 4
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| id | umjimathkievua-article-1713 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:11:12Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/fa/42909d7f7332805f08127b57b829defa.pdf |
| spelling | umjimathkievua-article-17132019-12-05T09:24:35Z Parameters for Ramanujan’s function $χ(q)$ of degree five and their explicit evaluation Параметри функцiї рамануджана $χ(q)$ п’ятого ступеня та їх явне знаходження Dharmendra, B. N. Дхармендра, Б. Н. We study the ratios of parameters for Ramanujan’s function $χ(q)$ and their explicit values. Вивчаються вiдношення параметрiв функцiї Рамануджана $χ(q)$ та їх явнi значення. Institute of Mathematics, NAS of Ukraine 2017-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1713 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 4 (2017); 520-529 Український математичний журнал; Том 69 № 4 (2017); 520-529 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1713/695 Copyright (c) 2017 Dharmendra B. N. |
| spellingShingle | Dharmendra, B. N. Дхармендра, Б. Н. Parameters for Ramanujan’s function $χ(q)$ of degree five and their explicit evaluation |
| title | Parameters for Ramanujan’s function $χ(q)$ of degree five and their explicit
evaluation |
| title_alt | Параметри функцiї рамануджана $χ(q)$ п’ятого ступеня та їх явне знаходження |
| title_full | Parameters for Ramanujan’s function $χ(q)$ of degree five and their explicit
evaluation |
| title_fullStr | Parameters for Ramanujan’s function $χ(q)$ of degree five and their explicit
evaluation |
| title_full_unstemmed | Parameters for Ramanujan’s function $χ(q)$ of degree five and their explicit
evaluation |
| title_short | Parameters for Ramanujan’s function $χ(q)$ of degree five and their explicit
evaluation |
| title_sort | parameters for ramanujan’s function $χ(q)$ of degree five and their explicit
evaluation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1713 |
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