TWO NEW RELATIONS BETWEEN Q-PRODUCT IDENTITIES, THETA

Honam Mathematical J. 39 (2017), No. 3, pp. 379–399 https://doi.org/10.5831/HMJ.2017.39.3.379

TWO NEW RELATIONS BETWEEN Q-PRODUCT IDENTITIES, THETA FUNCTION IDENTITIES AND COMBINATORIAL PARTITION IDENTITIES M. P. Chaudhary∗ , Ahmed Buseri Ashine, and Feyissa Kaba Wakene

Abstract. The objective of this research article is to establish two relationships between q-product identities, theta function identities and combinatorial partition identities, using elementary results.

1. Introduction Throughout this paper, N, Z, and C denote the sets of positive integers, integers, and complex numbers, respectively, and N0 := N ∪ {0}. The following q-notations are recalled (see, e.g., [4, Chapter 6]): The q-shifted factorial (a; q)n is defined by 1 (n=0) (1.1) (a; q)n := Qn−1 k ) (n ∈ N), (1 − aq k=0 where a, q ∈ C and it is assumed that a 6= q −m (m ∈ N0 ). We also write (a; q)∞ :=

∞ Y

(1 − aq k )

k=0

(1.2)

=

Q∞

k=1 (1

− aq k−1 ) (a, q ∈ C; |q| < 1).

It is noted that, when a 6= 0 and |q| = 1, the infinite product in (1.2) diverges. So, whenever (a; q)∞ is involved in a given formula, the constraint |q| < 1 will be tacitly assumed. The following notations are also Received March 30, 2017. Accepted July 25, 2017. 2010 Mathematics Subject Classification. 5A30, 11F27, 11P83 Key words and phrases. q-product identities, combinatorial partition identities, theta function identities and Jacob’s triple product identities. Research funding can be written here. *Corresponding author

M.P. Chaudhary∗ Ahmed Buseri Ashine and Feyissa Kaba Wakene

380

frequently used: (a1 , a2 , · · · , am ; q)n := (a1 ; q)n (a2 ; q)n · · · (am ; q)n

(1.3) and

(a1 , a2 , · · · , am ; q)∞ := (a1 ; q)∞ (a2 ; q)∞ · · · (am ; q)∞ .

(1.4)

Ramanujan defined the general theta function f(a, b) as follows (see, for details, [3, p. 31, Eq.(18.1)] and [5]; see also [1]): (1.5)

n(n−1) P 2 (an + bn ) f (a, b) = 1 + ∞ n=1 (ab) ∞ X n(n+1) n(n−1) = a 2 b 2 = f (b, a) (|ab| < 1).

n=−∞

We find from (1.5) that n(n+1)

n(n−1)

(1.6) f (a, b) = a 2 b 2 f (a(ab)n , b(ab)−n ) = f (b, a) (n ∈ Z). Ramanujan also rediscovered the Jacobi’s famous triple-product identity (see [3, p. 35, Entry 19]): (1.7) f (a, b) = (−a; ab)∞ (−b; ab)∞ (ab; ab)∞ , which was first proved by Gauss. Several q-series identities emerging from Jacobi’s triple-product identity (1.7) are worthy of note here (see [3, pp. 36-37, Entry 22]): (1.8)

φ(q) :=

P∞

n=−∞ q

n2

=1+2

P∞

= {(−q; q 2 )∞ }2 (q 2 ; q 2 )∞ = ψ(q) := f (q, q 3 ) =

(1.9) (1.10)

P∞

n=0 q

n=1 q

n2

(−q; q 2 )∞ (q 2 ; q 2 )∞ ; (q; q 2 )∞ (−q 2 ; q 2 )∞

n(n+1) 2

=

(q 2 ;q 2 )∞ ; (q;q 2 )∞

n(3n−1) P n 2 f (−q) := f (−q, −q 2 ) = ∞ n=−∞ (−1) q ∞ ∞ X X n(3n−1) n(3n+1) = (−1)n q 2 + (−1)n q 2 = (q; q)∞

n=0

n=1

Equation (1.10) is known as Euler’s Pentagonal Number Theorem. The following q-series identity: (1.11)

(−q; q)∞ =

1 (q;q 2 )∞

=

1 χ(−q)

provides the analytic equivalence of Euler’s famous theorem: The number of partitions of a positive integer n into distinct parts is equal to the

Two new relations between q-product identities

381

number of partitions of n into odd parts. We also recall the Rogers-Ramanujan continued fraction of R(q): 4

1

1

5

1

4

5

f (−q,−q ) (q;q )∞ (q ;q )∞ 5 5 R(q) := q 5 H(q) G(q) = q f (−q 2 ,−q 3 ) = q (q 2 ;q 5 )∞ (q 3 ;q 5 )∞

(1.12)

1

q 5 q q2 q3 · · · (|q| < 1). 1+ 1+ 1+ 1+ Here G(q) and H(q) are widely investigated Roger-Ramanujan identities defined by =

(1.13)

G(q) := =

(1.14) =

2

qn n=0 (q;q)n

P∞

=

1

=

(q; q 5 )∞ (q4; q 5 )∞

H(q) :=

1 = (q 2 ; q 5 )∞ (q 3 ; q 5 )∞

(q; q)∞

f (−q 5 ) f (−q 2 ,−q 3 ) (q; q 5 )∞ (q 4 ; q 5 )∞ (q 5 ; q 5 )∞

q n(n+1) n=0 (q;q)n

P∞

f (−q 5 ) f (−q,−q 4 ) (q 2 ; q 5 )∞ (q 3 ; q 5 )∞ (q 5 ; q 5 )∞

;

=

(q; q)∞

;

and the functions f(a, b) and f(-q) are given in (1.5) and (1.10), respectively. For a detailed historical account of (and for various investigated developments stemming from) the Rogers-Ramanujan continued fraction (1.12) and identities (1.13) and (1.14), the interested reader may refer to the monumental work [3, p. 77 et seq.] (see also [1, 4]). The following continued fraction was recalled in [6, p. 5, Eq. (2.8)] from an earlier work cited therein: For |q| < 1,

= (1.16)

(q 2 ;q 2 )∞ (q;q 2 )∞ 3 q q 2 (1 − q 2 )

(q 2 ; q 2 )∞ (−q; q)∞ =

(1.15)

1 q q(1 − q) 1− 1+ 1− 1+

(q;q 5 )∞ (q 4 ;q 5 )∞ (q 2 ;q 5 )∞ (q 3 ;q 5 )∞ 2

5

=

1−

q 5 q 3 (1 − q 3 ) ; 1+ 1 − · · ·

q q2 q3 q4 q5 q6 1 1+ 1+ 1+ 1+ 1+ 1+ 1+ 3

5

2

··· 3

4

5

6

q q q q q q (1.17) C(q) := (q(q;q;q5 ))∞∞(q(q4 ;q;q5 ))∞∞ = 1 + 1+ 1+ 1+ 1+ 1+ 1+ · · · Andrews et al. [2] investigated new double summation hypergeometric q-series representations for several families of partitions and further explored the role of double series in combinatorial partition identities by introducing the following general family: P s(n 2 )+tn r(l, u, v, w; n), (1.18) R(s, t, l, u, v, w) := ∞ n=0 q


382

where j P[ nu ] q uv(2 )+(w−ul)j (1.19) r(l, u, v, w : n) := j=0 (−1)j (q;q) uv ;q uv ) . n−uj (q j The following interesting special cases of (1.18) are recalled (see [2, p. 106,Theorem 3]; see also [1]):

(1.20) (1.21)

R(2, 1, 1, 1, 2, 2) = (−q; q 2 )∞ ; R(2, 2, 1, 1, 2, 2) = (−q 2 ; q 2 )∞ ; 2m

2m

∞ . (1.22) R(m, m, 1, 1, 1, 2) = (q(qm ;q;q2m ))∞ Here, in this paper, we aim to present certain interrelations between q-product identities, theta function identities and combinatorial partition identities associated with the identities in (1.8)-(1.10) and (1.20)(1.22).

2. The Main Results Here we state and prove certain interesting interrelations among qproduct identities and combinatorial partition identities. Theorem. Each of the following relations holds true: 2 3162 (q,q 2 ,q 2 ;q 2 )∞ R(3,3,1,1,1,2)R(6,6,1,1,1,2) × (3.1) q 26 (q 6 ,q 12 ,q 12 ;q 12 )∞ (q 18 ,q 36 ,q 36 ;q 36 )∞ 3 6 6 6 (q , q , q ; q )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 18 × + (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 9691326 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 2 + × q 16 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 3 6 6 6 (q , q , q ; q )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 10 × + (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 56044062 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 2 + × q 11 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 3 6 6 6 (q , q , q ; q )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 6 + × (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 349029 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 2 + 21 × q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞


383

3 6 6 6 (q , q , q ; q )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 14 + × (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 256122 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 18 + 19 × q (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 2 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ × + (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 784997406 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 10 + × q9 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 2 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ + × (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 4539569022 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 6 + × q4 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 2 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ + × (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 2 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 6 +87528332700q × (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 2 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 28271349 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 14 + × q 14 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 2 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ × + (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 45675 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 4 + 22 × q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 3 6 6 6 (q , q , q ; q )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 12 + × (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 2815884 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 4 + × q 17 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 3 6 6 6 (q , q , q ; q )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 8 × + (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 7735419 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 4 + × q 12 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞

384


3 6 6 6 (q , q , q ; q )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 4 + × (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 1303 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 4 + 27 × q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 3 6 6 6 (q , q , q ; q )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 16 × + (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 299673675 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 12 + × q8 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 4 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ + × (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 4 18475014924 (q 6 , q 12 q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ + × q3 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 3 6 6 6 (q , q , q ; q )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 8 × + (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 3 6 6 6 4 6 12 12 12 2 (q , q , q ; q )∞ (q , q , q ; q )∞ +50752084059q × (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 4 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ × + (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 8548983 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 16 + × q 13 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 4 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ × + (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 252 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 6 + 28 × q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 3 6 6 6 (q , q , q ; q )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 14 + × (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 33462 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 6 + 23 × q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 3 6 6 6 (q , q , q ; q )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 10 × + (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 424305 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 6 + 18 × q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞


3 6 6 6 (q , q , q ; q )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 6 + × (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 99144 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 6 + 13 × q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 3 6 6 6 (q , q , q ; q )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 2 × + (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 133923132 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 14 + × q7 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 6 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ + × (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 17783078742 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 10 + × q2 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 6 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ + × (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 225493073505 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 6 + × q3 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 6 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ × + (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 3 6 6 6 2 6 12 12 12 8 (q , q , q ; q )∞ (q , q , q ; q )∞ +52689186504q × (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 6 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ × + (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 24 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 8 + 29 × q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 3 6 6 6 (q , q , q ; q )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 12 + × (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 5640 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 8 + 24 × q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 3 6 6 6 (q , q , q ; q )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 8 × + (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 31131 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 8 + 19 × q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞

385

386


3 6 6 6 (q , q , q ; q )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 4 + × (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 1033121304 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 12 + × q (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 8 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ × + (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 3 6 6 6 8 6 12 12 12 4 (q , q , q ; q )∞ (q , q , q ; q )∞ +242783506440q × (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 8 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ + × (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 3 6 6 6 4 6 12 12 12 9 (q , q , q ; q )∞ (q , q , q ; q )∞ +1340087471451q × (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 8 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ + × (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 10 2 2 2 30 (q, q , q ; q )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) +q × (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 3 6 6 6 (q , q , q ; q )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 10 × + (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 198 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 10 + 25 × q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 3 6 6 6 (q , q , q ; q )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 6 × + (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 819 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 10 + 20 × q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 3 6 6 6 (q , q , q ; q )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 2 + × (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 3 6 6 6 10 6 12 12 12 5 (q , q , q ; q )∞ (q , q , q ; q )∞ +3486784401q × (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 10 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ × + (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 3 6 6 6 6 6 12 12 12 10 (q , q , q ; q )∞ (q , q , q ; q )∞ +690383311398q × (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞


387

10 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ + (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 3 6 6 6 2 6 12 12 12 15 (q , q , q ; q )∞ (q , q , q ; q )∞ +2855676424419q × (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 10 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ × + (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 36 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 12 + 26 × q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 3 6 6 6 (q , q , q ; q )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 4 + × (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 7 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 12 + 21 + q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 4 3 6 6 6 6 12 12 12 16 (q , q , q ; q )∞ (q , q , q ; q )∞ × +10167463313316q (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 12 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ × + (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2 12 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ +1977006755367q 21 + (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2 1 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 14 + 27 × q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 3 6 6 6 (q , q , q ; q )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 2 × + (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 14 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ +22876792454961q 22 × (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 3 6 6 6 (q , q , q ; q )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 2 × + (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 79184709 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 4 + + q5 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 2322 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 20 + 25 + q (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 367497 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 16 + 20 q (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ ×

388


(q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 2 × = 215233605q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 10 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 2 2 2 2 4 (q, q , q ; q )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) +1312127829q × (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 6 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 140772816 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 2 × + q6 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 3 6 6 6 (q , q , q ; q )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 2 × + (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 1080596700 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 2 × + q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 2 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 2 (q 6 , q 12 q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ +17433922005q 16 × (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 10 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 2 (q 6 , q 12 q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 11 +106282354149q × (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 6 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 2 (q 6 , q 12 q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 6 × +11402598096q (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 3 6 6 6 (q , q , q ; q )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 2 × + (q , q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 4 +14348907q 8 × (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 12 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 9


389

(q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 4 × +140831865q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 8 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 185827203 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 4 + × q2 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 4 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 46438758 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 4 + + q7 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 4 19 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ +94143178827q 2 × (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 2 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 4 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 17 +923997866265q × (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 8 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 4 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 12 × +1219212278883q (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 4 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 4 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ +304684691238q 7 + (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 6 2 2 2 2 (q, q , q ; q )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) +7971615q × (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 10 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 17301357 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 6 + × q3 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 6 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 3

390


8597097 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 6 × + q8 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 2 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 6 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 q 36 ; q 36 )∞ 23 +4236443047215q × (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 10 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 6 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 q 36 ; q 36 )∞ 18 × +9194650465437q (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 6 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 6 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 q 36 ; q 36 )∞ 13 × +4568849826777q (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 2 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 793881 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 8 + × q4 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 8 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 739206 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 8 + × q9 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 4 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 121338 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 8 + + 14 q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 8 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 q 36 ; q 36 )∞ 24 +34173973914201q × (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 8 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 8 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 q 36 ; q 36 )∞ 19 +31820394443526q × (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2)


391

(q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 4 + × (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 8 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 q 36 ; q 36 )∞ 14 +5223203032698q + (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 6561 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 10 + 5 × q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 10 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 11259 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 10 × + 10 q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 6 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 14688 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 10 × + 15 q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 2 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 10 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 q 36 ; q 36 )∞ +22876792454961q 30 × (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 10 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 10 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 q 36 ; q 36 )∞ 25 +39257705570859q × (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 6 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 10 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 q 36 ; q 36 )∞ 20 × +51213889281888q (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 2 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 1080 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 12 + 16 × q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 4 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞

392


567 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 12 × + 11 q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 8 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 12 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 q 36 ; q 36 )∞ 36 +305023899399480q × (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 4 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 12 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 q 36 ; q 36 )∞ 31 +160137547184727q × (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 8 + × (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 48 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 14 + 17 × q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 6 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 14 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 q 36 ; q 36 )∞ 32 × +1098086037838128q (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 6 × + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 4 2 2 2 18 36 36 36 5 (q, q , q ; q )∞ (q , q , q ; q )∞ +10130328342q + (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 1 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 24 + 30 + 5311752795 q (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ and (3.2)

R(6,6,1,1,1,2) R(18,18,1,1,1,2)

3

(q 6 ,q 12 ;q 12 )∞ (−q 18 ,−q 36 ;q 36 12 (q 18 ,q 36 ;q 36 ) (−q 6 ,q 12 ;q∞ ∞

6

R(1, 1, 1, 1, 1, 2) 2 27(q, q 2 ; q 2 )∞ (−q 3 , −q 6 ; q 6 )∞ = + R(2, 1, 1, 1, 2, 2)R(2, 2, 1, 1, 2, 2)(q 3 , q 6 ; q 6 )∞ R(3, 3, 1, 1, 1, 2) 2 6 9R(6, 6, 1, 1, 1, 2) (q, q 2 ; q 2 )∞ (−q 3 , q 6 ; q∞ + × R(3, 3, 1, 1, 1, 2) R(2, 1, 1, 1, 2, 2)R(2, 2, 1, 1, 2, 2)(q 3 , q 6 ; q 6 )∞ R(1, 1, 1, 1, 1, 2) 4 (q 6 ; q 6 )∞ (−q 18 , −q 36 ; q 36 )∞ 2 × + R(3, 3, 1, 1, 1, 2) (q 18 ; q 18 )∞ (−q 6 , q 12 ; q 12 )∞


393

(6, 6, 1, 1, 1, 2)(q; q)∞ (−q 3 , −q 6 ; q 6 )∞ 3 {R(1, 1, 1, 1, 1, 2)}2 3 × + R(2, 1, 1, 1, 2, 2)R(2, 2, 1, 1, 2, 2) R(3, 3, 3, 1, 1, 1, 2) 4 6 12 36 (−q 18 , −q 36 ; q∞ (q ; q )∞ 1 × 6 6 {(q ; q )∞ R(3, 3, 1, 1, 1, 2)}3 (q 18 ; q 18 )∞ (−q 6 , q 12 ; q 12 )∞ (q 12 ; q 12 )∞ Proofs: Applying the identity(1.10), and further using (1.20)-(1.22), we obtain the following identities : 92 3 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 2 3 P2 + 2 = 7 P (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ q2 2 7 (q 3 , q 6 , q 6 ; q 6 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (3.3) + 234q 2 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2)

(3.4)

18

1054Q

1054 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 18 = 45 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ q2

10 35 × 13294 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 295245 10 = 3 13294Q − 25 Q10 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ q2 (3.5) 10 25 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 5 2 − 3 × 295245q (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 5

37 .8542 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 6 199989 6 = 3 8542Q − 15 Q6 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ q2 (3.6) 15 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 6 7 2 − 3 × 199989q (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 7

2 4575 596 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ −78732 596Q2 + = −78732 × 5 2 2 2 18 36 36 36 Q2 q 2 (q, q , q ; q )∞ (q , q , q ; q )∞ 2 5 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (3.7) − 78732 × 4575.q 2 3 6 6 6 6 12 12 12 (q , q , q ; q )∞ (q , q , q ; q )∞

(3.8)

14

116343Q

116343 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 14 = 35 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ q2


394

1 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 4 94 4 P + 4 = 7 P q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 4 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 4 7 +9 q (3.9) (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 12 313 32 × 5075 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 32 5075Q12 − 12 = − Q q 15 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 12 2 2 2 18 36 36 36 15 15 (q, q , q ; q )∞ (q , q , q ; q )∞ (3.10) − 3 × 5075q (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 8 579555 35 × 11588 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 5 8 3 11588Q − = Q8 q 10 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (3.11) 8 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ − 35 × 579555q 10 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 4 3147 131 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 10 310 131Q4 − = 3 × − Q4 q 5 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (3.12) 4 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ − 310 × 3147q 5 (q 6 , q 12 , q 12 ; q 12 )∞ (q 3 , q 6 , q 6 ; q 6 )∞ 16

(3.13)

1303Q

1303 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 16 = 20 q (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞

6 96 9 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 6 9 P + 6 = 21 + P (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ q2 (3.14) 6 (q 3 , q 6 , q 6 ; q 6 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 7 21 2 +9 q (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 14

(3.15)

28Q

=

28 q

10

3718Q (3.16)

35 2

(q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞

14

885735 3718 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 10 − = 25 Q10 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ q2 25 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 10 2 − 885735q (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞


395

6 47145 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 640791 6 = 3 15715Q − 15 Q6 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ q2 6 15 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (3.17) − 1922373q 2 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 4

3

11793 136Q − Q2 2

(3.18)

2 34 × 136 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ = 5 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ q2 2 5 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 4 2 − 3 × 11793q (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞

98 3 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 8 8 3 P + 8 = 14 P q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (3.19) 8 (q 3 , q 6 , q 6 ; q 6 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 9 14 +3×8 q (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) (3.20) 12

8Q

8 = 15 q

(q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞

12

24627 1880 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 8 1880Q − = 10 Q8 q (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 8 2 2 2 18 36 36 36 10 (q, q , q ; q )∞ (q , q , q ; q )∞ (3.21) − 264627q (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 8

27378 10377 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 4 4 9 1153Q − = Q4 q5 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (3.22) 4 2 2 2 18 36 36 36 5 (q, q , q ; q )∞ (q , q , q ; q )∞ − 246402q (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞

P 10 +

(3.23)

910 P 10

10 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 25 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ q2 10 (q 3 , q 6 , q 6 ; q 6 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 10 25 +9 q 2 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) =

1


396

10

Q

(q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 10 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 10 8 25 2 −3 q (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞

1 38 − 10 = 25 Q q2

(3.24)

3

2

1251 22Q − Q6 6

(3.25)

198 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 6 = 15 2 2 2 18 36 36 36 q 2 (q, q , q ; q )∞ (q , q , q ; q )∞ 15 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 6 − 11259q 2 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞

1632 819 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 2 2 3 91Q − = 5 2 2 2 18 36 36 36 Q2 q 2 (q, q , q ; q )∞ (q , q , q ; q )∞ 5 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 2 − 14688q 2 (3.26) (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 2

P

12

12 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) + (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 12 (q 3 , q 6 , q 6 ; q 6 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ + 912 q 21 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2)

1 912 + 12 = 21 P q

(3.27)

6

2

30 Q − 4 Q 4

(3.28)

36 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 4 = 5 − q (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 4 2 2 2 18 36 36 36 5 (q, q , q ; q )∞ (q , q , q ; q )∞ − 1080q (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞

(3.29) 8 2 2 2 18 36 36 36 567 10 (q, q , q ; q )∞ (q , q , q ; q )∞ − 8 = −567q Q (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ P

14

(q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 14 + (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 14 (q 3 , q 6 , q 6 ; q 6 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 14 49 2 +9 q (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2)

914 1 + 14 = 49 P q2

(3.30)


397

(q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 2 − (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 15 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 6 − 48q 2 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞

1 48 Q − 6 = 5 Q q2 2

(3.31)

(3.32) (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 8 1 − 8 = −q 10 Q (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 916 1 (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 16 16 P + 16 = 28 + P q (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 16 (q 6 , q 12 , q 12 ; q 12 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 16 28 +9 q (3.33) (q, q 2 , q 2 ; q 2 )∞ R(3, 3, 1, 1, 1, 2)R(6, 6, 1, 1, 1, 2) 38 × 2284 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 12 312 8 12 3 2284Q + 12 = + Q q 15 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 12 2 2 2 18 36 36 36 20 15 (q, q , q ; q )∞ (q , q , q ; q )∞ (3.34) +3 q (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 8 310 × 3896 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 61965 8 = 3 3896Q − − Q8 q 10 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 8 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ (3.35) − 310 × 61965q 10 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 10

4 312 × 149 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 19062 312 149Q4 − = − Q4 q5 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 4 2 2 2 18 36 36 36 12 5 (q, q , q ; q )∞ (q , q , q ; q )∞ (3.36) − 3 × 19062q (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ 24

−Q

(3.37)

20

+ 2322Q

16

+ 367497Q

24 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ + (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 20 2322 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ + 25 + q (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 16 367497 (q 3 , q 6 , q 6 ; q 6 )∞ (q 6 , q 12 , q 12 ; q 12 )∞ + q 20 (q, q 2 , q 2 ; q 2 )∞ (q 18 , q 36 , q 36 ; q 36 )∞ 1 = 30 q

(3.3)- (3.37) into (2.1),by little algebra and moving negative terms to the right hand side we obtain (3.1). Further, we attempt to prove our second identity (3.2), to establish it


398

applying the identity(1.8)-(1.9), and further using (1.20) - (1.22), we obtain the following identities: (3.38) 27P R2 =

27(q, q 2 ; q 2 )∞ (−q 3 , −q 6 ; q 6 )∞ R(2, 1, 1, 1, 2, 2)R(2, 2, 1, 1, 2, 2)(q 3 , q 6 ; q 6 )∞

R(1, 1, 1, 1, 1, 2) R(3, 3, 1, 1, 1, 2)

2

2 6 9R(6, 6, 1, 1, 1, 2) (q, q 2 ; q 2 )∞ (−q 3 , q 6 ; q∞ × R(3, 3, 1, 1, 1, 2) R(2, 1, 1, 1, 2, 2)R(2, 2, 1, 1, 2, 2)(q 3 , q 6 ; q 6 )∞ 4 6 6 2 R(1, 1, 1, 1, 1, 2) (q ; q )∞ (−q 18 , −q 36 ; q 36 )∞ × 18 18 6 12 12 R(3, 3, 1, 1, 1, 2) (q ; q )∞ (−q , q ; q )∞

9P 2 R4 Q2 T = (3.39)

P 3 R6 T 2 Q4 =

(6, 6, 1, 1, 1, 2)(q; q)∞ (−q 3 , −q 6 ; q 6 )∞ R(2, 1, 1, 1, 2, 2)R(2, 2, 1, 1, 2, 2)

3

{R(1, 1, 1, 1, 1, 2)}2 R(3, 3, 3, 1, 1, 1, 2)

3 ×

(3.40) ×

1 × {(q 6 ; q 6 )∞ R(3, 3, 1, 1, 1, 2)}3

(−q 18 , −q 36 ; q 36 )∞ (q 18 ; q 18 )∞ (−q 6 , q 12 ; q 12 )∞

4 ×

(q 6 ; q 12 )∞ (q 12 ; q 12 )∞

(3.41) 6

3

Q T =

R(6, 6, 1, 1, 1, 2) R(18, 18, 1, 1, 1, 2)

3

(q 6 , q 12 ; q 12 )∞ (−q 18 , −q 36 ; q 36 (−q 6 , q 12 ; q 12 )∞ (q 18 , q 36 ; q 36 )∞

6

using (3.38)-(3.41) into (2.2), arranging the terms, we obtain (3.2). This completes the proof of our theorem.

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M.P. Chaudhary International Scientific Research and Welfare Organization New Delhi, India. E-mail:[email protected] Ahemed Buseri Ashine Department of Mathematics, College of Natural and Computational Sciences Madda Walabu University, Bale Robe, Ethiopia. Feyissa Kaba Wakene Department of Mathematics, College of Natural and Computational Sciences Madda Walabu University, Bale Robe, Ethiopia.