Honam Mathematical J. 39 (2017), No. 2, pp. 267–273 https://doi.org/10.5831/HMJ.2017.39.2.267
NOTE ON Q-PRODUCT IDENTITIES AND COMBINATORIAL PARTITION IDENTITIES M. P. Chaudhary∗ and Getachew Abiye Salilew
Abstract. The objective of this note is to establish three results between q-products and combinatorial partition identities in a elementary way. Several closely related q-product identities such as (for example)continued fraction identities and Jacobis triple product identities are also considered.
1. Introduction and Preliminaries 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., [9, 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 (1.2)
(a; q)∞ :=
∞ Y
(1 − aq k ) =
k=0
∞ Y
(1 − aq k−1 ) (a, q ∈ C; |q| < 1).
k=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 frequently used: (1.3)
(a1 , a2 , · · · , am ; q)n := (a1 ; q)n (a2 ; q)n · · · (am ; q)n
Received March 30, 2017. Accepted May 12, 2017. 2010 Mathematics Subject Classifications. 5A30, 11F27, 11P83. Key words and phrases. q- product identities, combinatorial partition identities, modular equation and Jacobi’s triple product identities. *Corresponding author
268
M. P. Chaudhary∗ and Getachew Abiye Salilew
and (1.4)
(a1 , a2 , · · · , am ; q)∞ := (a1 ; q)∞ (a2 ; q)∞ · · · (am ; q)∞ .
Ramanujan defined the general theta function f (a, b) as follows (see, for details, [3, p. 31, Eq.(18.1)] and [5]; see also [10]): f (a, b) = 1 + (1.5)
=
∞ X
(ab)
n=1 ∞ X
a
n(n−1) 2
n(n+1) 2
(an + bn )
n(n−1) 2
b
= f (b, a) (|ab| < 1).
n=−∞
We find from (1.5) that (1.6)
f (a, b) = a
n(n+1) 2
b
n(n−1) 2
f (a(ab)n , b(ab)−n ) = f (b, a) (n ∈ Z).
Ramanujan also rediscovered the Jacobi’s famous triple-product identity (see [4, 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 [4, pp. 36-37, Entry 22]): φ(q) : =
∞ X
q
n2
=1+2
n=−∞
(1.8) (1.9)
∞ X
2
qn
n=1
= {(−q; q 2 )∞ }2 (q 2 ; q 2 )∞ = 3
ψ(q) : = f (q, q ) =
∞ X
q
n(n+1) 2
=
n=0 2
f (−q) : = f (−q, −q ) =
∞ X
(−q; q 2 )∞ (q 2 ; q 2 )∞ ; (q; q 2 )∞ (−q 2 ; q 2 )∞ (q 2 ; q 2 )∞ ; (q; q 2 )∞
(−1)n q
n(3n−1) 2
n=−∞
(1.10)
=
∞ X
n
(−1) q
n(3n−1) 2
n=0
+
∞ X
(−1)n q
n(3n+1) 2
= (q; q)∞
n=1
Equation (1.10) is known as Eulers Pentagonal Number Theorem. The following q-series identity: (1.11)
(−q; q)∞ =
1 1 = 2 (q; q )∞ χ(−q)
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269
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 number of partitions of n into odd parts. We also recall the Rogers-Ramanujan continued fraction of R(q): 1
R(q) : = q 5
4 5 4 5 1 f (−q, −q ) 1 (q; q )∞ (q ; q )∞ H(q) 5 = q5 = q G(q) f (−q 2 , −q 3 ) (q 2 ; q 5 )∞ (q 3 ; q 5 )∞
1
(1.12)
=
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 G(q) : =
2 ∞ X f (−q 5 ) qn = (q; q)n f (−q, −q 4 )
n=0
(1.13)
1 (q 2 ; q 5 )∞ (q 3 ; q 5 )∞ (q 5 ; q 5 )∞ = ; (q; q 5 )∞ (q4; q 5 )∞ (q; q)∞ ∞ X q n(n+1) f (−q 5 ) 1 H(q) : = = = 2 5 2 3 (q; q)n f (−q , −q ) (q ; q )∞ (q 3 ; q 5 )∞ =
n=0
(1.14)
=
(q; q 5 )∞ (q 4 ; 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 [9, 10]). The following continued fraction was recalled in [6, p. 5, Eq. (2.8)] from an earlier work cited therein: For |q| < 1, (q 2 ; q 2 )∞ (q; q 2 )∞ 1 q q(1 − q) q 3 q 2 (1 − q 2 ) q 5 q 3 (1 − q 3 ) = ; 1− 1+ 1− 1+ 1− 1+ 1 − · · · (q; q 5 )∞ (q 4 ; q 5 )∞ 1 q q2 q3 q4 q5 q6 (1.16) = ··· ; (q 2 ; q 5 )∞ (q 3 ; q 5 )∞ 1+ 1+ 1+ 1+ 1+ 1+ 1+ (q 2 ; q 5 )∞ (q 3 ; q 5 )∞ q q2 q3 q4 q5 q6 (1.17) C(q) := = 1 + ··· . (q; q 5 )∞ (q 4 ; q 5 )∞ 1+ 1+ 1+ 1+ 1+ 1+ (1.15) (q 2 ; q 2 )∞ (−q; q)∞ =
M. P. Chaudhary∗ and Getachew Abiye Salilew
270
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: (1.18)
∞ X
R(s, t, l, u, v, w) :=
n
q s(2 )+tn r(l, u, v, w; n),
n=0
where n
(1.19)
[u] X r(l, u, v, w : n) := (−1)j j=0
j
q uv(2 )+(w−ul)j . (q; q)n−uj (q uv ; q uv )j
The following interesting special cases of (1.18) are recalled (see [2, p. 106,Theorem 3]; see also [10]): (1.20)
R(2, 1, 1, 1, 2, 2) = (−q; q 2 )∞ ;
(1.21)
R(2, 2, 1, 1, 2, 2) = (−q 2 ; q 2 )∞ ;
(1.22)
R(m, m, 1, 1, 1, 2) =
(q 2m ; q 2m )∞ . (q m ; q 2m )∞
Here, in this paper, we aim to present certain interrelations between q-product identities, modular equations and combinatorial partition identities associated with the identities in (1.8) and (1.20)-(1.22).
2. Sets of Preliminary Results For our purpose, here, we recall some known identities, as; φ(q) 4 4 4 4 (2.1) If P = 1ψ(−q) and Q = φ(q 3 ) , then Q + P Q = 9 + P . (2.2)
If P =
(2.3)
If P =
q 4 ψ(−q 3 ) ψ(−q) and qψ(−q 9 ) ψ(−q) 1
q 2 ψ(−q 5 )
φ(q) , then Q + P Q = 3 + P . φ(q 9 ) φ(q) = φ(q5 ) , then Q2 + P 2 Q2 = 5 +
Q=
and Q
P 2.
3. The Main Results Here we present some identities which give certain relationships between q-product identities and combinatorial partition identities as in the following theorem.
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Theorem 3.1. Each of the following identities is true: (−q 3 , −q 3 , −q 3 , −q 3 ; q 6 )∞ (q 2 , q 2 , q 2 , q 2 ; q 2 )∞ +9 q{R(2, 1, 1, 1, 2, 2)}4 (q 6 , q 6 , q 6 , q 6 ; q 6 )∞ R(1, 1, 1, 1, 1, 2)R(2, 1, 1, 1, 2, 2) 4 (−q 6 , −q 6 , −q 6 , −q 6 ; q 6 )∞ + = R(2, 2, 1, 1, 2, 2)R(3, 3, 1, 1, 1, 2) (−q 3 , −q 3 , −q 3 , −q 3 ; q 6 )∞ 4 2 2 2 2 2 R(1, 1, 1, 1, 1, 2) (q , q , q , q ; q )∞ + × R(2, 2, 1, 1, 2, 2)R(3, 3, 1, 1, 1, 2) q (−q 6 , −q 6 , −q 6 , −q 6 , −q 3 , −q 3 , −q 3 , −q 3 ; q 6 )∞ × . (−q 3 , −q 3 , −q 3 , −q 3 , q 6 , q 6 , q 6 , q 6 ; q 6 )∞ 1 (−q 9 ; q 18 )∞ (q 2 ; q 2 )∞ (3.2) +3 q R(2, 1, 1, 1, 2, 2)(q 18 ; q 18 )∞ R(1, 1, 1, 1, 1, 2)R(2, 1, 1, 1, 2, 2)(−q 18 ; q 18 )∞ + = R(2, 2, 1, 1, 2, 2)R(9, 9, 1, 1, 1, 2)(−q 9 ; q 18 )∞ 1 R(1, 1, 1, 1, 1, 2)(−q 18 , −q 9 ; q 18 )∞ (q 2 ; q 2 )∞ + q R(2, 2, 1, 1, 2, 2)R(9, 9, 1, 1, 1, 2)(−q 9 , q 18 ; q 18 )∞
(3.1)
and (3.3)
1 (−q 5 , −q 5 ; q 10 )∞ (q 2 , q 2 ; q 2 )∞ +5 q {R(2, 1, 1, 1, 2, 2)}2 (q 10 , q 10 ; q 10 )∞ R(1, 1, 1, 1, 1, 2)R(2, 1, 1, 1, 2, 2) 2 (−q 10 , −q 10 ; q 10 )∞ = + R(2, 2, 1, 1, 2, 2)R(5, 5, 1, 1, 1, 2) (−q 5 , −q 5 ; q 10 )∞ 2 1 R(1, 1, 1, 1, 1, 2)(q 2 ; q 2 )∞ (−q 10 , −q 10 , −q 5 , −q 5 ; q 10 )∞ + q R(2, 2, 1, 1, 2, 2)R(5, 5, 1, 1, 1, 2) (−q 5 , −q 5 , q 10 , q 10 ; q 10 )∞
Proof. First of all, we have to prove our first identity (3.1). To prove it apply (1.8) [for q = q 3 ] and (1.9) [for q = −q and q = −q 3 ], we obtain (3.4)
P =
ψ(−q) 1
q 4 ψ(−q 3 )
=
(−q 3 ; q 6 )∞ (q 2 ; q 2 )∞ 1 6 6 q 4 R(2, 1, 1, 1, 2, 2)(q ; q )∞ 1
and (3.5)
Q=
φ(q) R(1, 1, 1, 1, 1, 2)R(2, 1, 1, 1, 2, 2)(−q 6 , q 3 ; q 6 )∞ = φ(q 3 ) R(2, 2, 1, 1, 2, 2)(−q 3 , q 6 ; q 6 )∞
using (3.4) and (3.5) into (2.1), by arranging the powers of suitable terms and after little algebra, we obtain (3.1).
M. P. Chaudhary∗ and Getachew Abiye Salilew
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Further, we attempt to prove our second identity (3.2), to prove it use (1.8) [for q = q 9 ] and (1.9) [for q = −q and q = −q 9 ], we get (3.6) (3.7)
ψ(−q) 1 (−q 9 ; q 18 )∞ (q 2 ; q 2 )∞ = qψ(−q 9 ) q R(2, 1, 1, 1, 2, 2)(q 18 ; q 18 )∞ φ(q) R(1, 1, 1, 1, 1, 2)R(2, 1, 1, 1, 2, 2)(−q 18 , q 9 ; q 18 )∞ Q= = φ(q 9 ) R(2, 2, 1, 1, 2, 2)(−q 9 , q 18 ; q 18 )∞ P =
using (3.6) and (3.7) into (2.2), by arranging the powers of suitable terms, after simplification, we obtain (3.2). Finally, we attempt to establish our third identity (3.3), to prove it apply (1.8) [for q = q 5 ] and (1.9) [for q = −q and q = −q 5 ], we get (3.8) (3.9)
(−q 5 ; q 10 )∞ (q 2 ; q 2 )∞ 10 10 q ψ(−q 5 ) q R(2, 1, 1, 1, 2, 2)(q ; q )∞ R(1, 1, 1, 1, 1, 2)R(2, 1, 1, 1, 2, 2)(−q 10 , q 5 ; q 10 )∞ φ(q) = Q= φ(q 5 ) R(2, 2, 1, 1, 2, 2)(−q 5 , q 10 ; q 10 )∞ P =
ψ(−q)
1 2
=
1
1 2
using (3.8) and (3.9) into (2.3), arranging the suitable powers of terms, and after simplification, we obtain (3.3). References [1] C. Adiga, T. Kim, M.S.M. Naika and H.S. Madhusuidhan, On Ramanujan’s cubic continued fraction and explicit evaluations of theta- fractions, Indian J. Pure Appl. Math, 2004. [2] G. E. Andrews, K. Bringman and K. Mahlburg, Double series representations for Schur’s partition function and related identities, J. Combin. Theory Ser. A 132 (2015), 102-119. [3] N. D. Baruah and J. Bora, Modular relations for the nonic analogues of the Rogers Ramanujan functions with applications to partitions, J. Number Theory, 128 (2008), 175-206. [4] B. C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, Berlin, Heidelberg and New York, 1991. [5] M. P. Chaudhary, Generalization of Ramanujan’s identities in terms of q-products and continued fractions, Global J. Sci. Frontier Res. Math. Decision Sci. 12(2) (2012), 53-60. [6] M. P. Chaudhary, Generalization for character formulas in terms of continued fraction identities, Malaya J. Mat. 1(1)(2014), 24-34. [7] M. P. Chaudhary, Some relationships between q -product identities, combinatorial partition identities and continued-fractions identities III, Pacic J. Appl. Math. 7(2)(2015), 87-95. [8] M. P. Chaudhary and J. Choi, Note on modular relations for Roger-Ramanujan type identities and representations for Jacobi identities, East Asian Math. J. 31(5)(2015), 659-665.
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[9] H. M. Srivastava and J. Choi, Zeta and q -Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012. [10] H. M. Srivastava and M. P. Chaudhary, Some relationships between q - product identities, combinatorial partition identities and continued-fractions identities, Adv. Stud. Contemporary Math. 25(3)(2015), 265-272.
M. P. Chaudhary Department of Mathematics, International Scientific Research and Welfare Organization, New Delhi 110018, India. E-mail:
[email protected] Getachew Abiye Salilew Department of Mathematics, College of Natural and Computational Science, Madda Walabu University, Bale Robe, Ethiopia. E-mail:
[email protected] 
