New modular relations involving cubes of the Göllnitz–Gordon functions

Introduction

The aim of this paper is to present several new modular relations involving cubes of the Göllnitz-Gordon functions and provide new proofs of some known modular relations for the Göllnitz-Gordon functions due to Gugg [11] . The advantage of our method is that the identities to be proved do not need to be known in advance. We start with an overview of the terminology and notation of q-series. Throughout this paper, we assume that |q| < 1 and for nonnegative integer n, we employ the standard notation n-1 ∞ (a; q )n =(1aq i ), (a; q )∞ =(1aq i ) i=0 i=0 and and (a1, a2, . . . , an; q )∞ = (a1; q )∞(a2; q )∞ • • • (an; q )∞.

Recall that Ramanujan's general theta function f (a, b) is defined by (1.3) Three special cases of (1.1) are defined by Three special cases of (1.1) are defined by In a manuscript of Ramanujan [15] , there are forty identities involving G(q ) and H (q ). The forty identities have been proved in a series of papers by Rogers [17] , Darling [8] , Watson [18] , Bressoud [6] , Biagioli [5] and Yesilyurt [24,25] . Berndt et al. [3] published an excellent monograph on the forty identities.

The following two functions analogous to (1.12) and (1.13) are the so-called Göllnitz-Gordon functions

The functions S (q) and T (q) satisfy the following identities [9,10]: Motivated by the similarity between the Rogers-Ramanujan functions and the Göllnitz-Gordon functions, Huang [13] and, Chen and Huang [7] derived many elegant modular relations for S(q ) and T (q ) analogous to Ramanujan's list of forty identities for the Rogers-Ramanujan functions. They extracted interesting partition theoretic results from some of the relations. Subsequently, Baruah et al. [1] found new proofs of modular relations which involve only S(q ) and T (q ) by using Schröter's formulas and some theta-function identities given in Ramanujan's notebooks [2] . In the process, they also found some new relations. Yan [22] also gave some proofs of modular relations due to Huang [13] and, Chen and Huang [7] . Xia and Yao [20,23] provided new proofs of some modular relations established by Huang [13] and, Chen and Huang [7] . In the process, they also dis-covered some new relations which involve only the Göllnitz-Gordon functions. Recently, Gugg [11] discovered two new elegant relations involving cubes of the Göllnitz-Gordon functions. We end this section by listing several modular relations involving only the Göllnitz-Gordon functions which will be proved in this paper. Baruah et al. [1] and Xia and Yao [20,23] gave new proofs of (2.28) and (2.29). Replacing 2 by -q in (2.26) and employing (1.8), (2.9) and (2.29), we obtain S 3 (q )S(q 3 )q 3 T 3 (q )T (q 3 ) Equating the even parts and the odd parts on both sides of (2.32), we see that S (-q )S (-q ) + q T (-q )T (-q ) = + 2q -3q 4 S(-q 6 )T (-q 6 )(S(-q 6 )T (-q 2 ) + q 2 T (-q 6 )S(-q 2 )) (2.33) and

6 24 8 48 S (-q )T (-q ) + q T (-q )S(-q ) = + 2q f2f8 2 f12 7 f48

3 -3q 2 S(-q 6 )T (-q 6 )(S(-q 6 )S(-q 2 ) + q 4 T (-q 6 )T (-q 2 )).

(2.34) Replacing q 2 by -q in (2.33) and utilizing (1.8), (2.9) and (2.29), we can obtain (1.24). In view of (1.8), (2.9), (2.28) and (2.34), we can derive (1.25) after replacing q 2 by -q in (2.34). Huang [13] established the following two identities: )T (q )q 2 T (q 5 )S(q ) = f

See also [1,20,23] . Identity (1.26) follows from (2.9), (2.39) and (2.42). Combining (2.9), (2.40) and (2.41), we arrive at (1.27).

Xia and Yao [20] also established the following 2-dissection formula for +q 9 S 3 (-q 2 )T (-q 18 ) -3q 10 S 2 (-q 2 )T (-q 2 )T (-q 18 ) + 3q 11 S(-q 2 )T 2 (-q 2 )T (-q 18 )q 12 T 3 (-q 2 )T (-q 18 )).

Proof of Theorem 1.3

Besides the functions ψ (q), ϕ(q ) and f (-q ), Ramanujan defined one further function χ (q) := (-q; q 2 )∞. (4.1) By the definition of fn, it is trivial to verify that

References

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