CHAPTER 2 Definitions and Preliminary Results We first recall Ramanujan's definitions for a general theta function and some of its important special cases. Set oo (2.1) f(a,b):= Yl ara(n+1)/26n("-1)/2, \ab\ 1. n = oo Basic properties satisfied by /(a, b) include [5, p. 34, Entry 18] (2.2) f(a,b)=f(b,a), (2.3) / ( l , a ) = 2 / ( a , a 3 ) , (2.4) / ( - l , o ) = 0 , and, if n is an integer, (2.5) /(a, b) = a n n + 1 )/V ( n ~ 1 ) / 2 /(a(ab) n , b{ab)~n). The basic property (2.2) will be used many times in the sequel without comment. The function /(a, 6) satisfies the well-known Jacobi triple product identity [5, p. 35, Entry 19] (2.6) /(a, b) = (-a a&)oo(-& ab)oo(ab] ab)^. The three most important special cases of (2.1) are oo (2.7) p(q) := f(q, q) = £ qn* = (-q q2)Uq2 q2)^ n— oo (2.8) m := /(,, ,3) = £ 9"(n+1)/2 = ^ T n=0 and (? 72)oo ' (2.9) / ( - g ) := /(-*, -g 2 ) = £ ( - l ) V ( 3 n " 1 ) / 2 ^ (9 ?)oo =: T 1/24 T7(T), n = —oo where g = exp(27rir), Im r 0, and 7 7 denotes the Dedekind eta-function. The product representations in (2.7)-(2.9) are special cases of (2.6). Also, after Ra- manujan, define (2.10) X(Z) :=("? Z2)oo. Using (2.6) and (2.9), we can rewrite the Rogers-Ramanujan identities (1.2) in the forms (2-11) G( q) = ^ f l and *(,) = 4 = * ^ .
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