SPIRIT OF RAMANUJAN 3 Exercise 1.1.7. Establish a second q-analogue of Pascal's formula, which is given for n 1 by = q" n-1 m 1 + n - 1 m Exercise 1.1.8. Prove that, for each pair of nonnegative integers m,n, J'=0 m + j . 3 . qJ = m + n + 1 ra + 1 We now define four primary arithmetical functions on which we focus throughout the monograph. Definition 1.1.9. If n is a positive integer, letp(n) denote the num- ber of unrestricted representations of n as a sum of positive integers, where representations with different orders of the same summands are not regarded as distinct. We call p(n) the partition function. For example, p(4) = 5, because there are 5 ways to represent 4 as a sum of positive integers, namely, 4 = 3+1 = 2+2 = 2+1+1 = 1+1+ 1 + 1. Readers should check that p(5) = 7 and p(6) = 11, but readers should not check (at least by hand) that p(200) = 3,972,999,029,388. A few moments of reflection convinces one that p(n) grows rapidly. More precisely, G. H. Hardy and Ramanujan [110], [192, pp. 276- 309] showed that, a s n - ^ o o , (1.1.6) p(n) 1 AnVS ex p 7T i.e., the ratio of the left and right sides of (1.1.6) tends to 1 as n tends to oo. The generating function for p(n), due to Euler, is given by (1 oo 1 .1.7) Y^p(n)qn = -. = - 1 l-qk where we define p(0) = 1. To see this, observe that the factor 1 = £*" 3=0
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