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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 | --Copyright The Numerical Algorithms Group Limited 1991.
--% Infinite products of Taylor series
-- We compute series expansions of various infinite products using INFPROD0
-- Author: Clifton J. Williamson
-- Date Created: 20 August 1992
-- Date Last Updated: 20 August 1992
-- Keywords: Taylor series, infinite product
-- References:
)clear all
-- the partition function
f : UTS(INT,x,0) := 1 - x
g : UTS(INT,x,0) := recip f
-- the coefficient of x ** n in the following series is the number of ways
-- that n can be partitioned as a sum of positive integers:
infiniteProduct g
-- Ramanujan's tau function
h := infiniteProduct(f ** 24)
-- the coefficient of x ** n in the following series is Ramanujan's
-- function tau(n)
delta := x * h
-- the function tau(n) is multiplicative, i.e. if gcd(m,n) = 1, then
-- tau(m * n) = tau(m) * tau(n)
coefficient(delta,21)
coefficient(delta,3) * coefficient(delta,7)
coefficient(delta,20)
coefficient(delta,4) * coefficient(delta,5)
coefficient(delta,65)
coefficient(delta,13) * coefficient(delta,5)
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