By Hei-Chi Chan
The purpose of those lecture notes is to supply a self-contained exposition of numerous interesting formulation chanced on by means of Srinivasa Ramanujan. principal leads to those notes are: (1) the overview of the Rogers-Ramanujan endured fraction -- a consequence that confident G H Hardy that Ramanujan used to be a "mathematician of the top class", and (2) what G. H. Hardy referred to as Ramanujan's "Most attractive Identity". This publication covers a number of comparable effects, comparable to a number of proofs of the well-known Rogers-Ramanujan identities and a close account of Ramanujan's congruences. It additionally covers various ideas in q-series.
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Extra info for An Invitation to Q-Series: From Jacobi's Triple Product Identity to Ramanujan's "Most Beautiful Identity"
With the above understood, we can evaluate in two diﬀerent ways the partition function Z(q, z) = z Q(S) q H(S) . 2) n=1 In the ﬁrst line, the ﬁrst (second) inﬁnite product comes from the positive (negative) levels. Note the minus sign in the exponent of q −ei . It is negative because it is measured relative to the vacuum state. Bosonic evaluation of Z(q, z) This is based on a diﬀerent way of describing admissible states. Let us deﬁne (with n ∈ Z) vac(n):= the state with all levels < n being occupied.
The commutator for x ⊗ tm and y ⊗ tn is deﬁned by [x ⊗ tm , y ⊗ tn ] = [x, y] ⊗ tm+n + mk x, y δm+n,0 . Step 3. To obtain the aﬃne Lie algebra ˆg, we need to slightly enlarge Lg by adding to it the operator d: ˆg := Lg ⊕ Ck ⊕ Cd, February 27, 2011 19:51 World Scientific Book - 9in x 6in 000˙HC Macdonald’s identities 42 where the non-vanishing commutators involving d are given by [d, x ⊗ tn ] = n x ⊗ tn . That is, d reads oﬀ the mode n of x ⊗ tn . With this understood, let us turn to the root space decomposition of ˆg.
Dn (a) satisﬁes Eq. 2) 000˙HC 55 and consider two cases depending on the parity of n. In this way, the ﬂoor function involved could be removed. -C. Chan (2010c)). 1. Consider the case a = 0. Here we write Dn (0) as Dn . Say we want to verify D10 = D9 + q 8 D8 . 4) First we write down the diﬀerence D10 − D9 (and we need Eqs. 4)): 8 9 8 9 8 5 8 5 q − q + q − q . 6) (note: we added the ﬁrst term, which is zero, for bookkeeping reason). But Eqs. 6) look rather diﬀerent! Thankfully, Eqs. 4) will come to our rescue.
An Invitation to Q-Series: From Jacobi's Triple Product Identity to Ramanujan's "Most Beautiful Identity" by Hei-Chi Chan