An Invitation to Q-Series: From Jacobi's Triple Product by Hei-Chi Chan

By Hei-Chi Chan

The purpose of those lecture notes is to supply a self-contained exposition of numerous interesting formulation came across via Srinivasa Ramanujan. imperative ends up in those notes are: (1) the review of the Rogers-Ramanujan persisted fraction -- a consequence that confident G H Hardy that Ramanujan was once a "mathematician of the top class", and (2) what G. H. Hardy referred to as Ramanujan's "Most attractive Identity". This booklet covers a number of comparable effects, corresponding to numerous proofs of the well-known Rogers-Ramanujan identities and an in depth account of Ramanujan's congruences. It additionally covers a number of suggestions in q-series.

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2) is verified. 2 Dn (a) satisfies Eq. 2) The situation is rather different in the present case. The verification is not straightforward. Part of the complication comes from the floor function in Dn . If one opts for directly verifying that Dn (a) satisfies Eq. 2. 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 .

15) n=1 Note that various factors in the first line appear in the order of the righthand side of Eq. 11). To express the right-hand side of Eq. 14), we need the following (for ˆ = Z2 Z. Hence (with details, see Carter (2005)). As mentioned above, W w ∈ Z2 ) w(ˆ ˆ ρ) − ρˆ = 2nα − n(2n + 1)δ, −(2n + 1)α − n(2n + 1)δ, if w = +1; if w = −1. Also we have (w) ˆ = 1 for the first case and −1 for the second. With this understood, the right-hand side of Eq. 14) is given by ˆ ρ)− ˆ ρˆ (w)e ˆ w( w=±1 n∈Z e−(2n+1)α−n(2n+1)δ e2nα−n(2n+1)δ − = n∈Z = n∈Z z −2n n(2n+1) q n∈Z − z 2n+1 q n(2n+1) n∈Z m m m(m−1)/2 = (−1) z q .

13) Taking the limit z → −1 in the last equation gives the desired identity. 6. 3. It will be useful in Chapter 17. 2 (A remarkable four-parameter q-series identity). K. Alladi, G. E. Andrews and A. Berkovich (2003) discovered the following remarkable four-parameter identity: (−Aq)∞ (−Bq)∞ (−Cq)∞ (−Dq)∞ Ai B j C k Dl = i,j,k,l · (1 − q a ) + q i,j,k,l− constraints a+bc+bd+Q q Tτ +Tab +Tac +···+Tcd −bc−bd−cd+4TQ−1 +3Q+2Qτ (q)a (q)b (q)c (q)d (q)ab (q)ac (q)ad (q)bc (q)bd (q)cd (q)Q (1 − q b ) + q a+bc+bd+Q+b+cd .

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