2019/06/27 by James Mc Laughlin, Laughlin, James Mc · 1 citation
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Advanced Combinatorial Mathematics
paper · pdf · doi:10.48550/arxiv.1906.11978
Andrews and Bressoud, Alladi and Gordon, and others, have proven, in a number of papers, that the coefficients in various arithmetic progressions in the series expansions of certain infinite q-products vanish. In the present paper it is shown that these results follow automatically (simply by specializing parameters) in an identity derived from a special case of Ramanujan's 1ψ1 identity. Likewise, a number of authors have proven results about the m-dissections of certain infinite q-products using various methods. It is shown that many of these m-dissections also follow automatically (again simply by specializing parameters) from this same identity alluded to above. Two identities that mat be considered as extensions of two Identities of Ramanujan are also derived. It is also shown how applying similar ideas to certain other Lambert series gives rise to some rather curious q-series identities, such as, for any positive integer m, \frac(q,q,a,(q)/(a),(b q)/(d), (dq)/(b), (aq)/(b d), (b d q)/(a);q)∞ (b,(q)/(b),d,(q)/(d),(a)/(b),(bq)/(a),(a)/(d),(dq)/(a);q)∞ = ∑ r=0m-1 qr \frac (qm,qm,a q2 r,\fracqm-2 ra,(b qm)/(d),(d qm)/(b), (a qm)/(b d),(b dqm)/(a);qm)∞ (b qr,\fracqm-rb,d qr,\fracqm-rd,(a qr)/(b),\fracb qm-ra,(a qr)/(d), \fracdqm-ra;qm)∞ and (aq;q)∞∑n=1∞ \fracn an qn(q;q)n = ∑r=1m(aqr;qm)∞∑n=1∞ \fracnan qn r(qm;qm)n. Applications to the Fine function F(a,b;t) are also considered.