2021/08/29 by Xiaojing Chen, Chen, Xiaojing, Wenchang Chu +1
Mathematics · #05A30 #11B65 #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2108.12796
openalex publication_date 2021/08/29 · openalex created_date 2021/09/13 · openalex updated_date 2026/07/28
By making use of the multiplicate form of the extended Carlitz inverse series relations, we establish two general `dual' theorems of Jackson's summation formula for well--poised 8ϕ7-series. Their duplicate forms under the partition pattern n=\lfloor\fracn2\rfloor+\lfloor\fracn+12\rfloor are explored and yield numerous q-series identities whose limiting cases as q→1 result in classical π-related Ramanujan--like series of convergence rate ``\frac116" including one for 1/π2 discovered by Guillera (2003). The triplicate dual formulae under the partition pattern n=\lfloor\fracn3\rfloor+\lfloor\fracn+13\rfloor+\lfloor\fracn+23\rfloor are examined via the ``reverse bisection method", which leads us to twenty new q-series identities together with their classical counterparts of convergence rate ``(-1)/(27)" when q→1.