Universal quaternary mixed sums involving generalized 3-, 4-, 5- and 8-gonal numbers via products of Ramanujan's theta functions
Abstract
Generalized m-gonal numbers are those pm(x)= [ (m - 2)x2 - (m - 4)x ]/2 where x and m are integers with m ≥ 3. If any nonnegative integer can be written in the form apr(h)+bps(l)+cpt(m)+dpu(n), where a,b,c,d are positive integers, then we call apr(h)+bps(l)+cpt(m)+dpu(n) a universal quaternary sum. In this paper, we determine the universality of many quaternary sums when r,s,t,u ∈ \3,4,5,8\, using the theory of Ramanujan's theta function identities
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