摘要：Ramanujan in his deathbed letter to GH Hardy concerned the asymptotic properties of modular forms and mock theta functions. For the mock theta function f ( q ) , he claimed that as q approaches an even order 2k root of unity ζ, lim q → ζ ( f ( q ) − ( − 1 ) k ( 1 − q ) ( 1 − q 3 ) ( 1 − q 5 ) ⋯ ( 1 − 2 q + 2 q 4 − ⋯ ) ) = O ( 1 ) , where ( 1 − q ) ( 1 − q 3 ) ( 1 − q 5 ) ⋯ ( 1 − 2 q + 2 q 4 − ⋯ ) = ∏ n = 1 ∞ 1 − q n ( 1 + q n ) 2 . Recently, Folsom, Ono and Rhoades have proved two closed formulas for the implied constant and formulated an open problem which is related to their two theorems. In this note, we give a new proof on the problem of the two theorems by using some results about the generating functions of convex compositions given by GE Andrews and Appell-Lerch sums. MSC:11F37, 11F03, 11F99.
关键词：modular forms ; mock theta functions ; generating functions
