On the \(k\)th smallest part of a partition into distinct parts (Q6631606)

In this article, the authors discuss properties of a classic theorem of \textit{K. Uchimura} [J. Comb. Theory, Ser. A 31, 131--135 (1981; Zbl 0473.05006)], in which he re-discovered Ramanujan's elegant identity [\textit{S. Ramanujan}, Notebooks of Srinivasa Ramanujan. Vols. 1, 2. Bombay: Tata Institute of Fundamental Research (1957; Zbl 0138.24201)]; further he gave a new representation and discussed a particular case for \(c=1\). In light of \textit{G. E. Andrews} et al. [Acta Arith. 158, No. 3, 199--218 (2013; Zbl 1268.05019)] extension of \(FFW(n)=d(n)\), the authors explore another important extension of \(FFW(n)=\sum_{\pi\in{D(n)}}(-1)^{\#(\pi)-1}s(\pi)\), which was first defined by \textit{Z. B. Wang} et al. [Am. Math. Mon. 102, No. 4, 345--347 (1995; Zbl 0831.11056)]. The authors also obtain a close form of Uchimura's formula for the sum of fixed powers of the \(s(n)\) [\textit{K. Uchimura}, Discrete Appl. Math. 18, 73--81 (1987; Zbl 0629.10005)], and also found a new representation of a more general sum given by \textit{A. Agarwal} et al. [Ann. Comb. 28, No. 2, 555--574 (2024; Zbl 1547.11116)]. In order to establish required findings, some related results are also discussed.