Analytic proof of the partition identity \(A_{5,3,3}(n)=B_{5,3,3}^{0}(n)\) (Q2426707)

For an even integer \(\lambda\), let \(A_{\lambda,k,a}(n)\) denote the number of partitions of \(n\) into parts such that no part \(\not\equiv 0\bmod(\lambda+1)\) may be repeated and no part is \(\equiv 0\), \(^+_-(a-\lambda/2)(\lambda+1)\bmod (2k-\lambda+1)(\lambda+1)\). For an odd integer \(\lambda\), let \(A_{\lambda,k,a} (n)\) denote the number of partitions of \(n\) into parts such that no part \(\not\equiv 0\pmod(\lambda+ 1)/2\) may be repeated, no part is \(\equiv\lambda+1\bmod (2\lambda+ 2)\) and no part is \(\equiv 0\), \(\pm(2a-\lambda)(\lambda+1)/2\bmod(2k -\lambda+1)(\lambda+1)\). Let \(B_{\lambda,k,a}(n)\) denote the number of partitions of \(n\) of the form \(b_1+\cdots+b_s\) with \(b_i\geq b_{i+1}\), no part \(\not\equiv 0 \bmod(\lambda+1)\) is repeated, \(b_i-b_{i+k-1}\geq\lambda+1\) with strict inequality if \(\lambda+1\) divides \(b_i\), \(f_j+\cdots+f_{\lambda-j+1}\leq a-j\) for \(1\leq j\leq(\lambda+1)/2\) and \(f_1+\cdots+f_{\lambda+1}\leq a-1\) where \(f_i\) is the number of appearances of \(i\) in the partition. Inspired by a result of \textit{G. E. Andrews, C. Bessenrodt} and \textit{J. B. Olsson} [Trans. Am. Math. Soc. 344, 597--615 (1994; Zbl 0806.05065)], \textit{Padmavathamma, M. Ruby Salestina} and \textit{S. R. Sudarshan} [in: Adhikari, S. D. (ed.) et al., Number theory, Ramanujan Math. Soc. Lect. Notes Ser. 1, 57--70 (2005; Zbl 1156.11340)] proved that \(A_{5,3,3}(n)= B^0_{5,3,3}(n)\), where \(B^0_{5,3,3}(n)\) denotes the number of partitions of \(n\) enumerated by \(B_{5,3,3}(n)\) with the added restrictions \(f_{6j+3}=0\), \(f_{6j+2}+f_{6j+4}\leq 1\), \(f_{6j+5}+ f_{6j+7}\leq 1\) for \(j\geq 0\), and \(f_{6j-1}+f_{6j}+f_{6j+6}+f_{6j+7}\leq 3\) for \(j\geq 1\). In the paper under review the authors give an analytic proof for a refinement of this partition identity.