Powers of the eta-function and Hecke operators
Let \(\eta(z)\) be the Dedekind eta function. For an integer \(s>0\), let \(\delta(s)= 24/\gcd(s, 24)\) be the smallest integer such that the \(q\)-expansion of \(\eta^s(\delta(s)z)\) has integer exponents. The authors study the action of Hecke operators on negative powers \(1/\eta^s(\delta(s)z)\) of the eta function. For prime numbers \(l\) they present explicit formulas for \((1/\eta^s(\delta(s)z))|T(l^2)\) if \(s\) is odd and \(l^2\equiv 1\pmod{\delta(s)}\), and for \((1/\eta^s(\delta(s)z))|T(l)\) if \(s\) is even and \(l\equiv 1\pmod{\delta(s)}\). The formulas are generated through the use of Faber polynomials \(J(m; x)\) which can be defined by the expansion NEWLINE\[NEWLINE\sum^\infty_{m=0} J(m; x)q^m= {E^2_4(z) E_6(z)\over\Delta(z)}\cdot {1\over j(z)- x}NEWLINE\]NEWLINE and which have the following property: Let \(j(m; z)\) be the unique modular function on \(\mathrm{SL}_2(\mathbb Z)\) such that \(j(m; z)= q^{-m}+ O(q)\), so in particular \(j(1; z)= j(z)-744\); then we have \(J(m; j(z))= j(m; z)\). As for applications, the authors obtain congruences for powers of the \(\Delta\)-function and formulas which enable a fast calculation of vector partition numbers \(p_s(n)\) of length \(s\) whose generating function is NEWLINE\[NEWLINE\sum^\infty_{n=0} p_s(n) q^n= \prod^\infty_{n=1} (1- q^n)^{-s}.NEWLINE\]NEWLINE The results are obtained by extending methods of \textit{A. O. L. Atkin} [Proc. Lond. Math. Soc. (3) 18, 563--576 (1968; Zbl 0313.10025)] and \textit{K. Ono} [Adv. Math. 228, No. 1, 527--534 (2011; Zbl 1260.11061)].
scientific article; zbMATH DE number 6023774