expanded in the form of Then for any Lambert series generating the sequence of , we have the corresponding inversion relation of the factorization theorem expanded above given by This work on Lambert series factorization theorems is extended in to more general expansions of the form where is any (partition-related) reciprocal generating function, is any arithmetic function, and where the modified coefficients are expanded by The corresponding inverse matrices in the above expansion satisfy so that as in the first variant of the Lambert factorization theorem above we obtain an inversion relation for the right-hand-side coefficients of the form Recurrence relations Within this section we define the following functions for natural numbers : We also adopt the notation from the previous section that where is the infinite q-Pochhammer symbol. Then we have the following recurrence relations for involving these functions and the pentagonal numbers proved in: Derivatives Derivatives of a Lambert series can be obtained by differentiation of the series termwise with respect to . We have the following identities for the termwise derivatives of a Lambert series for any where the bracketed triangular coefficients in the previous equations denote the Stirling numbers of the first and second kinds. We also have the next identity for extracting the individual coefficients of the terms implicit to the previous expansions given in the form of Now if we define the functions for any by where denotes Iverson's convention, then we have the coefficients for the derivatives of a Lambert series given by Of course, by a typical argument purely by operations on formal power series we also have that See also Erdős–Borwein constant Arithmetic function Dirichlet convolution References Analytic number theory Q-analogs Mathematical series