Product partitions and recursion formulae

Utilizing a method briefly hinted in the author’s paper written in 1991 jointly with V. C. Harris, we derive here a number of unpublished recursion formulae for a variety of product partition functions which we believe have not been considered before in the literature. These include the functionsp∗(n;k,h) (which stands for the number of product partitions ofn> 1 into k parts of which h are distinct), and p∗ (d)(n;m) (which stands for the number of product partitions of n into exactly m parts with at most d repetitions of any part). We also derive recursion formulae for certain product partition functions without the use of generating functions.