Thue–Morse Series and k-ary Partitions

We show that the product of the generating function of k-ary partitions with the Thue–Morse series whose coefficients are kth roots of unity defines a k-regular sequence, given explicitly as a product over the digits of the base-k expansions. We study this sequence and highlight similarities with classical congruence results for k-ary partitions. We provide an elementary proof that the unrestricted k-ary partition function is not k-regular. Finally, we study the inverse of the Thue–Morse series, derive explicit recurrence relations for its coefficients, and show that this inverse cannot be k-regular.