In recent years I discovered a few Euler product transforms which apply to basic hypergeometric series (q-series). I found that applying the transforms to well known results such as the q-binomial theorem, or to results like Heine's q-series version of the Gauss hypergeometric series, led me to new sums of Dirichlet series.
We use the notations for a given integer factorization into primes
and say that
which defines the Jordan totient function, and
which is the sum of powers of divisors function. After applying the discovered transform, we obtain trivially,
which are both well-known results. Continuing this way, we obtain the new results
which are relevant to the concept of "sum of powers of divisors" associated with points on a finite cubic or hypercubic lattice. Further results related to these, which involve the
Liouville function
, are
which are obtainable from known techniques. However, the application of the transform extends to the new results
These two results, together with the following two here, are Dirichlet series analogues of the q-binomial theorem
Some examples of cases of the D-analogue of the Gauss hypergeometric series summation (and also for the "Heine q-summation") are
These and other results are presented in my papers.
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