Translations AND dilations of infinite series

Question

Sometimes, when working with infinite series, it's useful to add "dilated" or "translated" versions of the infinite series, term by term, back to the original. There are ways of making this rigorous for each.

If you want to add a "translated" copy of a series to the original one, you can:

1. attach the series coefficients to a power series
2. multiply by $x^n$ to translate by a shift of n
3. add back to the original power series
4. take the limit as $x \to 1$ to recover the original series

If you want to add a "dilated" copy of a series to an original one, you can:

1. attach the series coefficients to a Dirichlet series
2. multiply by $\frac{1}{n^s}$ to dilate by a factor of n
3. add back to the original Dirichlet series
4. take the limit as $s \to 0$ to recover the original series

My question: What if you want to be able to translate and shift a series? Is there a clever thing you can "attach" the series coefficients to that will let you do these sorts of manipulations rigorously?

But is there some way this is commonly done?

Phrased equivalently in terms of convolution: the Cauchy and Dirichlet convolution rings are very useful. You can also toss both operations together to come up with a neat algebraic structure in which Dirichlet convolution distributes over Cauchy convolution (minus some quirks about how 0 is handled) and both distribute over addition. Is there some sort of natural choice of "formal series" for which there are two operations, one of which might be multiplication, that behave analogously to this three-operation algebraic structure?

Answer 1

Instead of converting a sequence $A=(a_n)_{n=0}^{\infty}$ into some kind of function, I would just specify what transformation you want to do on the series itself.

For example, a translation by $m$ could be written $T_m(A) =(0,0,...(m\ 0's), a_0, a_1, ...) $ or, via the indices, $T_m(A)_i =0 \ if\ i < m; \ otherwise\ A_{m-i} $.

For the "dilation", assuming I understand what you mean, $D_n(A)_i =a_{i/n} \ if \ n|i; \ otherwise\ 0 $.

This way you avoid spurious limiting processes and concentrate on what you want to do with the elements of the sequence.