Rational functions over $\mathbb{Z}$ yielding series that converge to a given real number

Question

I wonder what can be said about the set of rational functions over $\mathbb{Z}$ for which summation of the values at positive integers converges to a given real number. More precisely, this is about describing the following set for a given $c\in\mathbb{R}$ $$\mathcal{S}(c)=\left\{f\in\mathbb{Z}(X)\,\,\Big|\,\,\sum_{n=1}^{\infty}f(n)=c\right\}$$

For example, we have $1/X^2\in\mathcal{S}(\zeta(2))$ by definition.

If $\mathcal{S}_c$ is non-empty then it is infinite. This can be seen using telescopic sums, which show that with $f\in\mathcal{S}(c)$ this set also contains

• $T_1f:=X(f(X)-f(X+1))$, for example $$T_1\frac{1}{X^2}=\frac{2X+1}{X^3+2X^2+X}\in\mathcal{S}(\zeta(2))$$
• $T_2f:=\frac12((X+1)f(X)-Xf(X+1))$, for example $$T_2\frac{1}{X^2}=\frac{3X^2+2X+1}{2X^4+4X^3+2X^2 }\in\mathcal{S}(\zeta(2))$$

(I am aware that there is some abuse of notation going on here...)

Some questions that come to my mind are

1. How are different functions in $\mathcal{S}(c)$ related to each other? Is it always via elementary manipulation like $T_1,T_2$ above or, say, building $r_1f_1+\dots+r_nf_n\in\mathcal{S}(c)$ out of $f_i\in\mathcal{S}(c)$, where $r_i\in\mathbb{Q}$ with $\sum_ir_i=1$ (again abuse of notation)

2. Do all rational functions in $\mathcal{S}(c)$ have the same difference of degree between denominator and numerator polynomials? (e.g. is it 2 for all $f\in\mathcal{S}(\zeta(2))$?

3. What can be said about the set $\{c\in\mathbb{R}\,|\,\mathcal{S}(c)\neq\emptyset\}$? Is it true that it contains no non-zero rational number?

I'd be happy to learn what theory could be of help with such questions.

Answer 1

$S(c)$ is non-empty for all nonzero rational $c$. Note that if $$f(x)={1\over x(x+1)}={1\over x}-{1\over x+1}$$ then $\sum_1^{\infty}f(n)=1$, so if $c=a/b$, $a\ne0$, and $f(x)={a\over bx(x+1)}$ then $\sum_1^{\infty}f(n)=c$.

The degree difference is not constant for given $c$ since, e.g., $$\sum{1\over n(n+1)(n+2)(n+3)}={1\over18}$$

Note also that $S(0)$ is not just the zero function, as it contains, e.g., $$f(x)={1\over18x(x+1)}-{1\over x(x+1)(x+2)(x+3)}$$

A couple of examples with irrational sum:

$$\sum_1^{\infty}{1\over2n(2n-1)}=\log2,\qquad\sum_0^{\infty}{4\over(4n+1)(4n+2)(4n+3)}=\log2$$

$$\sum_0^{\infty}{8\over(4n+1)(4n+3)}=\pi,\qquad\sum_1^{\infty}{16\over(4n-3)(4n-1)(4n+1)}=\pi-2$$ so the degree difference is not constant for these values of $c$, either.