# Sonin's identity

The following identity attributed to N.Y. Sonin states the following:

Suppose $$f\in C^2[a,b]$$. Let $$\rho(x)=\frac12-\{x\}$$, where $$\{x\}$$ is the fractional part of $$x$$, and $$\sigma(x)=\int^x_0\rho(t)\,dt$$. Then $$\sum_{a< n\leq b}f(n)=\int^b_a f(t)\,dt +\rho(b-)f(b)-\rho(a)f(a)-\big(\sigma(b)f'(b)-\sigma(a)f'(a)\big) +\int^b_a\sigma(t)\,f''(t)\,dt$$ where summation runs over all integers between $$a$$ and $$b$$.

This looks like integration by parts and Abel summation kind of thing. I tried to apply Riemann-Stieltjes formula directly but this did not quite work. Hints (or a sketch of proof) would be appreciated.

\begin{align} \int_a^b \sigma(t) f''(t) \, dt &= [\sigma(t) f'(t)]_a^b - \int_a^b \rho(t) f'(t) \, dt \\&= [\sigma(t) f'(t)]_a^b - [\rho(t) f'(t)]_{a+}^{b-} + \int_{(a,b)} f(t) \, d\rho(t) . \end{align} Also if $$a < x < b$$, then $$\lfloor x\rfloor = - \lfloor a \rfloor + \sum_{a where $$H(x) = I_{x \ge 0} .$$ So $$\rho(t) = \tfrac12 - x + \lfloor x\rfloor = \tfrac12 - \lfloor a \rfloor - x + \sum_{a < n < b} H(x-n).$$ Finally, if $$a, then $$\int_a^b f(t) \, dH(t-n) = f(n) .$$
• First, notice that $$\rho$$ is a $$1$$-periodic function, and that $$\rho'(t)=-1$$ for $$x\in [k,k-1)$$, $$k\in\mathbb{Z}$$. For $$k\leq \alpha, use integration by parts twice (once with $$u=f(t)$$ and $$dv=\rho'(t)\,dt$$; and another with $$u=f'(t)$$ and $$dv=\sigma'(t)\,dt=\rho(t)\,dt$$) to get
\begin{align} -\int^\beta_\alpha f(t)\,dt &= \int^\beta_\alpha f(t)\rho'(t)\,dt\\ &=\rho(\beta-)f(\beta)-\rho(\alpha)f(\alpha)-\int^\beta_\alpha \rho(t)\,f'(t)\,dt\\ &=\rho(\beta-)f(\beta)-\rho(\alpha)f(\alpha)-\big(\sigma(\beta)f'(\beta)-\sigma(\alpha)f(\alpha)-\int^\beta_\alpha \sigma(t)\,f''(t)\,dt\big) \end{align} The use of a left limit in $$\rho(\beta-)$$ comes form Lebesgue's integration by parts formula.
• Applying this formula to each subintervals $$[k,k+1]\subset(a,b]$$ with $$k\in\mathbb{Z}$$, and the potentially the fractional intervals $$(a,\lfloor a\rfloor +1]$$, and $$[\lfloor b\rfloor,b]$$, and