On Asymptotics of the Jump of Highest Derivative for a Polynomial Spline

In this article, we obtain $2[n/2]+2$ terms ($[\boldsymbol{\cdot}]$ stands for the integer part) of the asymptotic expansion of the error
$$ \bigl(S^{(n)}({}\,\overline{\kern-.3mm x}_i+0)-S^{(n)}({}\,\overline{\kern-.3mm x}_i-0)\bigr)\big/h-f^{(n+1)}({}\,\overline{\kern-.3mm x}_i), $$
where $S(x)$ is a periodic spline of degree $n\ge 0$ and deficiency 1 that interpolates a periodic sufficiently smooth function $f(x)$ at the nodes $x_i$ ($i=0,\pm1,\dots$) of a uniform mesh of width $h$. The nodes of the spline are the points ${}\,\overline{\kern-.3mm x}_i=x_i+h\bigl(1+(-1)^n\bigr)/4$.
The expansion coefficients are represented explicitly in terms of the values of the Bernoulli polynomials at 0 for $n$ odd and 1/2 for $n$ even.