The present paper is devoted to power series of SP type, i.e. with coefficients that are syntactically sum-product combinations. Apart from their applications to analytic knot theory and the so-called "Volume Conjecture", SP-series are interesting in their own right, on at least four counts: (i) they generate quite distinctive resurgence algebras (ii) they are one of those relatively rare instances when the resurgence properties have to be derived directly from the Taylor coefficients (iii) some of them produce singularities that unexpectedly verify finite-order differential equations (iv) all of them are best handled with the help of two remarkable, infinite-order integral-differential transforms, "mir" and "nir".
Power series with Taylor coefficients of sum-product type and algebraic differential equations /SP-series and the interplay between their resurgence and differential properties
Universal partial sums of Taylor series as functions of the centre of expansion
V. Nestorian证明，如果$Ω$是$\mathbb{C}$的一个不包含$0$的单连通子集，并且$S(Ω)$是所有函数$f\in \mathcal{H}(Ω)$的集合，具有集合$\left\{T_N(f)(z)\coloneqq\sum_{n=0}^N\dfrac{f^{(n)}(z)}{n!} (-z)^n : N = 0,1,2,\dots \right\}$在$\mathcal{H}(Ω)$中稠密的性质，则$S(Ω)$是$\mathcal{H}(Ω)$中稠密的$G_δ$集合。在$Ω$是一个不含$0$的开放圆盘$D(z_0,r)$的特殊情况下，我们肯定地回答这个猜想。
Taylor's series expansions for real powers of functions containing squares of inverse (hyperbolic) cosine functions, explicit formulas for special partial Bell polynomials, and series representations for powers of circular constant
