I'm quite curious about the following phenomena, that still puzzle me although I have a proof, and I'd be really glad if someone may shred some light, showing an interpretation or a generalization. I will sketch the computations at request, which consist on a manipulation of the integral formula of the remainder of the Taylor expansion.

1. Let $v\in C^\infty(\mathbb{R})$, vanishing at $0$ with some order $p\in\mathbb{N} _ +$ . In other words, the formal Taylor series of $v$ at $0$ belongs to the ideal $x^p\mathbb{R}[[x]]$ . Then, the function $w(x):=v(x)/x^p$ is also (extensible to) a $C^\infty(\mathbb{R})$ function, and we can express the $k-$ order derivative of $w$ at $x$ (say $x\ge0$) in terms of the derivatives of order $k+p$ of $v$ on $[0,x]$ as follows:

$$\frac{w^{(k)}(x)}{k!}=\frac{\int_0^x (x-s)^{p-1}s^k\, \frac{v^{(k+p)}(s)}{(k+p)!}ds}{\int_0^x (x-s)^{p-1}s^k\, ds} \, . $$ In other terms, the $k-$th Taylor coefficient of $w$ at $x$ is an integral mean of the $(k+p)-$th Taylor coefficients of $v$, weighted with a Beta distribution on $[0,x]$. This is quite clear if $x=0$ and $v$ is analytic there, and not immediately obvious in general, but has it a special meaning, or is it an instance of a more general principle?

2. Let $u\in C^\infty(\mathbb{R})$, and assume that the formal Taylor series of $u$ at $0$ belongs to $\mathbb{R}[[x^2]]$ . Then, the function $w(x):=u( \sqrt { x } ) $ is also (extendible to) a $C^\infty(\mathbb{R})$ function, and we can express the $k-$ order derivative of $w$ at $x^2$ in terms of the $2k-$ order derivatives of $v$ on $[0,x]$ as follows:

$$w^{(k)}(x^2)=\frac{\, (2x)^{-2k+1}}{(k-1)!\, }\, \int_0^x (x^2-t^2)^{k-1}u^{(2k)}(t) dt\, $$ (this may also be written as an equality relating Taylor coefficients by means of an integral mean).

3. There is also a more general statement for a function $w(x):=u(x^{1/p})$ for $p\in\mathbb{N} _ +$, assuming that the formal Taylor series of $u$ is in $\mathbb{R}[[x^p]]$; the $k-$th Taylor coefficient of $w$ at $x^p$ is then an integral mean of the $kp-$th Taylor coefficients of $v$, supported on $[0,x]$, with certain densities depending on $p$ and $n$ recursively defined. Is there a more general statement connecting analogously operations in $\mathbb{R}[[x]]$ and $C^\infty$ functions via integral means of their Taylor formal series?