Taylor series of an implicit function

Question

Suppose the function $s:[-\delta, \delta] \to \mathbb{R}$, $\delta > 0$, is defined implicitly by

$$s(t) = 1 - c\beta t (s(t))^{\beta}$$

for some $c > 0$, $0 <\beta < 1$. Can an explicit description of the coefficients of the Taylor series of $s$ around $0$ be given?

Answer 1

You can find the coefficients recursively. Let $s(t)=\sum_{n=0}^\infty s_n\,t^n$. Setting $t=0$ we get $s(0)=s_0=1$. Derivate with respect to $t$ the defining equation and again set $t=0$: $$ s'=-c\,\beta\,s^\beta-c\,\beta^2\,t\,s^{\beta-1}\,s'\implies s_1=s'(0)=-c\,\beta. $$ Derivating again with respect to $t$: $$ s''=-2\,c\,\beta^2\,s^{\beta-1}\,s'-c\,\beta^2\,(\beta-1)\,t\,s^{\beta-2}\,s'-c\,\beta^2\,t\,s^{\beta-1}\,s''\implies s_2=\frac{s''(0)}{2}=c^2\,\beta^3. $$ Repeat the process until you have enough coefficients.

Another approach is to write $$ s^\beta=\Bigl(1+\sum_{n=1}^\infty s_n\,t^n\Bigr)^\beta=\sum_{k=0}^\infty\binom{k}{\beta}\Bigl(\sum_{n=1}^\infty s_n\,t^n\Bigr)^k, $$ and from here obtain a system of equations for the coefficients. I have implemented it in Mathematica, obtaining $$\begin{align} s_3&=\frac12\,c^3\,\beta^4(1- 3 \beta)\\ s_4&=\frac13\,c^4\,\beta^5 (1 - 6\,\beta + 8\,\beta^2) \end{align}$$

Conjecture $$ s_n=\frac1{n-1}\,c^n\,\beta^{n+1}\,P_n(\beta), $$ with $P_n$ a polynomial on $\beta$ of degree $n-2$ and $P_n(0)=1$.

Mathematica code

n is the number of computed coefficients, b is $\beta$ and the coeficients are a[1],...,a[n].

n=6;
coef=Array[a,n];
s=1+coef.t^Range[n];
coef2=Rest[CoefficientList[Normal[Series[1-c b t s^b,{t,0,n}]],t]];
Solve[coef==coef2, coef]//Simplify