Is there a closed-form formula for coefficients in Maclaurin expansion of the function $\bigl[{}_2F_1\bigl(\frac{1}{2},\frac{1}{2};2;t\bigr)\bigr]^m$?

Question

At the site https://mathoverflow.net/a/423802, Professor Emeritus Gerald A. Edgar gave that \begin{equation*} {}_2F_1\biggl(\frac{1}{2},\frac{1}{2};2;t\biggr) =\frac{4}{\pi}\biggl[\biggl(1-\frac{1}{t}\biggr)K\bigl(\sqrt{t}\ \bigr) +\frac{1}{t}E\bigl(\sqrt{t}\ \bigr)\biggr], \end{equation*} which confirms that the Gauss hypergeometric series ${}_2F_1\bigl(\frac{1}{2},\frac{1}{2};2;t\bigr)$ is not an elementary function, where $K(t)$ and $E(t)$ denote the complete elliptic integrals of the first and second kinds respectively. This gives an answer to my question at https://mathoverflow.net/q/423800.

Suggested by Max Muller (https://stackexchange.com/users/510306/max-muller), I ask the following question separately.

Can one write out a closed-form formula for the general term of the coefficients in the Maclaurin power series expansion of the power function \begin{equation*} \biggl[{}_2F_1\biggl(\frac{1}{2},\frac{1}{2};2;t\biggr)\biggr]^m, \quad m\ge1? \end{equation*} In other words, does there exist a closed-form expression for all the coefficients $C_{m,n}$ in the Maclaurin power series expansion \begin{equation*} \biggl[{}_2F_1\biggl(\frac{1}{2},\frac{1}{2};2;t\biggr)\biggr]^m =\sum_{n=0}^{\infty}C_{m,n}\frac{t^n}{n!} \end{equation*} for $m\ge1$?

The motivations of these problems can be found in the paper

1. Wei-Shih Du, Dongkyu Lim, and Feng Qi, Several recursive and closed-form formulas for some specific values of partial Bell polynomials, Advances in the Theory of Nonlinear Analysis and its Applications 6 (2022), no. 4, 528--537; available online at https://doi.org/10.31197/atnaa.1170948.

By the way, I can derive a recursive relation for the coefficients $C_{m,n}$. However, I am very interested in a possible closed-form formula for all the coefficients $C_{m,n}$.

Answer 1

remark
I worked with ${}_2F_1(\frac12,\frac12;2;16t)^m = \sum_{n=0}^\infty D_{m,n} t^n$ to get integer coefficients $D_{m,n} = 16^nC_{m,n}$. A recurrence for $D_{2,n}$ is $$ \left( 256\,{n}^{3}+768\,{n}^{2}+768\,n+256 \right) D_{2,n} + \left( -32\,{n}^{3}-192\,{n}^{2}-368\,n-232 \right) D_{2,n+1} + \left( {n}^{3}+9\,{n}^{2}+26\,n+24 \right) D_{2,n+2}=0 ,\\ D_{2,0} =1,\quad D_{2,1} =4 . $$ Maple's command rsolve does not find a closed form solution for this.

Answer 2

After writing this answer I realized OP made mention of the recursive definition of the coefficients. I will leave this here as an extended comment.

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Powers of power series can themselves be expressed as power series. In particular, if $$ F\left({1/2,1/2\atop 2};t\right)=\sum_{n=0}^\infty\frac{(1/2)_n(1/2)_n}{(2)_n(1)_n}t^n, $$ then using this expression we obtain $$ F^m\left({1/2,1/2\atop 2};t\right)=\sum_{n=0}^\infty c_{m,n}t^n, $$ where $c_{m,0}=1$ and $$ c_{m,n}=\frac{1}{n}\sum_{k=1}^n(mk-n+k)\frac{(1/2)_k(1/2)_k}{(2)_k(1)_k}c_{m,n-k.} $$ Implementing in Mathematica with

c[m_, 0] := 1;
c[m_, n_] := 
  1/n Sum[(m k - n + k) Pochhammer[1/2, k]^2/(Pochhammer[2, k] k!)
      c[m, n - k], {k, 1, n}];
Fm[m_, t_, K_] := Sum[c[m, n] t^n, {n, 0, K}];

we find for the first six terms $$ \left( \begin{array}{cc} n &c_{m,n}\\0 & 1 \\ 1 & \frac{m}{8} \\ 2 & \frac{1}{128} m (m+5) \\ 3 & \frac{m (m (m+15)+59)}{3072} \\ 4 & \frac{m (m (m (m+30)+311)+1128)}{98304} \\ 5 & \frac{m (m (m (m (m+50)+965)+8590)+30084)}{3932160} \\ \end{array} \right) $$

Here is a plot for $m=2$ comparing the square of the hypergeometric function against the series expansion truncated to the first six terms