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

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 $C_{m,n}$.

@user A closed-form formula means a closed-form expression defined at https://en.m.wikipedia.org/wiki/Closed-form_expression — qifeng618, Mar 30 '23 at 18:24
Since you know the expression $C_{1,n}$ (which is a closed-form expression), any $C_{m,n}$ can be written as a finite sum of finite products of such terms multiplied by the corresponding multinomial coefficients. — user, Mar 30 '23 at 19:10
@GEdgar That's true but usually the letter $m$ is used to designate integers. — user, Mar 30 '23 at 20:50
What I need is the type of the Maclaurin power series expansions at https://math.stackexchange.com/a/4657809, https://math.stackexchange.com/a/4661280, https://math.stackexchange.com/a/4379986, https://math.stackexchange.com/a/4379999, https://math.stackexchange.com/a/4380027 — qifeng618, Mar 31 '23 at 17:17
I need a closed-form expression for $C_{m,n}$, in which there are no too many sums and products. One example is: For $k\in\mathbb{N}$ and $|x|<1$, \begin{equation} \biggl(\frac{\arcsin x}{x}\biggr)^{k} =1+\sum_{m=1}^{\infty} (-1)^m\frac{Q(k,2m)}{\binom{k+2m}{k}}\frac{(2x)^{2m}}{(2m)!}, \end{equation} where \begin{equation} Q(k,m)=\sum_{\ell=0}^{m} \binom{k+\ell-1}{k-1} s(k+m-1,k+\ell-1)\biggl(\frac{k+m-2}{2}\biggr)^{\ell} \end{equation} for $k\in\mathbb{N}$ and $m\ge2$. — qifeng618, Mar 31 '23 at 20:46
That the function isn't presented in terms of elementary standard functions doesn't prove that it isn't an elementary function. — IV_, Apr 01 '23 at 11:51
@IV_ What is the meaning of your comment? Confirm the sentence “which confirms that the Gauss hypergeometric series … is not an elementary function”? Deny it? It seems that you neither confirm it nor deny it. Why don’t you confirm or deny it directly, explicitly, and clearly? Thanks! — qifeng618, Apr 01 '23 at 23:29

score 2 · Answer 1 · answered Mar 30 '23 at 19:36

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.

score 0 · Answer 2 · answered Mar 30 '23 at 21:26

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.

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

I need a closed-form expression for $C_{m,n}$. For example, for $k\in\mathbb{N}$ and $|x|<1$, $$ \biggl[\frac{(\arccos x)^{2}}{2(1-x)}\biggr]^k =1+(2k)!\sum_{n=1}^{\infty} \frac{Q(2k,2n)}{(2k+2n)!}[2(x-1)]^{n} $$ and $$ \biggl[\frac{(\operatorname{arccosh}x)^{2}}{2(x-1)}\biggr]^k =1+(2k)!\sum_{n=1}^{\infty} \frac{Q(2k,2n)}{(2k+2n)!}[2(x-1)]^{n}, $$ where $$ Q(k,m)=\sum_{\ell=0}^{m} \binom{k+\ell-1}{k-1} s(k+m-1,k+\ell-1)\biggl(\frac{k+m-2}{2}\biggr)^{\ell} $$ for $k\in\mathbb{N}$ and $m\ge2$. These series expansions can be found at https://doi.org/10.1515/dema-2022-0157 — qifeng618, Mar 31 '23 at 22:15
I need a closed-form expression for all the coefficients $C_{m,n}$, in which there are no too many sums and products, its form should be as simpler as possible, famous sequences, such as the Stirling numbers and polynomials, the Bernoulli numbers and polynomials, and central factorial numbers and polynomials, can be employed in the closed-from expression. — qifeng618, Mar 31 '23 at 22:21

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

2 Answers2