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

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 $$C_{m,n}$$.

• What exactly do you mean by a "closed-form formula"?
– user
Mar 30 at 17:54
• @user A closed-form formula means a closed-form expression defined at en.m.wikipedia.org/wiki/Closed-form_expression Mar 30 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 at 19:10
• OP does not say $m$ is an integer. Mar 30 at 19:18
• @GEdgar That's true but usually the letter $m$ is used to designate integers.
– user
Mar 30 at 20:50

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.

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 doi.org/10.1515/dema-2022-0157 Mar 31 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. Mar 31 at 22:21