# Is the Gauss hypergeometric series ${}_2F_1\bigl(\frac{1}{2},\frac{1}{2};2;t\bigr)$ an elementary function?

For $$\alpha,\beta\in\mathbb{C}$$ and $$\gamma\in\mathbb{C}\setminus\{0,-1,-2,\dotsc\}$$, Gauss' hypergeometric function $${}_2F_1(\alpha,\beta;\gamma;z)$$ can be defined by the series $$$$\label{Gauss-HF-dfn} {}_2F_1(\alpha,\beta;\gamma;z)=\sum_{n=0}^{\infty}\frac{(\alpha)_n(\beta)_n}{(\gamma)_n}\frac{z^n}{n!},\quad |z|<1.$$$$ The following special cases are well-known: \begin{align*} {}_2F_1(a,b;b;z)&=\frac{1}{(1-z)^a},\\ {}_2F_1(1,1;2;z)&=-\frac{\ln(1-z)}{z},\\ {}_2F_1\biggl(\frac12,1;\frac32;z^2\biggr)&=\frac1{2z}\ln\frac{1+z}{1-z},\\ {}_2F_1\biggl(\frac12,1;\frac32;-z^2\biggr)&=\frac{\arctan z}{z},\\ {}_2F_1\biggl(\frac12,\frac12;\frac32;z^2\biggr)&=\frac{\arcsin z}{z},\\ {}_2F_1\biggl(\frac12,\frac12;\frac32;-z^2\biggr)&=\frac{\ln\bigl(z+\sqrt{1+z^2}\bigr)}{z}. \end{align*} See Chapter 5 and page 109 in the book [1] below.

Lemma 2.6 in the paper [2] below reads that, for $$0\ne|t|<1$$ and $$n=1,2,\dotsc$$, $$\begin{equation*} {}_2F_1\biggl(\frac{1-n}{2}, \frac{2-n}{2};1-n;\frac1{t^2}\biggr) =\frac{t}{2^n\sqrt{t^2-1}\,} \biggl[\biggl(1+\frac{\sqrt{t^2-1}\,}{t}\biggr)^n -\biggl(1-\frac{\sqrt{t^2-1}\,}{t}\biggr)^n\biggr]. \end{equation*}$$

Corollary 4.1 in the paper [3] below states that, for $$n=0,1,2,\dotsc$$, $$\begin{multline}\label{Gauss-HF-Spec-Value} {}_2F_1\biggl(n+\frac{1}{2},n+1;n+\frac{3}{2};-1\biggr) =\frac{(2n+1)!!}{(2n)!!}\frac{\pi}{4}\\ +\frac{2n+1}{2^{2n}}\sum_{k=1}^{n} (-1)^{k} \binom{2n-k}{n} \frac{2^{k/2}}{k}\sin\frac{3k\pi}{4}. \end{multline}$$

My question is: can one find an elementary function $$f(t)$$ such that $$\begin{equation*} {}_2F_1\biggl(\frac{1}{2},\frac{1}{2};2;t\biggr)=f(t), \quad |t|\le1? \end{equation*}$$ In other words, is the Gauss hypergeometric series $$_2F_1\bigl(\frac{1}{2},\frac{1}{2};2;t\bigr)$$ an elementary function?

References

1. N. M. Temme, Special Functions: An Introduction to Classical Functions of Mathematical Physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996; available online at http://dx.doi.org/10.1002/9781118032572.
2. Feng Qi, Qing Zou, and Bai-Ni Guo, The inverse of a triangular matrix and several identities of the Catalan numbers, Applicable Analysis and Discrete Mathematics 13 (2019), no. 2, 518--541; available online at https://doi.org/10.2298/AADM190118018Q.
3. Feng Qi and Mark Daniel Ward, Closed-form formulas and properties of coefficients in Maclaurin's series expansion of Wilf's function composited by inverse tangent, square root, and exponential functions, arXiv (2022), available online at https://arxiv.org/abs/2110.08576v2.
• The motivation and background of this problem 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, vol. 6 (2022), no. 4, 528--537; available online at doi.org/10.31197/atnaa.1170948. Oct 11, 2022 at 1:50
• I think it would help you most if you'd ask the second question separately. Since you've already accepted an answer for this one, it is unlikely the second question will garner a lot of attention Mar 29 at 10:35
• @MaxMuller Thank you for your suggestion. I have asked the second question at math.stackexchange.com/q/4669567 separately. Mar 30 at 16:42

Maple does it in terms of complete elliptic integrals $$\rm{K}$$ and $$\rm{E}$$ ... $${\mbox{_2F_1}\left(\frac12,\frac12;\,2;\,t\right)}={\frac {4\left( t-1 \right){\rm K} \left( \sqrt {t} \right) +4{\rm E} \left( \sqrt {t} \right) }{t\pi}} \tag1$$ But that does not show it is elementary. In fact, I suspect it is not elementary.

Recall the known formulas: $$\rm{K}\big(\sqrt{t}\big) = \frac{\pi}{2}\; {}_2F_1\left(\frac12 , \frac12 ; 1 ; t\right) , \\ \rm{E}\big(\sqrt{t}\big) = \frac{\pi}{2}\; {}_2F_1\left(-\frac12 , \frac12 ; 1 ; t\right) .$$ By themselves, they are not elementary. $$(1)$$ should follow from these two and a contiguous formula for the hypergeometrics. $${c\;\mbox{_2F_1}(a-1,b;\,c;\,x)} + \left( x-1 \right) c\;{\mbox{_2F_1}(a,b;\,c;\,x)} + \left( b-c \right) x\;{\mbox{_2F_1}(a,b;\,c+1;\,x)} =0$$

• Dear Peofessor Edgar, I recited your answer in Section 4.2 of my 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, vol. 6, no. 4, 528--537 (2022); available online at doi.org/10.31197/atnaa.1170948 ]. Thank you very much for your answer to my question. @GeraldEdgar Mar 30 at 17:02

That the function isn't presented in terms of elementary standard functions doesn't prove that it isn't an elementary function.

If the function is an elementary function, it's a Liouvillian function. I want to point out some perhaps applicable methods for looking if this function is an elementary function.

a)
We could check the known algorithms for solving recurrence equations.
We could check the known algorithms for symbolic summation, e.g. Gosper's algorithm.
see e.g. my answer at https://math.stackexchange.com/questions/1743544/are-there-some-techniques-which-can-be-used-to-show-that-a-sum-does-not-have-a/2329022#2329022

b)
$$_2F_1\left(\frac{1}{2},\frac{1}{2};2;t\right)=\frac{2}{\pi}\int_0^1\sqrt{\frac{1-\alpha}{\alpha-\alpha^2t}}d\alpha$$ We see, this is at least a Liouvillian function. We could check with Risch algorithm if $$\int\sqrt{\frac{1-\alpha}{\alpha-\alpha^2t}}d\alpha$$ is an elementary function.

Risch algorithm can decide if an indefinite integral is elementary or not. FRICAS has implemented Risch algorithm the furthest, but not yet completely. If a FRICAS result is elementary, we have an evidence. But if it is not obviously elementary, we therefore have to look further. see Abbasi, N. M.: Computer Algebra Independent Integration Tests

FRICAS 1.3.8:

integrate(((1-a)/(a-a^2*t))^(1/2),a)

(1)

   -
6 t
*
weierstrassZeta
2
4 t  - 4 t + 4
--------------
2
3 t

,
3       2
8 t  - 12 t  - 12 t + 8
-----------------------
3
27 t
,
weierstrassPInverse
2
4 t  - 4 t + 4
--------------
2
3 t
,
3       2
8 t  - 12 t  - 12 t + 8
-----------------------
3
27 t
,
(3 a - 1)t - 1
--------------
3 t
+
(- 4 t + 2)
*
weierstrassPInverse
2
4 t  - 4 t + 4
--------------
2
3 t
,
3       2
8 t  - 12 t  - 12 t + 8
-----------------------
3
27 t
,
(3 a - 1)t - 1
--------------
3 t


/

      +-+

2 |1
3 t  |-
\ |t



Type: Union(Expression(Integer),...)

I don't know how to read the last part of this result.

It looks like the calculated antiderivative is not an elementary function, because the Weierstrass functions are not elementary functions. But give the arguments of the Weierstrass functions elementary functions?

We have to apply the integration limits now. Does that give an elementary function?

c)
Further approaches are the methods about elementary and Liouvillian integrals/solutions of linear differential equations of Davenport, Rosenlicht, Singer, Ulmer 1981 - 1992 and others.

• Concretely speaking, do you have a confirmative answer to my question? By the theory you stated here, is the Gauss hypergeometric series $_2F_1\bigl(\frac{1}{2},\frac{1}{2};2;t\bigr)$ an elementary function? If yes, can you write out the elementary function? Thanks. @IV_ Mar 31 at 19:10
• I don’t find any useful stuff in your answer and comments, although I believe that what you wrote are all right. Could you please give a full and direct solution? Otherwise, don’t ask me to do this or that. Please answer yes or no: confirm or deny. Apr 1 at 18:41
• @qifeng618 You should look more carefully at the references given in this answer. Khovanskii's book and the last references are relevant to the question since the Gauss hypergeometric function is a solution to a linear differential equation. Apr 1 at 22:38
• @FrançoisBrunault I read carefully the paper by Choi in Item b) and found nothing related to my question. If there were some answer to my question in those monographs, I hope to know yes or no about the question at first, because you are experts in the areas and I need your expertise help. Thank you very much. Apr 1 at 23:06
• @IV_ My question is a very concrete problem, your answers are several theories of mathematics. By your expertise, could you please confirm or deny my question directly, explicitly, and clearly? Thanks! Apr 1 at 23:13