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Through wolfram and wiki, I've learnt that these elementary functions have a representation as hypergeometric series:

$$_2F_1(\color{red}{1,1;2;-x})=\ln(x+1)$$

$$_2F_1(\color{red}{\color{red}{\frac{1}{2},\frac{1}{2};\frac{3}{2};x^2})}=\arcsin(x)$$

$$_2F_1(\color{red}{1,1;1;x})= \dfrac{1}{1-x}$$

I have never seen the hypergeometric series representation for $\sin(x)$ and $\cos(x)$. I wonder if trigonometric and hyperbolic functions have hypergeometric series' representation as those three functions? Do you know any table from mathematical handbook or references that list these special values of the Gauss hypergeometric function?

James Warthington
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  • Do you know any complex variable theory? There are theorems about the uniqueness of power series representations. – almagest Feb 07 '21 at 19:25
  • @almagest No, I haven't learnt complex analysis. Trust me, I wish I could. I have met tons of problems that can be answered by a course of complex analysis. Do you know where I can find those theorems? – James Warthington Feb 07 '21 at 19:26
  • @almagest I am principally interested in finding the particular value of hypergeometric function. – James Warthington Feb 07 '21 at 19:27
  • You could try the NIST Handbook of Mathematical Functiopns (a fairly large reference book). Chapter 15, specifically p386. But whiilst I applaud racing into new things, it might be better to do a basic course on Complex Variable theory first. – almagest Feb 07 '21 at 19:32
  • @almagest you are right, I should take that course. – James Warthington Feb 07 '21 at 19:33
  • @almagest Is it plain wrong to say that all elementary functions have a hypergeometric series representation? – James Warthington Feb 07 '21 at 19:36
  • "Almost all of the elementary functions of mathematics are either hypergeometric or ratios of hypergeometric functions." from: https://www.cambridge.org/core/books/special-functions/hypergeometric-functions/DB1B95210DC4E223D15E4112088ED29D – James Warthington Feb 07 '21 at 19:39
  • Related. – K.defaoite Feb 07 '21 at 19:50

2 Answers2

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Any sufficiently well-behaved function has a unique power series representation at a given point (such as the origin). The theory about that is known as Complex Variable theory and is one of the spectacular jewels of mathematics. I strongly recommend taking a basic course in it.

Unfortunately, maths is such a vast field that the hypergeometric and other "classical" functions have got squeezed out of most undergraduate math courses. There are relatively few modern textbooks on them. The old classic is Whittaker and Watson (A course in Modern Analysis - but it is far from modern in its flavour).

If you really want to look up details the NIST Handbook of Mathematical Functions is a great (but large and fairly expensive) reference book. Chapter 15, p386 gives plenty of examples of elementary functions that can be written as special cases of the hypergeometric function.

But not all "elementary" functions can be written as hypergeometric functions (and anyway "elementary" is not a preciselyy defined term). To go into this properly you have to look at the associated differential equations. But I strongly recommend against racing ahead until you have done a little more of the undergraduate syllabus.

almagest
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  • Let me tell how naive I am and how clueless a math student I am. I found this series in a book by Eugene Catalan: https://math.stackexchange.com/questions/3496698/how-to-derive-this-series/3496842#3496842 – James Warthington Feb 07 '21 at 21:28
  • So I have been tinkering with hypergeometric series and obtain many simple identities that can be easily verified by Wolfram. I thought hypergeometric function is just a magical formula which produce, at special arguments, power series of elementary functions. Any relationship between hypergeometric series and differential equations or more advanced Math topic is unknown to me. I am that naive and stupid. – James Warthington Feb 07 '21 at 21:31
  • But I have never seen a hypergeometric series representation of $\sin(x)$ and $\cos(x)$. So I asked it here. Naturally, I wonder if almost all elementary function possess a hypergeometric series representation. – James Warthington Feb 07 '21 at 21:32
  • @JamesWarthington please don't criticize yourself like that; I'd say the fact that you are even somewhat familiar with hypergeometric series is something to be proud of! Remember: one of the best ways of learning is through your mistakes. – A-Level Student Feb 09 '21 at 10:37
  • @A-LevelStudent Hey man, my favourite poster. Thank you for your kind word. I am still very impressed with your answer regarding proving the Euler formula. I wish you all the best. – James Warthington Feb 09 '21 at 10:55
  • @JamesWarthington :) Thanks, you too! – A-Level Student Feb 09 '21 at 10:56
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From ${_0F_1}(;\frac12;z)=\cosh(2\sqrt{z})$ and ${_0F_1}(;\frac32;z)=\sinh(2\sqrt{z})/(2\sqrt{z})$ we find

$$ \sin x=x \cdot {_0F_1}\!\left(;\frac32;-\frac{x^2}{4}\right)\\ \cos x={_0F_1}\!\left(;\frac12;-\frac{x^2}{4}\right) $$

Gary
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Roman
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    The OP is asking for representations in terms of the Gauss hypergeometric function. – Gary Jul 03 '23 at 08:23
  • Fair point, hidden at the end of the question; but the title asks for a "hypergeometric series" which $_0F_1$ delivers. – Roman Jul 03 '23 at 08:56
  • How about Mellin–Barnes integrals? $\sin(x)=\sqrt{\pi}G\left({{},{}},\left(\begin{array}{c}\frac12 \ 0 \end{array}\right),\frac{x}{2},\frac12\right)=\frac{\sqrt{\pi}}{2} H\left({{},{}},\left(\begin{array}{c}\left{\frac12,\frac12\right} \ \left{0,\frac12\right} \end{array}\right),\frac{x}{2}\right)$ is what Mathematica can do with MeijerGReduce and FoxHReduce, and similar with $\cos(x)$ etc. – Roman Jul 03 '23 at 13:14