## One Variable Hypergeometric Functions

For more than 200 years some of the greatest mathematicians in history have studied the hypergeometric functions in their one variable context.   You’ll find them covered in all sorts of mathematical works ranging from Euler to Riemann. However it was Barnes and Mellin who initially studied the integral representations whilst Goursat investigated their special properties.

It can be difficult to classify the different type of hypergeometric functions however there are three main categories:

• Functions defined as integrals.
• Solutions to systems of differential equations.
• Functions to series whose coefficients satisfy specific recursion properties.

Of course readers are probably familiar with the Mellin Barnes integral which defines these specific integrals. One variable functions have been understood for many years but mathematicians have now developed these approaches to cover multiple variables which of course yield slightly different results.    There are some interesting programmes on the UK Open University covering this, which may be available online – this link helps access the BBC since it was blocked abroad.

So while the study of the single variable functions is well over two centuries old, the study of the multivariate function is more recent. There has been something of a resurge in interest in the study of hypergeometric functions particularly in the last few years. The interest mainly is around the connections between hypergeometric functions and other areas of mathematics.

These include interesting areas such as investigating connections with such areas as algebraic geometry, symmetry and number theory for example. One of the key developments has come in the 80s and 90s through the work of Kapranov and Zelevinsky among others. In recent years this has been expanded by Saito, Sturmfels and Takayama.

## The Breakthrough of Irrational Numbers

Many centuries ago, Pythagoras was one of the first people to make the claim that the Universe was governed by numbers.  However at the time, numbers were understood to be whole numbers and ratios of these numbers (fractions).   However mathematics had a surprise in store for this famous mathematician – there are numbers which don’t come under either of these two categories.    They are called irrational numbers and they can be difficult to comprehend as they cannot be written down as decimals or even as recurring decimals.

For example you can in fact easily define an recurring decimal in a relatively straight forward way.   Even infinitely recurring decimals can easily be expressed as a fraction if needed.  However if you try and express and irrational number as a decimal you end up with a number which continues indefinitely without a regular or consistent pattern. The concept of these ‘irrational numbers’ was a huge breakthrough for mathematicians who were trying to look beyond whole numbers and their associated fractions.   They were considered a discovery though as Leopold Kronecker stated in the 19th century – “God made the integers, all the rest are the work of man.”

The most famous irrational number is of course, Pi.  In schools it is usually approximated to a more friendly 3.14 or 3 1/7 however the true value of Pi is nearer 3.14159265358979323846 however even this is only an approximation of it’s true value.   In reality Pi can never actually be written down exactly as the decimal places go on forever without and consistent pattern.  It still useful though and these numbers are frequently used as RNG seed numbers for things like encryption and ciphers like this Smart DNS software here.

Here’s a starter of a few hundred decimal places for Pi-

3.141592653589793238462643383279502884197169399375105820974944592307816406286 208998628034825342117067982148086513282306647093844609550582231725359408128481 117450284102701938521105559644622948954930381964428810975665933446128475648233 786783165271201909145648566923460348610454326648213393607260249141273724587006 606315588174881520920962829254091715364367892590360011330530548820466521384146 951941511609433057270365759591953092186117381932611793105118548074462379962749 567351885752724891227938183011949129833673362440656643086021394946395224737190 702179860943702770539217176293176752384674818467669405132000568127145263560827 785771342757789609173637178721468440901224953430146549585371050792279689258923 542019956112129021960864034418159813629774771309960518707211349999998372978049 951059731732816096318595024459455346908302642522308253344685035261931188171010 003137838752886587533208381420617177669147303598253490428755468731159562863882 353787593751957781857780532171226806613001927876611195909216420198938095257201 065485863278865936153381827968230301952035301852968995773622599413891249721775 283479131515574857242454150695950829533116861727855889075098381754637464939319 255060400927701671139009848824012858361603563707660104710181942955596198946767 837449448255379774726847104047534646208046684259069491293313677028989152104752 162056966024058038150193511253382430035587640247496473263914199272604269922796 782354781636009341721641219924586315030286182974555706749838505494588586926995 690927210797509302955321165344987202755960236480665499119881834797753566369807 426542527862551818417574672890977772793800081647060016145249192173217214772350 141441973568548161361157352552133475741849468438523323907394143334547762416862 518983569485562099219222184272550254256887671790494601653466804988627232791786 085784383827967976681454100953883786360950680064225125205117392984896084128488 626945604241965285022210661186306744278622039194945047123713786960956364371917 287467764657573962413890865832645995813390478027590099465764078951269468398352 595709825822620522489407726719478268482601476990902640136394437455305068203496 252451749399651431429809190659250937221696461515709858387410597885959772975498 930161753928468138268683868942774155991855925245953959431049972524680845987273 644695848653836736222626099124608051243884390451244136549762780797715691435997 700129616089441694868555848406353422072225828488648158456028506016842739452267 467678895252138522549954666727823986456596116354886230577456498035593634568174 324112515076069479451096596094025228879710893145669136867228748940560101503308 617928680920874760917824938589009714909675985261365549781893129784821682998948 722658804857564014270477555132379641451523746234364542858444795265867821051141 354735739523113427166102135969536231442952484937187110145765403590279934403742 007310578539062198387447808478489683321445713868751943506430218453191048481005 370614680674919278191197939952061419663428754440643745123718192179998391015919 561814675142691239748940907186494231961567945208095146550225231603881930142093 762137855956638937787083039069792077346722182562599661501421503068038447734549 202605414665925201497442850732518666002132434088190710486331734649651453905796 268561005508106658796998163574736384052571459102897064140110971206280439039759 515677157700420337869936007230558763176359421873125147120532928191826186125867 321579198414848829164470609575270695722091756711672291098169091528017350671274 858322287183520935396572512108357915136988209144421006751033467110314126711136 990865851639831501970165151168517143765761835155650884909989859982387345528331 635507647918535893226185489632132933089857064204675259070915481416549859461637 180270981994309924488957571282890592323326097299712084433573265489382391193259 746366730583604142813883032038249037589852437441702913276561809377344403070746 921120191302033038019762110110044929321516084244485963766983895228684783123552

and it keeps on going!

Euclid was one of the first mathematicians to really tackle the issue of irrationality in numbers in the 10th volume of ‘Elements’ where he tried to prove that there could be a number which couldn’t be expressed as a normal fraction.   He however started with something different from Pi and used the square root of 2, ie the number which when multiplied by itself is equal to 2.  His proof involved first assuming it could be written as a hypothetical fraction and then started to simply the expression.

However during his proof he demonstrated that unlike a normal fraction which can be simplified to a single form.  The fraction which was representing the square root of two could be simplified over and over again, in fact an infinite number of times, basically it could never be simplified.   This was his proof that this number was indeed irrational and the hypothetical fraction he used, could not in fact exist.