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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.