Hypergeometric Functions: From Euler to Appell and Beyond

Hello, thanks for coming. My name is Djibrian Ishkanian, I am from Boltram research and today I will talk about hypergeometric functions from Euler to Appel and beyond. In this presentation we'll talk about hypergeometric functions, hypergeometric series and so on. So hypergeometric series are important mathematical objects that appeared in the mid 17th century and played an important role in the development of science, mathematical and physical theories. Different well known scientists like Euler, Gauss, Riemann, Kummer and others worked with them treated these functions the differential equations they satisfy as well as some applications. Most of the elementary and special functions are grouped into the single variable hypergeometric class of functions where the generalized hypergeometric PFQ function is the main function. And also there are well known Gauss hypergeometric 2F1 and Gomer hypergeometric confluent hypergeometric polyflom functions. There are generalizations to doubly infinite series as well as hypergeometric functions of two variables, n variables and so on. So there are appel functions, horn functions, competitive fitting functions, Richelie ones and so on. And in this talk I will give an overview of the classical PFQ functions. The four Apple functions, three of which aplift 2, upper left 3 and upper left 4 were introduced in 13.3 version of Wolfram language. And also I will talk about the bilateral hypergeometric PFQ that will be available in version 14. Okay, so let's start from the beginning and talk about hypergeometric series. The definition is very simple. A hypergeometric series is a power series where the ratio of successive coefficients is a rational function of N. And here A and B are polynomials in N. So as a very simple example, the Taylor series of the exponential function is a hypergeometric series because we all know the Taylor series. And here if we will calculate the ratio of successive coefficients will be 1 over 1 plus n. And here if we are comparing with this formula, we see that A is 1 and B is N plus 1. So the generalized hypergeometric function pfq, it's a well known name from the literature, is defined by the following huge series. So it's an infinite series, it has two sets of parameters. Say these A's are the upper parameters and B's are the lower parameters. And if we'll take a look on the definition we will see these brackets with the indies and this is the procamer symbol or the rising factorial written here. And now if we'll take this definition of generalized hypergeometric PFQ function and calculate the ratio of successive terms. We'll get this expression and this generalized hypergeometric function PFQ function is implemented into Wolfram languages. Hypergeometric PFQ Its theory is rather well developed and it's a very powerful function and very important function. So what can you do with this pfq aside from the. From the development of theory and solving some physical problems. For example, you can solve this differential equation third order ordinary differential equation, which would generate three fundamental solutions written in terms of hypergeometric PFQs, where we don't have upper parameters and have two lower parameters vectors. Special cases of the generalized hypergeometric function PFQ have played an important role. And let's now focus on the function that is called 2F1 or Gauss hypergeometric function. This function is defined as this series and of course it's hypergeometric, the ratio of successive terms and it satisfies the Gauss differential equation, which is a second order differential equation. And an important point here is that it's in fact very general differential equation. It has a singular point located at 01 when these denominators are zeros and infinity. And this is a very important equation and it was used a lot in physics during the 20th century and has a very, very important applications. So if you will dissolve this equation, I will immediately get two fundamental solutions where one is a pure hypergeometric 2F1 function and the second one is hypergeometric 2F1 with shifted parameters and with some prefactor. The next important function, a well known function, is the 1F1 Kuhmer confluent hypergeometric function. It is defined by this series and it also has a lot of applications. As an example, the hydrogen atom radial wave function for continuous spectrum is written in terms of this function. Here it is. So in fact there are a lot of applications in physics, in statistics, in other areas, and also the theory itself of these functions has played an important role for the development of mathematics in general. To get some idea about these functions, to understand how they work, what are the features, functionality and so on. You can check the ref pages of these functions and we have a guide on that also. So now let's go to the next page and here I will talk about the elementary and special functions that are functions that are in fact functions of hypergeometric class. So in fact these elementary or special functions, they are most of them, they are fundamental solutions of some differential equations that are very similar to the differential equations that I presented. Before or the differential equations of special functions are specific cases and so on. So as an example, the well known Bessel functions, they solve this equation. Here we have Bessel J and Bessel Y. And if we will a bit change the form of this equation and try to dissolve this equation, we'll get another solution in Bessel I's. If we will make a very simple simplification of this result, just apply a full simplifier to that. We will need to generate the solution of this equation in terms of hypogeometric 0f1 functions. This is showing that the Bessel functions are in fact nothing more than specific cases of hypergeometric functions. And this is the situation for a lot of other well known functions, polynomials like Laguerre J, picture and so on. For another here I have another differential equation, this one. And when we're dissolving it, we're getting the solution in terms of hypergeometric U and Lagir L. And if we'll just directly substitute the hypergeometric 1F1, we will see that it's solving this equation. This is in fact the equation of hypergeometric 1F1 function. This was just to show that a lot of different functions that we use in physics and other areas of science, they are in fact directly connected with hypergeometric functions because they are solutions of hypergeometric differential equations. And now that was about the previous slides were about hypergeometric functions of one variable. And now we're turning to the hypergeometric functions of two variables. So here I have the definition of PFQ function, the generalized hyperg geometric function in the first slide. And this is just to turn to two variables here. So Apple functions are hypergeometric functions of two variables. Therefore, Apple F1, Apple F2, Apple F3 and Apple F4 and their definitions are written here. So here we see that we are going from one series, one infinite series to prove them. And we have two variables. The Apple F1 function was available in the system for long years, While in version 13.3 we have implemented the Apple F2 function, Apple F3 function, double F4 function, and they are implemented as upper left 2, upper left 3 and Apple left 4. An important point about Appel functions on their applications is that Appel functions solve systems of PDEs with polynomial coefficients. These are called horn PDEs. And here as an example, I show that the Appel F3 function with some parameters specified is solving this PDE. In fact, one interesting point here is that if we take a look in this pde. So we'll see a very similar structure to for example the Gauss hypergeometric differential equation and 2F1 function. And this is of course understandable because Apple functions, they are generalizing the hypergeometric F1 function. Of course, when we implemented these Apple functions, we also implemented all the functionality that we have for other more simple special functions. So you can easily plot family of Apple functions, you can generate series expansions. And for Apple functions F1, F2, F3 and F4, you will get it in terms of hyper geometric to F1s. This is also very understandable. Also Appel functions are very powerful because they are very general with a very hard theory. But this is very useful because we can use for integration. Here, this general integral is calculated in terms of Apple F1 function. And I have a small table showing the special cases of upper left two functions in terms of more simple functions. So here we have two, F1 and even elementary ones. But this is not all Appel functions are hypergeometric functions of two variables. But there are even more general hypergeometric functions of two variables. They are called complete FAA functions. So this is if Appel functions are generalizing the 2F1 function, then compared the FEA is generalizing the PFQ function and this compared the FE area function includes the UPEL functions of special case. It is very, very powerful because it can represent as an example the derivatives of PFQ with respect to parameters. Or it can be used to calculate multiple integrals of another giant monster function which is called mayerg. And Aside from compat FA function, there is a generalization 2n variables, in fact 2, 3 variables. And after that 2n variables. And these functions you can get them into literature. They're called Lori shalom functions. So when n is 2, they reduce to operate functions, when n is 1 n, they reduce to 2 f 1 Gauss hypergeometric function. Okay, this was about two and four variables. Now let's get back to the hypergeometric functions of one variable. But take a say another case. When we the definition of our function, this series is going not from zero to infinity, but from minus infinity to plus infinity. So this function is called a bilateral hypergeometric function. It's a doubly infinite hypergeometric function with a similar structure to the PFQ function with similar set of parameters, upper parameters, lower parameters. And this function, bilateral hypergeometric function, is implemented in version 14.0 a flow from language and it will appear there as bilateral hypergeometric pfq. We can Use this function as any other special function. We can calculate the numerical values. For example, here I took some random random parameters and calculated the value of the function. At this point you can of course plot this function. Here I am plotting the real and complex parts of it. But here, as this talk is a general talk, here I would like to stress that the bilateral hypergeometric function or php, a pHQ function, it has two completely different sub cases. The first case is the case when P is equal to Q. This means that the number of upper parameters is equal to the number of lower parameters. For example, we have A1, A2, B1, B2, or A1 and B1, something like that. In this case, our bilateral hypergeometric function is nothing else than sum of two ordinary PFQ functions. So you can check for example, this equality numerically as an example, and see that 2h2 is nothing else than sum of 3f2 with argument z and another 3f2 with argument 1c. And this is very understandable. This means that we are just dividing the double infinite series into two, one infinite series. This is going from minus infinity to minus one and from zero to infinity. So this is, this is a simple case. This is a simple sub case of but bilateral hypergeometric function. And the other case, the case when P is not equal to Q, this means that the number of parameters upper parameters is not equal to the number of lower parameters. This case is very hard and the theory is not very well developed on that. And to calculate the PHQ function, the bilateral hyper geometric function, for that case, we're using a regularization method for diverging series. It's called a Borel regularization. I left a link here so you can check what that is. And of course, for some specific sets of parameters, the bilateral hypergeometric PFQ is simplified to special or even elementary functions. For example, Here I took 1h1, 1 parameter, 1 upper parameter, 1 lower parameter, and the output is this function. The two H2K's can be simplified to hypergeometric 2f1 function, but not always for some specific sets of parameters. And aside from this, the bilateral hypergeometric PFQ is a very useful function for us because we use this function to improve our sum functionality. Because now we are able to calculate doubly infinite series applying sum very easily and in a very fast, very fast calculations. For this example, for this input we gained like around 100 times speedup, something like that. So this is very nice and very useful also for other functionality available in both language. Okay, I Think that this will be all for now. So, just to recall, I talked about the hypergeometric functions of one variable pfq function 2f function 1f1 function. I talked a bit about connection between elementary and special functions and the hypogeometric functions. Then I turned to hypergeometric functions of multiple variables, mentioning the Wolfram language 13.3 appel functions. And after that I talked about the bilateral hypergeometric PSQ that will be available in version. So thank you for coming and I will now check the chat if there are some questions and try to answer them about the applications of Gauss hypergeometric function. Okay, about the applications, just you can check the ref page. There is one example from the statistics and also I can say that there is another way of generalizing this Gauss hypergeometric function to the function group called HOIN functions which are also implemented in the Wolfram language. And there are also a lot of applications of that in physics, in quantum physics. Any upgrades on the MERD function? I'm not sure about that because last year I was working on OPEL functions and bilateral hypergeometrics can X integral, sine integral and log integral within a finite number of hypergeometric functions. I read hypergeometric functions can be used to solve quantic polynomials. But I'm wondering about here. In fact, I cannot answer to this question right now, but maybe we can take a look on that. There is one more question, generalizations of hypergeometric functions which seems to me not yet implemented. Yeah, of course, yeah, of course not all the hypergeometric functions are implemented. And here we have. We are facing the following problem, in fact, that all these functions when we are generalizing from one variable to two and more. The theory is very, very complex and not always it is possible to as an example design numerical algorithms to calculate everything. So at some time point I think that we will be able to solve these problems. But of course right now there are a lot of functions not implemented. Yeah, thank you. Jan wrote that error function is representable as 1F1 and shared the link on that. Okay, I think that I will wait like a couple of minutes more. If there will be any questions, try to answer and after that we'll finish this presentation. I will just add a bit to the comment of Jan about 2F1 and orthogonal polynomials. In fact, a couple of years ago I made another presentation showing up all the connections between two F1 and all the all the known orthogonal polynomials. So all they are grouped into this hypergeometric class of functions where the reference is about this function. About which function? About the bilateral or about the H function? H. Bilateral. Hypergeomagic. Yeah, there is a ref page. It will be available with the 14.0 version. Right now it's not available. Yes, I think that I'm ready to end this webinar. So. Okay. So thank you for coming and goodbye.