How do you solve a Gauss hypergeometric function?
This a hypergeometric equation with constants a, b and c defined by F = c, G = -(a + b + 1) and H = -ab and can therefore be solved near t = 0 and t = 1 in terms of the hypergeometric function. But this means that (0.8) can be solved in terms of the same function near x = A and x = B.
Why hypergeometric function is called hypergeometric?
We have seen that the hypergeometric series in (4) converges absolutely, when and, thus, defines a function: 2 F 1 ( a , b ; c ; z ) , which is analytic, when provided that c is neither zero nor a negative integer. This function is correspondingly called the hypergeometric function or Gauss’s hypergeometric function.
How do you write a generalized hypergeometric function?
In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation.
What do you mean by confluent hyper hypergeometric function?
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity.
What is the use of hypergeometric function?
In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE).
Who discovered hypergeometric distribution?
The term HYPERGEOMETRIC (to describe a particular differential equation) is due to Johann Friedrich Pfaff (1765-1825) (Kline, page 489).
How do you calculate hypergeometric probability?
The probability of getting EXACTLY 3 red cards would be an example of a hypergeometric probability, which is indicated by the following notation: P(X = 3). The probability of getting exactly 3 red cards is 0.325. Thus, P(X = 3) = 0.325.
How do you calculate hypergeometric distribution?
Hypergeometric Formula.. The hypergeometric distribution has the following properties: The mean of the distribution is equal to n * k / N . The variance is n * k * ( N – k ) * ( N – n ) / [ N2 * ( N – 1 ) ] .
Who invented hypergeometric function?
The term “hypergeometric series” was first used by John Wallis in his 1655 book Arithmetica Infinitorum. Hypergeometric series were studied by Leonhard Euler, but the first full systematic treatment was given by Carl Friedrich Gauss (1813).
What are the properties of hypergeometric distribution?
The hypergeometric distribution has the following properties: The mean of the distribution is equal to n * k / N . The variance is n * k * ( N – k ) * ( N – n ) / [ N2 * ( N – 1 ) ] .
Why do we use hypergeometric distribution?
The hypergeometric distribution is a discrete probability distribution. It is used when you want to determine the probability of obtaining a certain number of successes without replacement from a specific sample size.
What is the solution to the hypergeometric equation at x = 1?
Since z = 1 − x, the solution of the hypergeometric equation at x = 1 is the same as the solution for this equation at z = 0. But the solution at z = 0 is identical to the solution we obtained for the point x = 0, if we replace each γ by α + β − γ + 1.
What are Gauss equations?
The Gauss equations include, as particular cases, a number of differential equations encountered in applications; many ordinary linear second-order differential equations are reduced to (1) by transforming the unknown function and the independent variable [4].
What is a hypergeometric function?
Hypergeometric function. In mathematics, the Gaussian or ordinary hypergeometric function 2F 1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases.
Is the hypergeometric equation Riemann differential?
Verlag Wissenschaft. (1958) (Translated from Russian) MR0100119 The hypergeometric equation is a differential equation with three regular singular points (cf. Regular singular point) at 0, 1 and $ \\infty $ such that both at 0 and 1 one of the exponents equals 0. So it is a special case of the Riemann differential equation.