What is Hypergeometric: Definition and 77 Discussions
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). Every second-order linear ODE with three regular singular points can be transformed into this equation.
For systematic lists of some of the many thousands of published identities involving the hypergeometric function, see the reference works by Erdélyi et al. (1953) and Olde Daalhuis (2010). There is no known system for organizing all of the identities; indeed, there is no known algorithm that can generate all identities; a number of different algorithms are known that generate different series of identities. The theory of the algorithmic discovery of identities remains an active research topic.
Homework Statement
Find the general solution of Airy's equation f'' - zf=0 satisfying the initial conditions f(0)=1, f'(0)=0 as a power series expansion at z=0. Express the result in terms of the Gauss hypergeometric series.
The Attempt at a Solution
After subbing...
Homework Statement
A function g is \alpha-regularly varying around zero if for all \lambda > 0, \lim_{x\to 0} \frac{g(\lambda x)}{g(x)}=\lambda^{\alpha}
For real s and \alpha \in (0,1), define f:
f(s)=1-\alpha \int_{0}^{\infty} e^{\alpha t}...
I posted this in the Advanced Physics forum as well, but it occurred to me that this might be a more appropriate place. I'd delete it in Advanced Physics, but I can't see where to do that.
Homework Statement
I'm need to integrate the function
\frac{A}{(1+B^2x^2)^{\frac{C+1}{2}}}
which...
I am trying to calculate the following integral
I=\int_0^\infty\frac{x}{(x+ia)^2} {\mbox{$_2$F$_1\!$}}\left(\frac{1}{2},b,\frac{3}{2},-\frac{x^2}{c^2}\right) dx.
I tried several different ways but drew a complete blank. Is there a way of converting this nasty hypergeometric function into...
Hi all,
I need help with the following problem:
The urn contains 5 black and 8 red balls. You close your eyes and
remove balls from the urn one by one without replacement. What is
the probability that the last ball is black?
This looks to me like it is a hypergeometric distribution...
Homework Statement
A large company employs 20 individuals as statisticians, 7 of whom are women and 13 of whom are men. No two people earn the same amount.
What is the probability that 6 of the women earn salaries below the median salary of the group?
Homework Equations
If r is the...
Apologies in advance for the TeX in this post, I'm new and having difficulty with the formatting.
Homework Statement
I'm trying to understand the logic my professor uses to derive a second linearly independent solution to the hypergeometric DE:
z(1-z)\frac{d^2 w}{dz^2} + (\gamma - z(1+\alpha...
Hey,
I want to compute integrals of the following form
I= \int_{-i \infty}^{i\infty} (1-x^2)^\frac{d-1}{2} \prod_{i=1}^4 _2F_1(a_i,b_i,c_i;\frac{1-x}{2}) dx
where a_i,b_i,c_i are constants and c_i\in \mathbb{N}.
d is a positive integer.
For odd d I know that the integral will be zero by...
Hello,
I am wondering about integrals of the form
\int_0^1 {}_p F_q(\{a_1,\ldots,a_p\},\{b_1,\ldots,b_q\},y){}_{p'} F_{q'}(\{a'_1,\ldots,a'_{p'}\},\{b'_1,\ldots,b'_{q'}\},y)dy
integrals of product of hypergeometric functions.
I know that if the limits of integration were +/-...
Homework Statement
Suppose 9 mice are available for a study of a possible carcinogen and 4 of them will form a control group (i.e. will not receive the substance). Assuming that a random sample of 4 mice are selected, what is the probability that a particular mouse, Mike Mouse will be included...
Today, I use two softwares to calculate the value of a hypergeometric functions (2F1). One is Mathematica and another is Matlab. But they give me different results.
For examples:
(1) 2F1(0.5, 1., 1.5, 5) (Pay an attention to the sign of the image part.)
Mathematica's result...
Homework Statement
I have seen some hypergeometric function in the form:
2F1=(a,b;c;d),
Is there such thing as:
2F1=(a,b,c;d)
Homework Equations
The Attempt at a Solution
I don't understand why sometimes we have a comma and sometimes we have a semi-colon.
thank you
Hello,
for some calculation I need the behaviour of the hypergeometric function 2F1 near z=\tfrac{1}{2}. Specifically I need
_2 F_1(\mu,1-\mu,k,\tfrac{1}{2}+i x)
with x\in \mathbb{R} near 0, and 1/2\leq\mu\leq 2, 1\leq k \in \mathbb{N}.
Differentiating around x=0 and writing the...
I'm trying to show that:
F(a, b; z) = F(a-1, b; z) + (z/b) F(a, b+1 ; z)
where F(a, b; z) is Kummer's confluent hypergeometric function and
F(a, b; z) = SUMn=0[ (a)n * z^n ] / [ (b)n * n!]
where (a)n is the Pochhammer symbol and is defined by:
a(a+1)(a+2)(a+3)...(a+n-1)...
Homework Statement
I have three problems on my homework set that I can't figure out. I'll start with the longest one:
Show that by letting z=\zeta^{-1} and u=\zeta^{\alpha}v(\zeta) that the hypergeometric differential equation
z(1-z)\frac {d^2u}{dz^2} +
\left[\gamma-(\alpha+\beta+1)z...
Homework Statement
write the following serie in the form of hypergeometric function:
f(x)=\sum\frac{(-1*(x^2))^n}{(2^n)(2n-1)(2n+1)(2n+3)}
n changes from 0 to \infty
Homework Equations
hypergeometric function:
The Attempt at a Solution
guys i have thought about this for 2...
Hello:
I need to simplify the following if possible
_2F_1(a,b;c;-x^2) - _2F_1(a+1,b+1;c+1;-x^2)
In fact, a= 1/2 and c=3/2 and b>=1. In other words, the difference above that I am interested in is more specifically
_2F_1(.5, b; 1.5; -x^2) - _2F_1(.5+1, b+1...
hallo, i now spent an hour looking for a formula connecting the modified bessel functions I_n and K_n to the hypergeometrical series F(a,b;c;z).
has somedoby an idea?
thank you
The Heun equation is a generalization of the hypergeometric D.E. to the case
of four regular singular points. With the singular points at z=0,1,a,and inft,
the Heun equation takes the form,
z(z-1)(z-a)w''+(c_1*z^2+c_2*z+c_3)w'+(c_4*z+c_5)w=0
(a) In terms of k_1, k_2, k_3, k_4 and a...
Homework Statement
The Heun equation is a generalization of the hypergeometric D.E. to the case
of four regular singular points. With the singular points at z=0,1,a,and inft,
the Heun equation takes the form,
z(z-1)(z-a)w''+(c_1*z^2+c_2*z+c_3)w'+(c_4*z+c_5)w=0
(a) In terms of k_1, k_2...
Homework Statement
Balls are randomly withdrawn, one at a time without replacement, from an urn that initially has N white and M black balls. Find the probability that n white balls are drawn before m black balls, n <= N, m <= M.
Homework Equations
A hypergeometric random variable with...
Hello,
Im just wondering is there a simply calculated formula for the expected value and variance for the hypergeometric distribution. I know how to do it with long calculations. I know the expected value for the binomial is = np and the variance is = npq = np(1-p) .. Is there something like...
How do you derive hypergeometirc identities of the form
2F1(a,b,c,z)= gamma function. What I mean is that the hypergeometric function converges to a set of gamme functions function in terms of (a,b,c)
where z is not 1,-1, or 1/2 ?
The hypergeometric identities in the mathworld summary...
How would i go about showing the special case F(1, b, b; x) of the hypergeometic function is the geometric series and also how the geometric series is = 1/ (1 -x)
Cheers,
Dave
[Note: P and p throughout are probabilities]
Fact: The hypergeometric distribution is the precise probability of crawing from a dichotomous (S-F) population without replacement.
A family of probabilistic sieves can be constructed using this fact as follows.
Step 1: Suppose we know all...
I am having trouble with a problem that asks me to show that if I change the variable of integration of the following equation from t to t-1 the following
http://mathworld.wolfram.com/images/equations/EulersHypergeometricTransformations/equation1.gif
(disregard that z in the denominator, that...
Show directly that the set of probabilities associated with the hypergeometric distribution sum to one.
=> I am thinking that this tells me to prove that since this is a probability distribution function, it really should sum to 1. Is that what the problem asking me to do? =)
I got...