Questions tagged [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. 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.
2
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0answers
45 views
Integral $\int_0^x \ln^{n}\left(2\sin\frac{t}2\right)\,dt$
Evaluate $$\mathcal{L}_n(x)=\int_0^x \ln^n\left(2\sin\frac{t}2\right)\,dt\qquad n\in\Bbb N_0, x\in[0,\pi/2]$$
In this answer to a question of mine, @TitoPiezasIII claims that
$$\frac{(-1)^n}{n!}\...
1
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0answers
29 views
How do we prove that $\sum_{i=1}^{n} X_{i}$ is a sufficient statistic in the present context?
Let $n$ items be drawn in order without replacement from a shipment of $N$ items of which $N\theta$ are bad. Let $X_{i} = 1$ if the $i$-th drawn is bad and $X_{i} = 0$ otherwise. Show that $\sum X_{i}$...
1
vote
1answer
19 views
Finding a substitution to an equation $ \frac{(C x)^2}{2} y''(x) + D x (1 - E x) y'(x) - F y(x) = 0 $ to transform it to Kummer's equation
It seems that the ordinary differential equation
$$
\frac{(C x)^2}{2} y''(x) + D x (1 - E x) y'(x) - F y(x) = 0
$$
can be presented (suggested by Wolfram alpha) with a certain choice of $a$ and $b$ (...
1
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1answer
57 views
Transformations relating 3F2 at z with 3F2's at 1/z
I am searching for some transformations for a 3F2 hypergeometric function which send the argument z to 1/z. I am aware of the one given in NIST book (p. 410, Formula 16.8.8) in the special case q=2 (...
2
votes
2answers
50 views
Please provide a function approximating the following hypergeometric series?
Stipulation: Would prefer polynomial asymptotic with shrinking error term and no (Riemann) Zeta functions.
Series: $${_1F}_2(1;m+\frac{1}{2};m+1;\frac{1}{4}) =\ ?$$
Put differently, it looks like: $$...
1
vote
1answer
32 views
About a solution to ODE $ (c x)^2/2 y''(x) + a x (1 - b x) y'(x) - d y(x) = 0 $
The ordinary differential equation
$$
\frac{(c x)^2}{2} y''(x) + a x (1 - b x) y'(x) - d y(x) = 0
$$
should have two independent solutions which are given by a confluent hypergeometric function $U$ ...
4
votes
0answers
141 views
A twisted hypergeometric series $\sum_{n=1}^\infty\frac{H_n}{n}\left(\frac{(2n)!}{4^n(n!)^2}\right)^2$
I was given that $$S=\sum_{n=1}^\infty\frac{H_n}{n}\left(\frac{(2n)!}{4^n(n!)^2}\right)^2=\frac{32}\pi G\ln2+\frac{64}\pi\Im\operatorname{Li}_3\left(\frac{1+i}2\right)-2\ln^22-\frac53\pi^2,$$ where $...
3
votes
1answer
81 views
Prove an transformation formula for Gauss hypergeometric function $_2F_1(a,b;c;z)$
In " Special functions: an introduction to classical functions of mathematical physics" by Nico M. Temme, at page 113 is reported this formula:
$$_2F_1(a,b;c;z)=\frac{\Gamma(c)\Gamma(b-a)}{\Gamma(b)\...
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0answers
25 views
The solution of associated legendre function in hypergeometric function.
Im learning about invers scattering theory that have been told in a book entitled soliton : an introduction by P. G. Drazin and R. S. Johnson. In the 46'th page there was a paragraph said that the ...
0
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0answers
20 views
Can you exactly integrate $x_i^2 x_j^2 e^{\sum_{k=1}^4 \lambda_k x_k^2}$ over $S^3$?
I am concerned with the following integral
$$
I = \int_{S^3} x_i^2 x_j^2 e^{\sum_{k=1}^4 \lambda_k x_k^2} d\mu(x)
$$
where the $x_{i,j,k}$ are components of $x \in S^3$, each $\lambda_k < 0$, and ...
1
vote
1answer
44 views
Finding constants in Wronskian of standard solutions of confluent hypergeometric function
I've got this question on special functions take-home exam. I will add some literature in the nearest time.
Let us have an equation $xy''+(c-x)y'-ay=0$ which is a confluent hypergeometric function ...
1
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1answer
46 views
Binomial sums of the form $\sum_{k=\ell+1}^{n}(-1)^{(k-\ell+1)}\binom{k}{\ell}$
I'm trying to compute the sum
$$\sum_{k=\ell+1}^{n}(-1)^{(k-\ell+1)}\binom{k}{\ell}$$
with $\ell \in [2,n-1]$ in an efficient way. I know that this reduces to something like
$$ 1 -\sum_{i=0}^{n/2 -1}...
0
votes
0answers
33 views
Hypergeometric solution in a particular case
I have a question to make in relation to the solution of the hypergeometric differential equation. Let us consider the aforesaid equation,
\begin{equation}
y(1-y)h'' + [c-(1+a+b)y]h' -abh=0,
\end{...
1
vote
2answers
70 views
Kummer transform of the confluent hypergeometric function of second kind
I can see the kummer transformation of the confluent hypergeometric function of first kind throught the integral representation. However, I failed to see that for the second kind. More specificially, ...
1
vote
0answers
17 views
What is the value of $ {}_2F_1\left[-\frac{p-1}{2},-\frac{p-1}{2};1;a^2\right] \mod p?$, where a is any fixed integer and $p$ is a prime number.
I can only relate it to a Legendre polynomioal $P_{(p-1)/2}(\frac{a^2+1}{a^2-1}).$
1
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1answer
29 views
Discontinuity of Hypergeometric function along the branch cut
I am trying to evaluate an expression involving the hypergeometric function evaluated near its (principal) branch cut discontinuity, which is placed on the real line from $1$ to infinity.
For $x>1$...
0
votes
0answers
11 views
Hypergeometric function: parameters as integer values
I would like to know if exists a simple form to express
$ \, _2F_1(n-1,2-n;2;u) $
or
$ \, _2F_1(3-n,n;2;u) $
in terms of some sort of polynomic expression. n is a integer number.
$ n=0, -1, -2, ....
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votes
1answer
24 views
Convert a polynomial to Hypergeometric equation
How can I convert this polynomial to Hypergeometric equation.
$$Pn=n!\sum_{r=0}^{\frac{n}{2} } (-1)^r\frac{(p-1)!}{(p-n+r-1)!} \frac{1}{r! (n-2r)!}2x^{n-2r}$$
3
votes
2answers
77 views
Hypergeometric series for $\mathrm{Cl}_2(\pi/3)$
I am trying to find a hypergeometric series for $\mathrm{Cl}_2(\pi/3)$, where
$$\mathrm{Cl}_2(x)=-\int_0^x\log\left|2\sin\frac{t}2\right|dt=\sum_{k\geq1}\frac{\sin kx}{k^2}$$
Is the Clausen ...
-1
votes
1answer
65 views
Simplification of $\sin(\pi^x)$ , with $x$ being a positive irrational number
How to simplify if $a > 0$ and $\cos(a) < 0$
Was a previous post. Correction, it was suppose to be if a > 0 & cos(a) > 0.
An answer was given. https://math.stackexchange.com/a/1274372/...
0
votes
1answer
16 views
Please help with a Hypergeometric probability question [closed]
Need help with this question, my textbook has the solution but I don't know how to get it.
15 coins in a bag.
Three 5 rand coins
Five 2 rand coins
Seven 1 rand coins
What is the prob that if I pull ...
13
votes
1answer
279 views
Yet another difficult logarithmic integral
This question is a follow-up to MSE#3142989.
Two seemingly innocent hypergeometric series ($\phantom{}_3 F_2$)
$$ \sum_{n\geq 0}\left[\frac{1}{4^n}\binom{2n}{n}\right]^2\frac{(-1)^n}{2n+1}\qquad \...
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votes
0answers
35 views
Pfaff formula in the degenerate case
The Pfaff transformation for hypergeometric functions is true under the assumption that the parameters are not negative integers. But, as far as I understand, it also holds sometimes in the degenerate ...
5
votes
1answer
180 views
Integration of $\int^{1}_{-1} \frac {1}{3} \sinh^{-1} \left( \frac {3\sqrt 3}{2} (1-t^2) \right) dt$
Recently I came across with respect to this post of mine hyperbolic solution to the cubic equation for one real root given by
$$
t=-2\sqrt \frac {p}{3} \sinh \left( \frac {1}{3} \sinh^{-1} \left( \...
0
votes
1answer
65 views
Unexpected (incorrect) solution to Lagrange Inversion solution to $x^4 - x^3 - x^2 - x - 1 = 0$ about the solution near $x = 2$
I am developing generalized hypergeometric solutions for a set of such polynomials. With this example we can write $x^4 - x^3 - x^2 - x - 1 = \frac{x^5 - 2 x^4 + 1}{x - 1}$.
Lagrange Inversion ...
1
vote
1answer
22 views
Behavior of the Gaussian Hypergeometric function when one of its arguments approaches $0$ or $1$
For two positive integers $a,b$, denote by $_2F_1(a,1-b;a+1;z)$ the Gaussian Hypergeometric function whose first three parameters are fixed at $a,1-b$ and $a+1$, respectively. such function is linked ...
2
votes
0answers
39 views
Jacobi polynomials and Gram determinants
On page 294, Andrews, Askey and Roy - Special functions. For sequences of (independent) functions $\lbrace \phi(x) \rbrace_{n=0}^{\infty}$ and $\lbrace \psi(x) \rbrace_{n=0}^{\infty}$, a sequence $\...
2
votes
0answers
28 views
Simplify Sum of Hypergeometric Functions
Write the following "sum" in terms of $\,_2F_1(a,b;c;z)$:
$$ _2 F_1(a,b+1;c+1;z) +\, _2 F_1(a,b-1;c-1;z) -\, _2 F_1(a,b;c;z).$$
Attempt: I played around with identities in here. In particular, to ...
1
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1answer
39 views
Reduction of hypergeometric function for integer parameters
Consider integers $a,b,c,d>0$ and the hypergeometric function
$${}_3F_2(1-d,b,a+b;a+b+c,b+1;1)$$
I don't know much about hypergeometric functions but I understand that when the parameters are ...
3
votes
1answer
36 views
Hypergeometric representation of Fresnel $S(x)$
I am trying to find a representation for the Fresnel integral
$$S(x)=\int_0^x\sin\frac{\pi t^2}{2}\,\mathrm dt$$
Then with $$\sin x=\sum_{n\geq0}\frac{(-1)^n}{(2n+1)!}x^{2n+1}$$
We have
$$S(x)=\sum_{...
4
votes
1answer
116 views
Integral involving incomplete beta function
I have the following integral,
$$\int_{0}^1x^{a-1}(1-x)^{b-1}B_x(c,d)dx$$
where $B_x(c,d) = \int_{0}^xt^{c-1}(1-t)^{d-1}dt$ is the incomplete beta function, and $a,b,c,d>0$.
Question: Does this ...
0
votes
1answer
70 views
Expressing $G_{m,m+1}^{m+1,0}\left(x\middle| \begin{array}{c}1,\cdots,1 \\0,0,\cdots,0\\\end{array}\right)$ as a power series.
I have this family of MeijerG functions:
$$
G_{m,m+1}^{m+1,0}\left(x\left|
\begin{array}{c}
1,\cdots,1 \\
0,0,\cdots,0 \\
\end{array}
\right.\right)
$$
which I'd like to express in terms of a power ...
9
votes
0answers
157 views
About the product of two Elliptic integrals
Let $z,x\in\left(0,1\right)$. It is possible to prove that $$\int_{0}^{1}\int_{0}^{1}\frac{1}{\sqrt{hy\left(1-h\right)\left(1-y\right)}}\frac{dydh}{\sqrt{\left(1+zhy\right)^{2}-4xzhy}}=\frac{4}{\pi^{2}...
2
votes
1answer
34 views
Special polynomials and an identity of hypergeometric series
Motivation:
I have a few polynomials and am trying to find a representation for them in terms of special functions. I'm more interested in the techniques here, so I won't give any too particular ...
1
vote
2answers
82 views
Transform differential equation into hypergeometric differential equation
I would like to know if this differential equation can be transformed into the hypergeometric differential equation
$ 4 (u-1) u \left((u-1) u \text{$\varphi $1}''(u)+(u-2) \text{$\varphi $1}'(u)\...
0
votes
0answers
20 views
Trying to simply an equation to find the limit of the hypergeometric distribution
My textbook states that,
$\frac{((1-p)N)^{(n-x)}}{N^{(x)}(N-x)^{(n-x)}} = (1-p)^{(n-x)}$
where $a^{(b)} = aP_b$
I tried expanding the numerator and denominator, and then factoring out the $(1-p)*$ ...
0
votes
1answer
30 views
Simplifying this hypergeometric function with variable b
The student-t CDf has a hypergeometric function as a component
$$_2F_1\left(\frac{1}{2}, \frac{\nu + 1}{2}; \frac{3}{2}; -\frac{x^2}{\nu}\right)$$
where $\nu$ is the distributions degree of freedom.
...
18
votes
4answers
503 views
Evaluating $\int_0^1\frac{x^{2/3}(1-x)^{-1/3}}{1-x+x^2}dx$
How can we prove $$\int_0^1\frac{x^{2/3}(1-x)^{-1/3}}{1-x+x^2}\mathrm{d} x=\frac{2\pi}{3\sqrt 3}?$$
Thought 1
It cannot be solved by using contour integration directly. If we replace $-1/3$ with $-2/...
0
votes
0answers
67 views
Integral involving the logarithm of a confluent hypergeometric function
I am trying to find the solution of the integral
\begin{align}
I =\int_{0}^{\infty}e^{-t}t^{\alpha+1}\left[ _{1}F_{1}(-m; \alpha +1;t)\right]^{2}\log\left[ _{1}F_{1}(-m; \alpha +1;t)\right]^{2}dt
\...
1
vote
0answers
41 views
Indefinite integral involving the product of two generalized Laguerre polynomials
I am trying to find the indefinite integral
\begin{align}
\int{x^{\alpha +1}e^{-x}\left(L_{m}^{\alpha}(x)\right)^{2}dx}
\end{align}
where $L_{m}^{\alpha}(x)$ is the generalized Laguerre Polynomial, ...
1
vote
0answers
21 views
Sum involving hypergeometric 2F2 function
I'm trying to simplify the following sum:
$$
\sum_{i=0}^n\frac{z^i}{(n-i)!}\,\frac{1}{(1+a)_i\,(1-a)_i}\sum_{j=0}^i(-1+a)_j\,(-1-a)_j\frac{(-z)^j}{j!},
$$
where $n=1,2,\ldots$, $z>0$, $0<a<1$,...
0
votes
0answers
46 views
hypergeometric function and its asymptotic expansion near z=1
my dear fellows,
I have a question to make. Given the hypergeometric function $_{2}F_{1}[a,b,c,z]$ in the interval $z \in (1, \infty)$. What is the proper asymptotic expansion of the aforesaid ...
8
votes
2answers
156 views
Is it true that $\int_0^1 \big(K(k^{1/2})\big)^2\,dk = \frac{7}2\zeta(3)$?
Define the complete elliptic integral of the first kind as,
$$K(k) = \tfrac{\pi}{2}\,_2F_1\left(\tfrac12,\tfrac12,1,\,k^2\right)$$
Part I. From the link above, we find some of the evaluations below,
...
4
votes
1answer
99 views
Symmetry in function given by double sum
I had to deal with this function:
$$ f_n(x_1,x_2)=(x_2-x_1)^{n-1}\sum_{m=0}^{n-1}\sum_{j=0}^{n-m-1}C(n,m,j)\left(\frac{x_2}{x_2-x_1}\right)^m\left(\frac{x_2(1-x_1)}{x_2-x_1}\right)^j $$
where
$$C(n,...
1
vote
0answers
29 views
Integration of Hypergeometric 2F1 on the real axis with arbitrary extrema of integrations
I have to compute the integral
$$
\int_x^1 dz \, z^\alpha (1-z)^\beta \; _2F_1(a_1,a_2;b_1;z)
$$
where $x$ is some positive real number $0 \le x<1$. If $x=0$ then the answer is a $_3F_2$ (see ...
2
votes
2answers
58 views
Infinite binomial sum
Let
$\displaystyle\pi_{lr}\left(p\right) :=
{l \choose r}p^{r}\left(1 - p\right)^{l - r}\quad$ ( i.e., the binomial probability with parameters $\displaystyle l$ and $\displaystyle r$ ).
I'...
8
votes
0answers
166 views
What is $\int_0^1 \left(\tfrac{\pi}2\,_2F_1\big(\tfrac13,\tfrac23,1,\,k^2\big)\right)^3 dk$?
As in this post, define the ff:
$$K(k)=K_2(k)={\tfrac{\pi}{2}\,_2F_1\left(\tfrac12,\tfrac12,1,\,k^2\right)}$$
$$K_3(k)={\tfrac{\pi}{2}\,_2F_1\left(\tfrac13,\tfrac23,1,\,k^2\right)}$$
$$K_4(k)={\...
0
votes
3answers
36 views
Looking for the Closed Form of a Two-Variable Geometric Sum
Is there any closed form of the equation $$\sum_{i=0}^n a^{n-i} \cdot b^i$$ for real values $a$ and $b$ and integer $n \ge 0$?
1
vote
0answers
37 views
What is going on with these asymptotics for $\mathrm{Re}\left\{ \ _2F_1\left(\tfrac{1}{2}, \tfrac{5}{2} ; 2 ; x \right) \ \right\}$
I am interested in the large $x$ asymptotics for the function
$$
\mathrm{Re}\left\{ \ _2F_1\left(\tfrac{1}{2}, \tfrac{5}{2} ; 2 ; x \right) \ \right\}
$$
When I check the series expansion at $x = \...
0
votes
1answer
38 views
Hypergeometric functions and modular forms
May I please ask if it is possible to write Hypergeometric functions in terms of Jacobi theta functions? I am trying to bring the following Hypergeometric expression (pg.9, eq 4.3) into a known ...
