# 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.

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### derivative of hypergeometric function

I am doing an integral in Mathematica and I find the solution contains derivatives of hypergeometric functions. I would like (ideally) a simple analytic form for these. I have tried HypExp mathematica ...
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### Computation of incomplete confluent hypergeometric function of the first kind

In summary I need to compute "upper incomplete confluent hypergeometric function of the first kind": $U(a, b, x, z) = \sum_{n=0}^{\infty} \frac{ x^n }{n!} \frac{\Gamma(a + n, z)}{\Gamma(b + n)}$ ====...
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### $\int_{0}^{e}\frac{\sin(\operatorname{W(x)})-1}{\sin(\operatorname{W(x)})+1}\frac{\sin(\operatorname{2W(x)})-1}{\sin(\operatorname{2W(x)})+1}dx=?$

Solve the following integral : $$\int_{0}^{e}\frac{\sin(\operatorname{W(x)})-1}{\sin(\operatorname{W(x)})+1}\frac{\sin(\operatorname{2W(x)})-1}{\sin(\operatorname{2W(x)})+1}dx=?$$ My attempt: We ...
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### How to calculate the limit of the variance of the moment estimate of the geometric distribution?

Let $\overline X$ be the sample variance, so that a low-order moment estimate of the geometric distribution can be obtained:$\;\hat{p}=1/\overline X$. I want to verify whether this estimator is an ...
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### How to calculate this limit related to hypergeometric functions

How to calculate this limit $$\lim_{n\rightarrow+\infty} np^n \sum_{k=n}^{+\infty} \frac1k \binom{k-1}{n-1} (1-p)^{k-n} ,\quad\,where\;0<p<1$$
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### Moment Estimation on Geometric Distribution

I recently had trouble calculating the moment estimates for the parameter $p$ of the geometric distribution: $$P(X=k)=(1-p)^{k-1}p,\quad k=1,2,\cdots$$ We know that there are two kinds of moment ...
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I am studying computer algebra. To be more precise, I am trying to understand the problem of hypergeometric summation. Definition1: A function $F(n,k)$ is hypergeometric in both arguments if $\frac{... 0answers 31 views ### Associated Legendre Function of Second Kind The associated Legendre Function of Second kind is related to the Legendre Function of Second kind as such: $$Q_{n}^m(z)= (-1)^m (1-z^2)^{m/2} \frac{d^m}{dz^m}(Q_{n}(z))$$ The recurrence relations ... 0answers 19 views ### Independent Check of Wolfram's Evaluation of the Clausen ($_3F_2$) Hypergeometric Function Could someone be kind enough to independently verify the calculations of the Generalised Hypergeometric Function (Hypergeometric3F2) at http://functions.wolfram.com/webMathematica/FunctionEvaluation.... 1answer 82 views ### How can I compute this integral involving$\Gamma$-functions? I would like to find a closed form for the following integral: $$I=\int_0^1 d\alpha\ \alpha^{\omega-5/2} (1-\alpha)^{-1/2} \int_0^\alpha d\beta\ \beta^{2\omega-3} (1-\alpha-\beta)^{5/2-2\omega} (\... 1answer 32 views ### Trigonometric formula coming from hypergeometric functions While playing with hypergeometric functions, I numerically stumbled upon the identity:$$\mathrm{cos}\left(\dfrac{\pi}{6} - \dfrac{1}{6} \mathrm{arctan}\left( \dfrac{3\sqrt{15}}{11} \right) \right) = \... 0answers 17 views ### Euler/Kummer Transformations for the Clausen and Appell Hypergeometric Functions The series expansion for the Clausen$_3F_2$Hypergeometric function defined by the series $$_3F_2(a,b,c;d,e;x) =\sum_{k=0}^\infty\frac{(a)_k(b)_k(c)_k}{(d)_k(e)_kk!}x^k$$ is unconditionally ... 1answer 15 views ### Eliminating Value from Bessel Function [closed] How I can get the value for "k" as a function of "r" from equality$J1(kr)=Dr$, where D is constant.$J1(kr)$is the Bessel function with n=1. 0answers 37 views ### General Solution of Hypergeometric equation in range$x \in (0,1) $I'm trying to find the general solution of the hypergeometric equation \begin{equation} x(1-x)\partial_x^2y+\left[\gamma-(\alpha+\beta+1)x \right]{\partial_x} y -\alpha\beta y=0, \end{equation} in the ... 1answer 42 views ### simplify${}_1F_1(2\alpha , 2 ,ik\pi) +{}_1F_1(2\alpha , 2 ,-ik\pi)$1)Is it possible to simplify $$J_{k,\alpha}={}_1F_1(2\alpha , 2 ,ik\pi) +{}_1F_1(2\alpha , 2 ,-ik\pi)$$$k\in \{1,2,\cdots\}$and$\alpha \in (0,1)$.${}_1F_1$is confluent hypergeometric ... 1answer 57 views ### Expressing the Hypergeometric Function$_3F_2(a,a,b;p,p;x) $in terms of$_2F_1()$Is it possible to express the Clausen Hypergeometric Function$_3F_2(a,a,b;p,p;x)$(the first two parameters and the last two are identical) in terms of the Gauss Hypergeometric Function$_2F_1()$and ... 0answers 25 views ### generalised gauss hypergeometric series Can${}_4\!F_3(a,a,a+1/2,a+1/2;b,b,b;z)$be written successively in terms of${}_2\!F_3$and${}_1\!F_2$?$a$and$b$are integers in this case. 0answers 21 views ### Literature request for polynomials$P_n(x,y)$generated as$(1-z)^x {}_2F_1(\alpha, \alpha+x-y;2s;t)=\sum_{n\geq 0}P_n(x,y)t^n$Consider the following generating function $$\phi(t) = (1-t)^x {}_2F_1(\alpha,\alpha+x-y; 2\alpha;t)=(1-t)^{y} {}_2F_1(\alpha,\alpha+y-x; 2\alpha;t)$$ where we have used the Euler transform${}_2F_1(...
Confluent Hypergeometric Function of the Second Kind has the Maclaurin series of $z^{1-b}$ and asymptotic series of $\frac{1}{z^a}$, so it appeared that if $b>1$ (for $z\ll1$) and $a>0$(for \$z\...
Consider the following terminology for Meijer g function : $$G_{p,q}^{m,n}\left(z\left|\begin{smallmatrix}a_1,.......,a_p\\ b_1,......,b_q\end{smallmatrix}\right.\right)$$ Now consider the ...