All Questions
Tagged with legendre-polynomials hypergeometric-function
14
questions
1
vote
2
answers
129
views
Calculation for negative integer order Associated Legendre Function
I am currently engaging with the following hypergeometric function as a result of attempting to find a solution for this probability problem for $n$ number of dice:
$$_2F_1\left (\frac{n+k}{2}, \frac{...
0
votes
0
answers
78
views
"Legendre-type" integrals involving $\frac{dt}{\sqrt{t^2-2t\cos(\theta)+1}}$
Summing Legendre polynomials $P_{l}(\cos\theta)$ often leads to expressions containing $\frac{1}{\sqrt{t^2-2t\cos\theta+1}}$, as this is the generating function for the Legendre polynomials. I want to ...
- 203
2
votes
0
answers
235
views
Integral of associated Legendre polynomials over the unit interval
I am looking for a closed-form expression for the integral of the associated Legendre polynomial $P_l^m$ over the unit interval ($l \ge m$ non-negative integers),
$$
I_l^m = \int_{0}^{1} P_l^m(x) \, ...
- 73
2
votes
1
answer
65
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How to merge odd series and even series of hypergeometric function of Legendre polynomials into one hypergeometric function?
On the Wolfram MathWorld page of Legendre Differential Equation, Legendre polynomials are represented as
$$
P_l(x) = c_n
\begin{cases}\begin{align*}
&_2F_1\left(-\frac{1}{2}(l), \frac{1}{2}(l + 1);...
- 314
6
votes
1
answer
198
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Fourier Legendre expansion of Beta kernel $x^a (1-x)^b$
Preliminaries. I have difficulty computing FL expansion of Beta kernel $f_{a,b}(x)=x^a (1-x)^b$ where $4a, 4b \in \mathbb{Z}$. Here are two important examples:
$a=s-1,b=0: x^{s-1}=\sum_{n=0}^\infty \...
- 8,654
6
votes
2
answers
419
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Proof for a combinatorial identity
I have the following formula, which I believe it's true since it works in Mathematica for all values of $N$ I have tried, but I don't know how to prove it:
$$\sum_{q=0}^{N} {N \choose q}^2 x^{q} = \...
- 759
5
votes
1
answer
255
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General method of evaluating $\small\sum_{n\geq 0}\left(\frac{4^n}{(2n+1)\binom{2n}{n}}\right)^2\frac{1}{n+k}$
Question: $
\mbox{How can we evaluate}\quad
\sum_{n \geq 0}\left[{4^{n} \over \left(\, 2n + 1\,\right)
\binom{2n}{n}}\right]^{2}{1 \over n + k}\quad
\mbox{for general $k$ ?.}
$
General methodology ...
- 8,654
4
votes
1
answer
259
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Evaluate hypergeometric $_6F_5\left(\{\frac12\}_6;1,\{\frac32\}_4;1\right)$
Background: I'm searching for $_pF_q$ representations for MZVs. In related article On the interplay between hypergeometric series, Fourier-Legendre expansions and Euler sums by M. Cantarini and J. D’...
- 8,654
10
votes
1
answer
578
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Evaluate $\int_0^1 x^{a-1}(1-x)^{b-1}\operatorname{Li}_3(x) \, dx$
Define
$\small f(a,b)=\frac1{B(a,b)}\int_0^1 x^{a-1}(1-x)^{b-1} \text{Li}_3(x) \, dx$$ $$=\frac a{a+b}{}_5F_4(1,1,1,1,a+1;2,2,2,1+a+b;1)$
Where $a>-1$ and $b>0$.
$1$. By using contour ...
- 8,679
1
vote
1
answer
140
views
How to convert a hypergeom function to the Legendre function?
Anyone can help me to convert the following maple pdsolve expressed by the hypergeom function to the $LegendreP(n,b,x)$ or $Q$ function?
\begin{equation}
dsolve\Big( (1-x^2)\cdot \frac{d^2 y(x)}{dx^...
- 55
25
votes
3
answers
976
views
An elementary proof of $\int_{0}^{1}\frac{\arctan x}{\sqrt{x(1-x^2)}}\,dx = \frac{1}{32}\sqrt{2\pi}\,\Gamma\left(\tfrac{1}{4}\right)^2$
When playing with the complete elliptic integral of the first kind and its Fourier-Legendre expansion, I discovered that a consequence of $\sum_{n\geq 0}\binom{2n}{n}^2\frac{1}{16^n(4n+1)}=\frac{1}{16\...
- 355k
1
vote
0
answers
514
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Hypergeometric relation from Legendre functions
When writing the relation for the Legendre polynomials
$$
P^{-\mu}_{\nu}(z) = (-1)^{\mu} {\Gamma(\nu-\mu+1) \over \Gamma(\nu+\mu+1)} P^{\mu}_{\nu}(z)
$$
in the Gauss hypergeometric representation, we ...
- 161
2
votes
2
answers
502
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Associated Legendre function hypergeometric representation; case of integer l, m
The associated Legendre functions for general l, m (i.e, l and m are not in general integers) can be written in terms of the hypergeometric function $_2F_1(a,b,c;x)$ thus: $$P^m_l(x) = (\frac{x+1}{x-1}...
- 126
3
votes
1
answer
655
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Identity relating hypergeometric function and Legendre polynomial
In my notes I have written down the following relation:
$_2F_1(a,a+\frac{1}{2};c;z)=2^{c-1}z^{(1-c)/2}(1-z)^{-a+(c-1)/2}L_{2a-c}^{1-c}\big(\frac{1}{\sqrt{1-z}}\big)\ ,$
where $_2F_1(a,b;c;z)$ is the ...
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