# Questions tagged [laguerre]

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15
questions

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### Laguerre polynomial and series

Let $L^\alpha_n(x)$ be Laguerre polynomials of type $n$.
Consider the sum
$$\sum^\infty_{j=0} \frac{1}{(b-j)}L^{m}_j(x)$$
where $b\not\in\Bbb N,x>0$ and $m\in \Bbb N$.
I have found this series ...

0
votes

0
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### Closed formula for Laguerre

Let $L^\alpha_n(x)$ be Laguerre polynomials of type $n$.
Assume $0<\beta<1$. Is there a closed formula for this sum
$$\sum^\infty_{j=0} \frac{1}{(b+j)^{1-\beta}}L^{m}_j(x)$$
where $b>0$ and $...

3
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0
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### Closed formula for $\sum\limits^\infty_{k=0}\frac1{(k+a)(k+b)} L^1_k(x)L^1_k(y) $

Let $ L^{\alpha}_{n}(x)=\sum^{n}_{k=0} \binom{n+\alpha}{n-k}\big(-1\big)^{k}\frac{x^{k}}{k!},\alpha>-1$ be Laguerre polynomials of type $ n$.
Is there a closed formula for $$\sum^{\infty}_{k=0}\...

1
vote

1
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### Integral involving Bessel and Laguerre function

Is there a formulas for the following integral
$$\int^\infty_0 e^{-ar^2}L^1_k(b r^2)J_1(cr)r^d dr $$
where $L^1_k$ is the Laguerre polynomials of type 1 and $J_1$ is the Bessel function with $a,...

2
votes

0
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### Subsequence of Laguerre polynomials

Let $m\geqslant1$ be a fixed integer.
Let $f(n)$ be A007814, the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $...

7
votes

1
answer

269
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### Reference for proof of an integral from the "Tables of Integral Transforms" involving a Gaussian and a Laguerre polynomial

I am looking for a proof of one of the integrals presented in Harry Bateman's Tables of Integral Transforms. The specific integral in question is presented on page 42 in chapter 8.9 as equation (3):
$$...

2
votes

0
answers

114
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### Deduce Sheffer's classification of orthogonal polynomials of A-type 0

Theorem 1.9 in Daniel Galiffa and Tanya Riston's paper, An elementary approach to characterizing Sheffer A-type 0 orthogonal polynomial sequences, 2015, presents without proof Isador Sheffer's ...

5
votes

2
answers

282
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### Proving that the primitives of the Laguerre functions are uniformly bounded

Let $(L_k)_{k\geq 0}$ be the Laguerre polynomials. These polynmials are orthogonal with respect to the inner product: $$\langle f,g\rangle = \int_0^\infty f(x)g(x)\mathrm e^{-x}\,\mathrm dx.$$
Hence, ...

3
votes

1
answer

143
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### Inequality for generalized Laguerre polynomials

Please. Does anybody know a proof of this inequality
$$\Big|\frac{n!\Gamma(\alpha+1)}{\Gamma(n+\alpha+1)} L^{\alpha}_n(x)\Big|\leq e^{\frac{x}{2}}$$ where $\alpha\geq0$ and $x\geq0$ and
$L^{\alpha}_n$ ...

2
votes

0
answers

93
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### On integrals of products of Laguerre polynomials

Let $L_k$ be the classical Laguerre polynomial with degree $k$ defined by
$$
L_k(x)=\left(\frac {d}{dx}-1\right)^{k}\bigl\{\frac{x^{k}}{k!}\bigr\}.
$$
Let me define for $a_1, a_2$ positive, $\alpha_1,...

2
votes

2
answers

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### On Mehler's formula for Hermite polynomials

In the reference article of Richard Askey and George Gasper published in the American Journal of Mathematics, Autumn, 1976, Vol. 98, No. 3 (Autumn,1976), pp. 709-737, they attribute on page 731 the ...

2
votes

1
answer

732
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### Integral involving associated Laguerre polynomial and Bessel function

In a quantum mechanics problem I encountered the following integral
\begin{equation*}
\int_0^\infty t^{\nu+1}J_\nu(\beta t)L_{\mu-\nu}^{2\nu}(t)e^{-t/2}dt\,,
\end{equation*}
where $L$ denotes the ...

5
votes

1
answer

976
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### Integral involving Laguerre, Gaussian and modified Bessel function

I am trying to prove that the integral
\begin{align}
\int_{0}^{\infty } e^{-\frac{r^2}{2B}} r^{l-n}
L_n^{l-n}\left(\frac{r^2}{C}\right) I_{l-n}\left(\rho r \right) r dr
\end{align}
has ...

1
vote

0
answers

674
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### need help on the proof of the Christoffel–Darboux formula of Laguerre Polynomial

I found the Christoffel–Darboux formula of the Laguerre polynomials on wikipedia, https://en.wikipedia.org/wiki/Laguerre_polynomials
\begin{align}
K_n^{(\alpha)}(x,y) &:= \frac{1}{\Gamma(\alpha+1)...

2
votes

2
answers

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### Zeroes of Laguerre polynomials

The simplest Laguerre polynomials are
$$
L_k(x)=(\frac{d}{dx}-1)^k\left(\frac{x^k}{k!}\right).
$$
I would like to find a simple reference for proving or disproving the following assertions.
(1) All ...