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Questions tagged [laguerre]

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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 ...
zoran  Vicovic's user avatar
0 votes
0 answers
88 views

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 $...
zoran  Vicovic's user avatar
3 votes
0 answers
60 views

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}\...
zoran  Vicovic's user avatar
1 vote
1 answer
158 views

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,...
Ryo Ken's user avatar
  • 11
2 votes
0 answers
90 views

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 $...
Notamathematician's user avatar
7 votes
1 answer
269 views

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): $$...
schade96's user avatar
2 votes
0 answers
114 views

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 ...
Andrius Kulikauskas's user avatar
5 votes
2 answers
282 views

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, ...
F_Dussap's user avatar
3 votes
1 answer
143 views

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 $L^{\alpha}_n$ ...
Kacdima's user avatar
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2 votes
0 answers
93 views

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,...
Bazin's user avatar
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2 votes
2 answers
1k views

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 ...
Bazin's user avatar
  • 15.3k
2 votes
1 answer
732 views

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 ...
Tom26's user avatar
  • 23
5 votes
1 answer
976 views

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 ...
Katie's user avatar
  • 53
1 vote
0 answers
674 views

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)...
ZHONG Shane's user avatar
2 votes
2 answers
2k views

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 ...
Bazin's user avatar
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