Questions tagged [laguerre]

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Infinite Summation for the inverse of generalized Laguerre polynomials

From many Books, we already know the value of the following serie: $$\sum_{n=0}^{+ \infty} t^{n} L_{n}^{\alpha}(x)=\frac{1}{(1-t)^{\alpha+1}}e^{-tx/(1-t)}$$ On the other hand, i didn't find out this ...
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0answers
77 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 $...
7
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1answer
142 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): $$...
2
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0answers
95 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 ...
5
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2answers
182 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, ...
2
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1answer
79 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 $L^{\alpha}_n$ ...
2
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0answers
79 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,...
2
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2answers
445 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 ...
2
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1answer
399 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 ...
5
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1answer
734 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 ...
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0answers
541 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)...
2
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2answers
1k 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 ...