Questions tagged [laguerre]
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12
questions
-1
votes
0answers
40 views
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 ...
2
votes
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
votes
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):
$$...
2
votes
0answers
95 views
Deduce Sheffer's classification of orthogonal polynomials of A-type 0
Theorem 1.9 in Daniel Galiffa and Tanya Riston's paper, An elementary approach to characterizing Sheffer A-type 0 orthogonal polynomial sequences, 2015, presents without proof Isador Sheffer's ...
5
votes
2answers
182 views
Proving that the primitives of the Laguerre functions are uniformly bounded
Let $(L_k)_{k\geq 0}$ be the Laguerre polynomials. These polynmials are orthogonal with respect to the inner product: $$\langle f,g\rangle = \int_0^\infty f(x)g(x)\mathrm e^{-x}\,\mathrm dx.$$
Hence, ...
2
votes
1answer
79 views
Inequality for generalized Laguerre polynomials
Please. Does anybody know a proof of this inequality
$$\Big|\frac{n!\Gamma(\alpha+1)}{\Gamma(n+\alpha+1)} L^{\alpha}_n(x)\Big|\leq e^{\frac{x}{2}}$$ where $\alpha\geq0$ and $x\geq0$ and
$L^{\alpha}_n$ ...
2
votes
0answers
79 views
On integrals of products of Laguerre polynomials
Let $L_k$ be the classical Laguerre polynomial with degree $k$ defined by
$$
L_k(x)=\left(\frac {d}{dx}-1\right)^{k}\bigl\{\frac{x^{k}}{k!}\bigr\}.
$$
Let me define for $a_1, a_2$ positive, $\alpha_1,...
2
votes
2answers
445 views
On Mehler's formula for Hermite polynomials
In the reference article of Richard Askey and George Gasper published in the American Journal of Mathematics, Autumn, 1976, Vol. 98, No. 3 (Autumn,1976), pp. 709-737, they attribute on page 731 the ...
2
votes
1answer
399 views
Integral involving associated Laguerre polynomial and Bessel function
In a quantum mechanics problem I encountered the following integral
\begin{equation*}
\int_0^\infty t^{\nu+1}J_\nu(\beta t)L_{\mu-\nu}^{2\nu}(t)e^{-t/2}dt\,,
\end{equation*}
where $L$ denotes the ...
5
votes
1answer
734 views
Integral involving Laguerre, Gaussian and modified Bessel function
I am trying to prove that the integral
\begin{align}
\int_{0}^{\infty } e^{-\frac{r^2}{2B}} r^{l-n}
L_n^{l-n}\left(\frac{r^2}{C}\right) I_{l-n}\left(\rho r \right) r dr
\end{align}
has ...
1
vote
0answers
541 views
need help on the proof of the Christoffel–Darboux formula of Laguerre Polynomial
I found the Christoffel–Darboux formula of the Laguerre polynomials on wikipedia, https://en.wikipedia.org/wiki/Laguerre_polynomials
\begin{align}
K_n^{(\alpha)}(x,y) &:= \frac{1}{\Gamma(\alpha+1)...
2
votes
2answers
1k views
Zeroes of Laguerre polynomials
The simplest Laguerre polynomials are
$$
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 ...