Laguerre transform

In mathematics, Laguerre transform is an integral transform named after the mathematician Edmond Laguerre, which uses generalized Laguerre polynomials $L_n^\alpha(x)$ as kernels of the transform.Debnath, Lokenath, and Dambaru Bhatta. Integral transforms and their applications. CRC press, 2014.Debnath, L. "On Laguerre transform." Bull. Calcutta Math. Soc 52 (1960): 69-77.Debnath, L. "Application of Laguerre Transform on heat conduction problem." Annali dell’Università di Ferrara 10.1 (1961): 17-19.McCully, Joseph. "The Laguerre transform." SIAM Review 2.3 (1960): 185-191.

The Laguerre transform of a function $f(x)$ is

: $L\{f(x)\} = \tilde f_\alpha(n) = \int_{0}^\infty e^{-x} x^\alpha \ L_n^\alpha(x)\ f(x) \ dx$

The inverse Laguerre transform is given by

: $L^{-1}\{\tilde f_\alpha(n)\} = f(x) = \sum_{n=0}^\infty \binom{n+\alpha}{n}^{-1} \frac{1}{\Gamma(\alpha+1)} \tilde f_\alpha(n) L_n^\alpha(x)$

Some Laguerre transform pairs

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! $f(x)\,$

! $\tilde f_\alpha(n)\,$

x^{a-1}, \ a>0\,

| $\frac{\Gamma(a+\alpha)\Gamma(n-a+1)}{n!\Gamma(1-a)}$

e^{-ax}, \ a>-1\,

| $\frac{\Gamma(n+\alpha+1)a^n}{n!(a+1)^{n+\alpha+1}}$

\sin ax, \ a>0, \ \alpha=0\,

| $\frac{a^n}{(1+a^2)^{\frac{n+1}{2}}} \sin \left[n\tan^{-1} \frac{1}{a} +\tan^{-1} (-a)\right]$

\cos ax, \ a>0,\ \alpha=0\,

| $\frac{a^n}{(1+a^2)^{\frac{n+1}{2}}} \cos \left[n\tan^{-1} \frac{1}{a} +\tan^{-1} (-a)\right]$

L_m^\alpha(x)\,

| $\binom{n+\alpha}{n} \Gamma(\alpha+1)\delta_{mn}$

e^{-ax}L_m^\alpha(x)\,

| $\frac{\Gamma(n+\alpha+1)\Gamma(m+\alpha+1)}{n!m!\Gamma(\alpha+1)}\frac{(a-1)^{n-m+\alpha+1}}{a^{n+m+2\alpha+2}}{}_2F_1\left(n+\alpha+1;\frac{m+\alpha+1}{\alpha+1};\frac{1}{a^2}\right)$ Howell, W. T. "CI. A definite integral for legendre functions." The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 25.172 (1938): 1113-1115.

f(x) x^{\beta-\alpha}\,

| $\sum_{m=0}^n (m!)^{-1}(\alpha-\beta)_m L_{n-m}^\beta(x)$

e^x x^{-\alpha}\Gamma(\alpha,x)\,

| $\sum_{n=0}^\infty \binom{n+\alpha}{n} \frac{\Gamma(\alpha+1)}{n+1}$

x^\beta,\ \beta>0\,

| $\Gamma(\alpha+\beta+1)\sum_{n=0}^\infty \binom{n+\alpha}{n} (-\beta)_n\frac{\Gamma(\alpha+1)}{\Gamma(n+\alpha+1)}$

(1-z)^{-(\alpha+1)}\exp\left(\frac{xz}{z-1}\right),\ |z|<1, \ \alpha\geq 0\,

| $\sum_{n=0}^\infty \binom{n+\alpha}{n}\Gamma(\alpha+1)z^n$

(xz)^{-\alpha/2}e^z J_\alpha\left[2(xz)^{1/2}\right],\ |z|<1, \ \alpha\geq 0\,

| $\sum_{n=0}^\infty \binom{n+\alpha}{n}\frac{\Gamma(\alpha+1)}{\Gamma(n+\alpha+1)}z^n$

\frac{d}{dx}f(x)\,

| $\tilde f_\alpha(n) - \alpha\sum_{k=0}^n \tilde f_{\alpha-1}(k) + \sum_{k=0}^{n-1}\tilde f_\alpha(k)$

x\frac{d}{dx}f(x), \alpha=0\,

| $-(n+1) \tilde f_0(n+1) + n\tilde f_0(n)$

\int_0^xf(t)dt, \ \alpha=0\,

| $\tilde f_0(n) - \tilde f_0(n-1)$

e^xx^{-\alpha}\frac{d}{dx}\left[e^{-x}x^{\alpha+1}\frac{d}{dx}\right]f(x)\,

| $-n\tilde f_\alpha(n)$

\left\{e^xx^{-\alpha}\frac{d}{dx}\left[e^{-x}x^{\alpha+1}\frac{d}{dx}\right]\right\}^kf(x)\,

| $(-1)^kn^k\tilde f_\alpha(n)$

L_n^\alpha(x), \alpha>-1\,

| $\frac{\Gamma(n+\alpha+1)}{n!}$

xL_n^\alpha(x), \alpha>-1\,

| $\frac{\Gamma(n+\alpha+1)}{n!}(2n+1+\alpha)$

\frac{1}{\pi} \int_0^\infty e^{-t} f(t) dt \int_0^\pi e^{\sqrt{xt}\cos\theta} \cos(\sqrt{xt}\sin\theta)g(x+t-2\sqrt{xt}\cos\theta)d\theta, \alpha=0\,

| $\tilde f_0(n) \tilde g_0(n)$

\frac{\Gamma(n+\alpha+1)}{\sqrt\pi\Gamma(n+1)} \int_0^\infty e^{-t}t^\alpha f(t) dt \int_0^\pi e^{-\sqrt{xt}\cos\theta} \sin^{2\alpha}\theta g(x+t+2\sqrt{xt}\cos\theta)\frac{J_{\alpha-1/2}(\sqrt{xt}\sin\theta)}{[(\sqrt{xt}\sin\theta)/2]^{\alpha-1/2}}d\theta\,

| $\tilde f_\alpha(n) \tilde g_\alpha(n)$ Debnath, L. "On Faltung theorem of Laguerre transform." Studia Univ. Babes-Bolyai, Ser. Phys 2 (1969): 41-45.

References

Category:Integral transforms

Category:Mathematical physics