Laguerre transform explained

L ^{\alpha(x)} _{n} as kernels of the transform.^{[1] [2] [3] [4]}

The Laguerre transform of a function

f(x) isL\{f(x)\} = \tilde f _{\alpha(n)} = \int ^{inf ty} _{0} e ^{-x} x ^{\alpha} L ^{\alpha(x)} _{n} f(x) dx

The inverse Laguerre transform is given by

L ^{-1} \{\tilde f _{\alpha(n)\}} = f(x) = \sum ^{inf ty} _{n =0} \binom{n + \alpha}{n} ^{-1} (1)/(\Gamma(\alpha +1 )) \tilde f _{\alpha(n)} L ^{\alpha(x)} _{n}

Some Laguerre transform pairs

f(x)	\tilde f _{\alpha(n)}
x ^{a-1} , a> 0	(\Gamma(a + \alpha)\Gamma(n - a +1 ))/(n!\Gamma( 1- a))
e ^{-ax} , a> -1	(\Gamma(n + \alpha +1 )a ^{n})/(n!(a +1 ) ^{n+\alpha+1})
\sin ax, a> 0 , \alpha =0	(a ^{n})/(( 1+ a ^{2 )} ^{(n +1)/(2)}) \sin \left[n\tan ^{-1} (1)/(a) + \tan ^{-1} ( - a)\right]
\cos ax, a> 0 , \alpha =0	(a ^{n})/(( 1+ a ^{2 )} ^{(n +1)/(2)}) \cos \left[n\tan ^{-1} (1)/(a) + \tan ^{-1} ( - a)\right]
L ^{\alpha(x)} _{m}	\binom{n + \alpha}{n} \Gamma(\alpha +1 )\delta _{mn}
e ^{-ax} L ^{\alpha(x)} _{m}	(\Gamma(n + \alpha +1 )\Gamma(m + \alpha +1 ))/(n!m!\Gamma(\alpha +1 )) ((a -1 ) ^{n-m+\alpha+1})/(a ^{n+m+2\alpha+2}) {} _{2F} ; _{1 \left(n + \alpha +1 ; (m + \alpha +1)/(\alpha +1)} (1)/(a ^{2}) \right) ^[5]
f(x) x ^{\beta-\alpha}	\sum ^{n} _{m =0} (m!) ^{-1} (\alpha - \beta) _{m} L ^{\beta(x)} _{n - m}
e ^{x} x ^{-\alpha} \Gamma(\alpha,x)	\sum ^{inf ty} _{n =0} \binom{n + \alpha}{n} (\Gamma(\alpha +1 ))/(n +1)
x ^{\beta, \beta>0}	\Gamma(\alpha + \beta +1 )\sum ^{inf ty} _{n =0} \binom{n + \alpha}{n} ( - \beta) _{n (\Gamma(\alpha +1 ))/(\Gamma(n + \alpha +1 ))}
( 1- z) ^{-(\alpha+1)} \exp\left( (xz)/(z -1) \right),	z	<1, \ \alpha\geq 0\,	\sum ^{inf ty} _{n =0} \binom{n + \alpha}{n}\Gamma(\alpha +1 )z ^{n}
(xz) ^{-\alpha/2} e ^{z} J ^{1 / 2} _{\alpha\left[ 2 (xz)} \right],	z	<1, \ \alpha\geq 0\,	\sum ^{inf ty} _{n =0} \binom{n + \alpha}{n} (\Gamma(\alpha +1 ))/(\Gamma(n + \alpha +1 )) z ^{n}
(d)/(dx) f(x)	\tilde f _{\alpha(n)} - \alpha\sum ^{n} _{k =0} \tilde f _{\alpha-1} (k) + \sum ^{n -1} _{k =0} \tilde f _{\alpha(k)}
x (d)/(dx) f(x), \alpha =0	- (n +1 ) \tilde f _{0(n+1)} + n\tilde f _{0(n)}
\int ^{xf(t)dt,} _{0} \alpha =0	\tilde f _{0(n)} - \tilde f _{0(n-1)}
e ^{xx} ^{-\alpha} (d)/(dx) \left[e ^{-x} x ^{\alpha+1} (d)/(dx) \right]f(x)	- n\tilde f _{\alpha(n)}
\left\{e ^{xx} ^{-\alpha} (d)/(dx) \left[e ^{-x} x ^{\alpha+1} (d)/(dx) \right]\right\} ^{kf(x)}	( -1 ) ^{kn} ^{k\tilde} f _{\alpha(n)}
L ^{\alpha(x),} _{n} \alpha> -1	(\Gamma(n + \alpha +1 ))/(n!)
xL ^{\alpha(x),} _{n} \alpha> -1	(\Gamma(n + \alpha +1 ))/(n!) ( 2 n +1+ \alpha)
(1)/(\pi) \int ^{inf ty} _{0} e ^{-t} f(t) dt \int ^{\pi} _{0} 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)}
(\Gamma(n + \alpha +1 ))/(\sqrt\pi\Gamma(n +1 )) \int ^{inf ty} _{0} e ^{-t} t ^{\alpha} f(t) dt \int ^{\pi} _{0} e ^{-\sqrt{xt} \cos\theta} \sin ^{2\alpha} \theta g(x + t +2 \sqrt{xt}\cos\theta) (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)} ^[6]

Notes and References

1. Debnath, Lokenath, and Dambaru Bhatta. Integral transforms and their applications. CRC press, 2014.
2. Debnath, L. "On Laguerre transform." Bull. Calcutta Math. Soc 52 (1960): 69-77.
3. Debnath, L. "Application of Laguerre Transform on heat conduction problem." Annali dell’Università di Ferrara 10.1 (1961): 17-19.
4. McCully, Joseph. "The Laguerre transform." SIAM Review 2.3 (1960): 185-191.
5. 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.
6. Debnath, L. "On Faltung theorem of Laguerre transform." Studia Univ. Babes-Bolyai, Ser. Phys 2 (1969): 41-45.