Laguerre transform

The integral transform

$$f ( n) = T \{ F ( x) \} = \ \int\limits _ { 0 } ^ \infty e ^ {- x} L _ {n} ( x) F ( x) d x ,\ \ n = 0, 1 \dots$$

where $L _ {n} ( x)$ is the Laguerre polynomial (cf. Laguerre polynomials) of degree $n$. The inversion formula is

$$T ^ {-1} \{ f ( n) \} = F ( x) = \ \sum _ { n= 0} ^ \infty f ( n) L _ {n} ( x) ,\ \ 0 < x < \infty ,$$

if the series converges. If $F$ is continuous, $F ^ { \prime }$ is piecewise continuous on $[ 0 , \infty )$ and $F ( x) = O ( e ^ {ax} )$, $x \rightarrow \infty$, $a < 1$, then

$$T \left \{ \frac{d F ( x) }{dx} \right \} = \ \sum _ { k=0} ^ { n } f ( k) - F ( 0) ,\ \ n = 0 , 1 \dots$$

$$T \left \{ x \frac{d F ( x) }{dx} \right \} = - ( n + 1 ) f ( n + 1 ) + n f ( n) ,\ n = 0 , 1 , \dots.$$

If $F$ and $F ^ { \prime }$ are continuous, $F ^ { \prime\prime }$ is piecewise continuous on $[ 0 , \infty )$ and $| F ( x) | + | F ^ { \prime } ( x) | = O ( e ^ {ax} )$, $x \rightarrow \infty$, $a < 1$, then

$$T \left \{ e ^ {x} \frac{d}{dx} \left [ x e ^ {-x} \frac{d F ( x) }{dx} \right ] \right \} = - n f ( n) ,\ n = 0 , 1 , . . ..$$

If $F$ is piecewise continuous on $[ 0 , \infty )$ and $F ( x) = O ( e ^ {ax} )$, $x \rightarrow \infty$, $a < 1$, then for

$$G ( x) = \int\limits _ { 0 } ^ { x } F ( t) d t ,$$

$$g ( n) = T \left \{ \int\limits _ { 0 } ^ { x } F ( t) d t \right \} = f ( n) - f ( n - 1 ) ,\ n = 1 , 2 \dots$$

and for $n = 0$,

$$g ( 0) = f ( 0) .$$

Suppose that $F$ and $G$ are piecewise continuous on $[ 0 , \infty )$ and that

$$| F ( x) | + | G ( x) | = O ( e ^ {ax} ) ,\ \ x \rightarrow \infty ,\ a < \frac{1}{2} ,$$

$$T \{ F \} = f ( n) ,\ T \{ G \} = g ( n) .$$

Then

$$T ^ {-1} \{ f ( n) g ( n) \} =$$

$$= \frac{1} \pi \int\limits _ { 0 } ^ \infty e ^ {-t} F ( t) \int\limits _ { 0 } ^ \pi e ^ {\sqrt {xt } \cos \theta } \cos ( \sqrt {xt } \sin \theta ) \times$$

$$\times G ( x + t - 2 \sqrt {xt } \cos \theta ) d \theta d t .$$

The generalized Laguerre transform is

$$f _ \alpha ( n) = T _ \alpha \{ F ( x) \} =$$

$$= \ \int\limits _ { 0 } ^ \infty e ^ {- x} x ^ \alpha L _ {n} ^ \alpha ( x) F ( x) d x ,\ n = 0 , 1 \dots$$

where $L _ {n} ^ \alpha ( x)$ is the generalized Laguerre polynomial (see [4]).

References

[1]	A.G. Zemanian, "Generalized integral transformations" , Interscience (1968)
[2]	J. McCully, "The Laguerre transform" SIAM Rev. , 2 : 3 (1960) pp. 185–191
[3]	L. Debnath, "On Laguerre transform" Bull. Calcutta Math. Soc. , 52 : 2 (1960) pp. 69–77
[4]	Yu.A. Brychkov, A.P. Prudnikov, "Operational calculus" Progress in Math. , 1 (1968) pp. 1–74 Itogi Nauk. Mat. Anal. 1966 (1967) pp. 7–82