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