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Special Families of Polynomials 13
By virtue of theorem 1.1.1, we have the three-term recursion formula
(1.6.5) L (α)
n (x) =
where L (α)
0
2n + α − 1 − x
n
(x) = 1 and L(α) 1 (x) = 1 + α − x.
L (α) n + α − 1
n−1 (x) −
n
L (α)
n−2 (x), ∀n ≥ 2,
Various equations relate Laguerre polynomials corresponding to different values of
the parameter α. Some examples are (see also szegö (1939))
(1.6.6)
d
dx L(α) n+1 = −L(α+1) n , n ∈ N, α > −1,
(1.6.7) L (α)
n+1 = L(α+1) n+1 − L(α+1) n , n ∈ N, α > −1.
From (1.6.7) one obtains
(1.6.8) L (α+1)
n
=
n
k=0
L (α)
k , n ∈ N, α > −1.
An asymptotic formula relates Laguerre and Jacobi polynomials, namely:
(1.6.9) L (α)
n (x) = lim
β→+∞
P (α,β)
n
1 − 2x
β
, ∀x ∈ [0,+∞[.
This can be checked with the help of the differential equations (1.3.1) and (1.6.1).
A statement similar to that of theorem 1.4.1 also holds (see szegö(1939), p.171).
Theorem 1.6.1 - For any α > −1 and n ≥ 4, the successive values of the relative
maxima of |L (α)
n (x)| are decreasing when x < α+ 1
2
and increasing when x > α+ 1
2 .
The plots of figures 1.6.1 and 1.6.2 show respectively L (0)
n , 1 ≤ n ≤ 6, and L (0)
7 .
The size of the window is [0,20] × [−600,600] in figure 1.6.1 and [0,20] × [−900,900] in
figure 1.6.2. When n grows, graphical representations of Laguerre polynomials are dif-
ficult to obtain. An initial gentle behavior explodes to severe oscillations, for increasing
values of x.