1 28 November 2018
A Factorial Identity Resulting from the Orthogonality Relation
of the Associated Laguerre Polynomials
Spiros Konstantogiannis
28 November 2018
Abstract
Plugging the closed-form expression of the associated Laguerre
polynomials into their orthogonality relation, the latter reduces to a
factorial identity that takes a simple, non-trivial form for even-degree
polynomials.
Keywords: factorials, combinatorics, Laguerre polynomials
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2
1. Introduction
The associated Laguerre polynomials ( )L xνλ are λ -degree polynomial solutions to
the associated Laguerre differential equation
( ) ( ) ( ) ( )1 0xy x x y x y xν λ′′ ′+ + − + =
for , 0,1,...ν λ = [1, 2]. If 0ν = , the associated Laguerre polynomials reduce to the
Laguerre polynomials ( )L xλ [1, 2].
The polynomials ( )L xνλ are given by the closed-form expression [1, 3]
( ) ( )0
1!
nn
n
vL x x
nn
λνλ
λλ=
+− = −
∑ ,
where vn
λλ
+ −
is the binomial coefficient, i.e.
( )( ) ( )
!! !
vn n n
λ λ νλ λ ν
+ + = − − +
The polynomials ( )L xνλ satisfy the orthogonality relation [1-3]
( ) ( ) ( )0
exp 0dxx x L x L xν ν νλ λ
∞
′− =∫ ,
for λ λ′≠ .
2. The factorial identity
Using the closed-form expression of the associated Laguerre polynomials, the
orthogonality integral takes the form
( ) ( ) ( )0
expdxx x L x L xν ν νλ λ
∞
′− =∫
( ) ( ) ( )0 00
1 1exp
! !
m nm n
m n
v vdxx x x x
m nm n
λ λν λ λ
λ λ
∞ ′
= =
′+ +− − = − = ′− −
∑ ∑∫
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( ) ( )( ) ( )
( ),
, 0,00
1exp
! !
m nm n
m n
v vdxx x x
m nm n
λ λν λ λ
λ λ
+′∞+
=
′+ +− = − = ′− −
∑∫
( ) ( )( ) ( )
( ),
, 0,0 0
1exp
! !
m nm n
m n
v vdxx x
m nm n
λ λνλ λ
λ λ
+′ ∞+ +
=
′+ +− = − ′− −
∑ ∫
That is
( ) ( ) ( ) ( ) ( )( ) ( )
( ),
, 0,00 0
1exp exp
! !
m nm n
m n
v vdxx x L x L x dxx x
m nm n
λ λν ν ν ν
λ λ
λ λλ λ
+′∞ ∞+ +
′=
′+ +− − = − ′− −
∑∫ ∫
The integral on the right-hand side is easily calculated using the Gamma function,
since
( ) ( ) ( )0
exp 1 !m ndxx x m n m nν ν ν∞
+ + − = Γ + + + = + +∫ ,
and then
( ) ( ) ( ) ( ) ( )( ) ( )
( ),
, 0,00
1exp !
! !
m n
m n
v vdxx x L x L x m n
m nm n
λ λν ν ν
λ λ
λ λν
λ λ
+′∞
′=
′+ +− − = + + = ′− −
∑∫
( ) ( )( ) ( )
( )( ) ( ) ( )
( ) ( )
( ),
, 0,0
1 ! !!
! ! ! ! ! !
m n
m n
v vm n
m n m m n n
λ λ λ λν
λ ν λ ν
+′
=
′− + += + + =
′− + − +∑
( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )
( ),
, 0,0
1 !! !
! ! ! ! ! !
m n
m n
m nv v
m n m n m n
λ λ νλ λ
λ λ ν ν
+′
=
− + +′= + +
′− − + +∑
That is
( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )
( ),
, 0,00
1 !exp ! !
! ! ! ! ! !
m n
m n
m ndxx x L x L x v v
m n m n m n
λ λν ν ν
λ λ
νλ λ
λ λ ν ν
+′∞
′=
− + +′− = + +
′− − + +∑∫
Then, since ( ) ( )! ! 0v vλ λ′+ + ≠ , the orthogonality relation of the associated Laguerre
polynomials reduces to the following factorial identity
( ) ( )( ) ( ) ( ) ( )( ) ( )
( ),
, 0,0
1 !0
! ! ! ! ! !
m n
m n
m nm n m n m n
λ λ νλ λ ν ν
+′
=
− + +=
′− − + +∑ (1)
where , , 0,1,...λ λ ν′ = and λ λ′≠ .
If 0λ′ = , then 0n = too, and (1) reads
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( ) ( )( ) ( ) ( ) ( )
} ( )( )
0! 10 0
0 0
1 ! 110 0!0! ! 0 0 ! ! 0 ! ! ! !
m m
m m
mm m m m m
λ λνλ ν ν ν λ
=≠ ≠
= =
− + −= ⇒ =
− − + + −∑ ∑ ,
and since 1 !ν is non-zero,
( )( )
0
0
10
! !
m
m m m
λ
λ
≠
=
−=
−∑ (2)
The series in (2) has +1λ terms. If λ is odd, the series has an even number of terms,
while m and mλ − have different parity, i.e. if m is even/odd then mλ − is
odd/even, and thus
( )( ) ( )( )
( )( )
( )( )
1 1 1! ! ! !! !
m m m
m m m mm m
λ λ
λ λλ λ λ
− −− − −= = −
− −− − −
Then, the terms with 0m = and m λ= are opposite, as are the terms with 1m = and
1m λ= − , as are the terms with 2m = and 2m λ= − , etc. Thus, in this case, the
series consists of ( )1 2λ + pairs of opposite terms, and the identity (2) is rather
trivial. However, if λ is even, m and mλ − have the same parity, and also the series
has an odd number of terms, thus it does not consist of pairs of opposite terms.
Therefore, in the case where λ is even, the identity (2) is not trivial. Moreover,
setting 2λ λ→ , with 1,2,...λ = , the series in (2) is written as
( )( )
( )( ) ( )
( )( ) ( )( )
2 2 12
0 0 1
1 1 1! 2 ! 2 ! 2 2 ! 2 1 ! 2 2 1 !
m m m
m m mm m m m m m
λ λ λ
λ λ λ
−
= = =
− − −= + =
− − − − −∑ ∑ ∑
( ) ( )( ) ( ) ( )( )0 1
1 12 ! 2 ! 2 1 ! 2 1 !m mm m m m
λ λ
λ λ= =
= −− − − +∑ ∑ ,
and (2) takes the form
( ) ( )( ) ( ) ( )( )0 1
1 12 ! 2 ! 2 1 ! 2 1 !m mm m m m
λ λ
λ λ= =
=− − − +∑ ∑ (3)
with 1, 2,...λ =
Let us verify (3) for 1, 2,3λ = .
For 1λ = , we have
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( ) ( )( )1
0
1 1 1 1 1 10!2! 2!0! 2 22 ! 2 1 !m m m=
= + = + =−∑
and
( ) ( )( ) ( )( )1
1
1 1 1 11!1!2 1 ! 2 1 1 ! 1! 2 1 1 1 !m m m=
= = =− − + − +∑
For 2λ = , we have
( ) ( )( )2
0
1 1 1 1 2 1 1 1 1 3 4 10!4! 2!2! 4!0! 4! 2!2! 12 4 12 12 12 32 ! 2 2 !m m m=
= + + = + = + = + = =−∑
and
( ) ( )( )2
1
1 1 1 2 11!3! 3!1! 3! 32 1 ! 2 2 1 !m m m=
= + = =− − +∑
For 3λ = , we have
( ) ( )( )3
0
1 1 1 1 1 2 2 1 10!6! 2!4! 4!2! 6!0! 6! 2!4! 3*4*5*6 2*3*42 ! 2 3 !m m m=
= + + + = + = + =−∑
1 15 16 4 2 23*4*5*6 3*4*5*6 3*4*5*6 3*5*6 3*5*3 45
= + = = = =
and
( ) ( )( )3
1
1 1 1 1 2 1 1 11!5! 3!3! 5!1! 5! 3!3! 3*4*5 2*3*2*32 1 ! 2 3 1 !m m m=
= + + = + = + =− − +∑
1 1 3 5 8 2 23*4*5 3*4*3 3*4*5*3 3*4*3*5 3*4*5*3 3*5*3 45
= + = + = = =
3. References
[1] Arfken, Weber, and Harris, Mathematical Methods for Physicists (Elsevier Inc.,
Seventh Edition, 2013), pp. 892 – 895.
[2] Mary L. Boas, Mathematical Methods in the Physical Sciences (John Wiley &
Sons, Inc., Third Edition, 2006), pp. 610 – 611.
[3] http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html.