1 22 November 2018

A Factorial Identity Resulting from the Orthogonality Relation

of the Associated Laguerre Polynomials

Spiros Konstantogiannis

spiroskonstantogiannis@gmail.com

22 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

22 November 2018

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), p. 892 – 895.

[2] Mary L. Boas, Mathematical Methods in the Physical Sciences (John Wiley &

Sons, Inc., Third Edition, 2006), p. 610 – 611.

[3] http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html.