ordinary generating function for linear recursive relations

ordinary generating function for linear recursive relations

Let

(

a

n

)

be a linear

recursive sequence

with values in a (commutative) ring

R

, i.e. there exist

constants

β

1

,

…

,

β

k

∈

R

such that for any

n

>

k

we have

a

n

=

β

1

⋅

a

n

-

1

+

⋯

+

β

i

⋅

a

n

-

i

+

⋯

+

β

k

⋅

a

n

-

k

.

Now consider the ordinary generating function for this

sequence

f

⁢

(

t

)

=

∑

n

⩾

0

a

n

⋅

t

n

which is a

formal power series

in the ring of formal power series

R

⁢

[

[

t

]

]

. We will try to find the

closed form

of

f

⁢

(

t

)

.

First write down first

k

elements of

this

power series

:

f

⁢

(

t

)

=

∑

n

=

0

k

a

n

⋅

t

n

+

∑

n

>

k

a

n

⋅

t

n

and note that for

n

>

k

we can use our recursive relation:

f

⁢

(

t

)

=

∑

n

=

0

k

a

n

⋅

t

n

+

∑

n

>

k

(

β

1

⋅

a

n

-

1

+

⋯

+

β

k

⋅

a

n

-

k

)

⋅

t

n

which gives us

f

⁢

(

t

)

=

∑

n

=

0

k

a

n

⋅

t

n

+

(

β

1

⋅

t

⋅

∑

n

>

k

a

n

-

1

⋅

t

n

-

1

)

+

⋯

+

(

β

k

⋅

t

k

⋅

∑

n

>

k

a

n

-

k

⋅

t

n

-

k

)

.

(1)

Now focus on those sums on the right. For any

j

we have

∑

n

>

k

a

n

-

j

⋅

t

n

-

j

=

(

∑

n

⩾

0

a

n

⋅

t

n

)

-

(

∑

n

=

0

k

-

j

a

n

⋅

t

n

)

=

f

⁢

(

t

)

-

(

∑

n

=

0

k

-

j

a

n

⋅

t

n

)

.

Inserting this into (1) gives us

f

⁢

(

t

)

=

∑

n

=

0

k

a

n

⋅

t

n

+

(

β

1

⋅

t

⋅

(

f

⁢

(

t

)

-

∑

n

=

0

k

-

1

a

n

⋅

t

n

)

)

+

⋯

+

(

β

k

⋅

t

k

⋅

(

f

⁢

(

t

)

-

∑

n

=

0

0

a

n

⋅

t

n

)

)

which can be simplified as follows:

f

⁢

(

t

)

=

∑

n

=

0

k

a

n

⋅

t

n

+

∑

j

=

1

k

(

β

j

⋅

t

j

⋅

(

f

⁢

(

t

)

-

∑

n

=

0

k

-

j

a

n

⋅

t

n

)

)

.

Taking the components with

f

⁢

(

t

)

to the left side gives us

f

⁢

(

t

)

-

∑

j

=

1

k

β

j

⋅

t

j

⋅

f

⁢

(

t

)

=

∑

n

=

0

k

a

n

⋅

t

n

-

∑

j

=

1

k

(

β

j

⋅

t

j

⋅

∑

n

=

0

k

-

j

a

n

⋅

t

n

)

,

which finally gives us the

closed formula

for

f

⁢

(

t

)

(note that all sums are finite):

f

⁢

(

t

)

=

∑

n

=

0

k

a

n

⋅

t

n

-

∑

j

=

1

k

(

β

j

⋅

t

j

⋅

∑

n

=

0

k

-

j

a

n

⋅

t

n

)

1

-

∑

j

=

1

k

β

j

⋅

t

j

.

Note that in the denominator we have a power series with the constant term

1

, thus (by general properties of formal power series) it is

invertible

in

R

⁢

[

[

t

]

]

. Thus we proved the following

theorem

:

Theorem.

If

(

a

n

)

is a

recursive

sequnce given by

a

n

=

β

1

⋅

a

n

-

1

+

⋯

+

β

k

⋅

a

n

-

k

for all

n

>

k

then the ordinary generating function

f

⁢

(

t

)

=

∑

n

⩾

0

a

n

⋅

t

n

has its closed form given by

f

⁢

(

t

)

=

∑

n

=

0

k

a

n

⋅

t

n

-

∑

j

=

1

k

(

β

j

⋅

t

j

⋅

∑

n

=

0

k

-

j

a

n

⋅

t

n

)

1

-

∑

j

=

1

k

β

j

⋅

t

j

.

Remark.

Note that if we replace

R

with (for example) the reals

ℝ

then the theorem is still valid if we treat those power series as

functions

. Of course such equalites hold only there where those functions are defined.

Title

ordinary generating function for linear recursive relations

Canonical name

OrdinaryGeneratingFunctionForLinearRecursiveRelations

Date of creation

2013-03-22 19:20:07

Last modified on

2013-03-22 19:20:07

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

7

Author

joking (16130)

Entry type

Theorem

Classification

msc 13H05

Classification

msc 13B35

Classification

msc 13F25

Classification

msc 13J05

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