A generalization ofLucas polynomial sequence

G.-S.Cheon, Hana Kim and L.W. Shapiro

Sungkyunkwan University

2009. 10. 8

09 Brownbag Seminar

Contents

Delannoy numbers

Weighted Delannoy numbers

Riordan array

Generalized Lucas polynomial sequence

Combinatorial interpretations and examples

),,( cbaDw

Delannoy numbers

→ ↑ ↗=)0,1( =)1,0( =)1,1(

d

dd dn

dk

kdkn

dk

knD 2),(00∑∑≥≥

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ −+⎟⎟⎠

⎞⎜⎜⎝

⎛=

)0,0(

),( nk

Weighted Delannoy numbers

- weighted path

→ ↑ ↗

The weight of a weighted path

),,( cba

=)0,1( =)1,0( =)1,1( cab

Theorem 1

The total sum of the weights of all -weighted paths from to on the lattice plane is

: the weighted Delannoy numbers

where

),( knDw

),,( cba)0,0( ),( nk

ddndk

dw cba

kdkn

dk

knD −−

≥⎟⎟⎠

⎞⎜⎜⎝

⎛ −+⎟⎟⎠

⎞⎜⎜⎝

⎛= ∑

0:),(

)1,1(),1()1,(),( −−+−+−= kncDknbDknaDknD wwww

.)0,(,),0(,1)0,0( nw

nww bnDanDD ===

Riordan array

L.W. Shapiro (1991)

A Riordan array

where

Summation property (SP)

,1)( 221 K+++= zgzgzg

K+++= 33

22)( zfzfzzf

knkn zfzgzd ))()((][, =

))(),(()( , zfzgd kn =R

∑=

=n

k

nkkn zfhzgzhd

0, ))(()(][

),,( cbaDw

where

0,, ][),,( Nknknw dcbaD ∈=

⎩⎨⎧

<≥−

=.if0,if),(

, knknkknD

d wkn

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

++++++

+=

L

L

4222224

33

22

)43(66)43()32()32(

2

1

),,(

aabcaababccabcbbaabcaabcbb

aabcbab

cbaDw

Theorem 2

is a Riordan array given by

Lemma 3

The generating function (GF) for the row

sums of is given by

⎟⎠⎞

⎜⎝⎛

−+

−=

bzczaz

bzcbaDw 1

,1

1),,(

),,( cbaDw

.)(1

1)( 2czzbaz

−+−=φ

),,( cbaDw

)(zφ

Generalized Lucas polynomial sequence

A.F. Horadam (1996)

Polynomial sequence

where

or .

)}({ xWn

0=d

)2(),()()()()( 21 ≥+= −− nxWxqxWxpxW nnn

ddd xcxqxcxpxcxWcxW 321100 )(,)(,)(,)( ====

1

( )*

Lucas polynomial sequence of 1st kind

If then

,

where

and

Lucas polynomial sequence of 2nd kind

If then

)}({ xWn

1)(,1 10 == xWc

)()(,2 10 xpxWc ==

)()()()()(

xvxuxvxuxW

nn

n −−

=

)()()( xvxuxW nnn +=

)()()( xpxvxu =+ ).()()( xqxvxu −=)}({ xwn

=:)(xwn

Given ,

by substituting and into

such that , we obtain a

Riordan array .

Let be the -th row sum of the Riordan

array .n

)(),( xbbxaa == )(xqc =

)(xpba =+

))(),((:),,( xqxpDcbaD ww =

)(xWn

),,( cbaDw

))(),(( xqxpDw

][)(),( xxqxp R∈

.)()(1

1)( 20 zxqzxp

zxW n

nn −−

=∑≥

Theorem 4

Theorem 5

where and

⎡ ⎤.))(())(()( 2

2/)1(

0

kknn

kn xqxp

kkn

xW −−

=∑ ⎟⎟

⎠

⎞⎜⎜⎝

⎛ −=

),()()()()( 21 xWxqxWxpxW nnn −− +=

).()(1 xpxW =1)(0 =xW

where

and

Let and If

and then

and for)()( xwxw nn =

)()()( xpxvxu =+

).()()( xqxvxu −=

)()( xpxp = )()( xqxq = )()( 1 xWxW nn +=

)()()()()(

xvxuxvxuxW

nn

n −−

=

)()(:)( xvxuxw nnn +=

.,2,1,0 K=n

).(:)(0 xpxw =

We call and generalized Lucas

polynomial sequences of the first kind and of

the second kind, resp.

Theorem 6

⎡ ⎤.))(())((

11

)(2/

0

12∑=

+−

⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−+⎟⎟⎠

⎞⎜⎜⎝

⎛−−

=n

k

kknn xqxp

kkn

kkn

xw

)}({ xwn)}({ xWn

).1:(,)()(1

)(1)( 12

2

01 =

−−+

= −≥

−∑ wzxqzxp

zxqzxw n

nn

)}({ xWn

Theorem 7

Let be the generalized Lucas polynomial

sequence of the first kind. Then

where is the sum of weights of

-weighted paths from to using the

steps and for which

and

∑=

−+ ≥=n

kknkn nxxW

0,1 )0()()( ω

)(, xknk −ω ))(),(),(( xcxbxa

)0,0( ),( knk −

)1,0(),0,1( )1,1( )()()( xpxbxa =+

).()( xqxc =

Combinatorial interpretations and examples

Special cases of the Lucas polynomial sequences

Fibonacci polynomials

Pell polynomials

Jacobsthal polynomials

Fermat polynomials

Chebyshev polynomials of the 2nd kind

)}({ xWn

)(xp )(xq

xx2

x2

x2

x3

1

1

1

1−

2−

)(xWn

)(xFn

)(xPn

)(xJn

)(xnF

)(xUn

)(xa )(xc)(xb

xx

x

x

x

x2

x2

11

1

1−

2−

0

0

x

Example (Fibonacci polynomial )• the case of ; i.e, the Fibonacci

polynomials are the row sums of the Riordan array :

• the G.F. for

)(xFn

.1

)()(0

2∑≥ −−

==n

nnn zxz

zzxFxF

1,0, === cbxa

)1,0,(xDw

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

++++

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

MMML

)()()()()(

3121

1

11111

310002000010000000001

5

4

3

2

1

42

3

2

42

3

2

xFxFxFxFxF

xxxx

xx

xxxx

xx

• Combinatorial interpretation for .

Consider . Since when and , we may take and . Then

.

- weighted paths and their weights

)(xFn

34 2)( xxxF += )()()( 344 xWxWxF ==

xxbxaxp =+= )()()( 1)()( == xcxq

0)(,)( == xbxxa 1)( =xc

∑=

−=3

03,4 )()(

kkk xxF ω

)1,0,(x

x x 3x

Theorem 8

Let be the Lucas polynomial sequence

of the second kind. Then

)}({ xwn

∑ ∑=

−

=−−− ≥+=

n

k

n

kknkknkn nxxqxxw

0

2

02,, )2().()()()( ωω

Special cases of the Lucas polynomial sequences

Lucas polynomials

Pell-Lucas polynomials

Jacobsthal-Lucas polynomials

Fermat-Lucas polynomials

Chebyshev polynomials of the 1st kind

)}({ xwn

)(xp )(xq

xx2

x2

x2

x3

1

1

1

1−

2−

)(xwn

)(xLn

)(xQn

)(xjn

)(xfn

)(xTn

)(xa )(xc)(xb

xx

x

x

x

x2

x2

11

1

1−

2−

0

0

x

Example (Pell-Lucas polynomial )

• the G.F. for

• Combinatorial interpretation for .

Consider . Since

and , we may take

and . Then .

- weighted paths and their weights

22 42)( xxQ +=

)(xQn

)1(,211)()(

02

2

≥−−

+==∑

≥

nzxz

zzxQxQn

nnn

xxbxaxp 2)()()( =+=

1)()( == xcxq xxbxxa == )(,)(

1)( =xc )(1)()( 0,0

2

02,2 xxxQ

kkk ωω∑

=− ⋅+=

)1,,( xx

)(xQn

2x 2x 2x2x 1 1

Thank you for listening.