GENERATING FUNCTIONS FOR PELL AND PELL-LUCAS NUMBERS

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CO

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ORDINATOR

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GENERATING FUNCTIONS FOR PELL AND PELL-LUCAS NUMBERS

DR. NARESH PATEL

ASST. PROFESSOR

DEPARTMENT OF APPLIED SCIENCES

INSTITUTE OF ENGINEERING & TECHNOLOGY

DEVI AHILYA UNIVERSITY

INDORE – 452 017

ABSTRACT

In this paper, I have derived a list of generating functions for Pell and Pell-Lucas numbers. Exponential generating functions are used to derive combinatorial identities as well as hybrid identities. Generalizations of the results are the main features of this paper.

KEYWORDS

Exponential Functions, Generating Functions, Hybrid Identities.

INTRODUCTION

ELL AND PELL-LUCAS NUMBERS

Define the sequences

{ }

U

n and

{ }

V

n for all integers n by

1 2 0 1

,

0,

1,

,

2,

.

n n n

U

pU

U

V

pV

V

p

− −

=

+

=





=

+

=



For p = 1, we write

{ } { }

U

n

=

F

n and

{ } { }

V

n

=

L

n , which are the Fibonacci and Lucas numbers respectively. Their Binet forms, obtained by using

standard techniques for solving linear recurrences, are

α

β

α

−

=

n n

n

F

and

n n

n

L

=

α

+

β

,

where α and β are the roots of

2

1 0

x

− − =

x

_.

For p = 2, we write







=

+

=

+

=

− −

.

2

,

2

,

2

,

1

,

0

,

2

1 0

2 1

1 0

2 1

Q

P

n n n

Here

{ }

P

n and

{ }

Q

n are the Pell and Pell-Lucas Sequences respectively. Their Binet forms are given by

δ

γ

δ

γ

−

=

n n

n

P

and

n n n

Q

=

γ

+

δ

,

where γ and δ are the roots of 2

2

1

0

x

−

x

− =

_{that is}

γ

=

1

+

2

_and

δ

=

1

−

2

_.

GENERATING FUNCTIONS

Let

a

0,

a

1,

a

2,…be a sequence of real numbers. Then the function

2

0 1 2

0

( )

....

_n n

....

_n n

n

g x

a

a x

∞

=

= +

+

+ +

+ =

∑

is called

the generating function for the sequence

{ }

a

n . We can also define generating functions for the finite sequence

a

0_,

a

1_,…,

a

n_{by letting}

a

i

=

0

_for

i

>

n

_;

thus

2

0 1 2

( )

....

n n

g x

= +

a

a x

+

a x

+ +

a x

is called the generating function for the finite sequence

a

0,

a

1,…,

a

n.

For example,

2

( )

1 2

3

.... (

1)

n

....

g x

= +

x

+

x

+ + +

n

x

+

_{is the generating function for the sequence of positive integers and}

2

(

1)

( ) 1 3

6

....

2

n

n n

f x

= +

x

+

x

+ +

+

x

+

is the generating function for the sequence of triangular numbers 1, 3, 6, 10, ….

Also

2

0

1

( )

1

....

1

n n

n

g x

x

∞

=

= + + + + + =

=

−

∑

; (1)

Thus

1

1 x







−







_{is the generating function for the infinite sequence of ones, whereas}

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2 1

1

( )

1

....

1

n

x

f x

x

−

=

= + + + +

−

; Thus

1

n

x



−







−





is the generating for the finite sequence of n ones.

EQUALITY OF GENERATING FUNCTIONS

Two generating functions 0

( )

_n n

n

f x

a x

∞

=

∑

and 0

( )

_n n

n

g x

b x

∞

=

∑

are equal if

a

n

=

b

n for

n

≥

0

; that is

0 0

n n

n n n n

n n

a x

b x

a

b

∞ ∞

= =

=

⇒

=

∑

for

n

≥

0

.

ADDITION AND MULTIPLICATION OF GENERATING FUNCTIONS

Let 0

( )

_n n

n

f x

a x

∞

=

∑

and 0

( )

_n n

n

g x

b x

∞

=

∑

be two generating functions. Then

(

)

0

( )

_n _n n

n

f x

g x

a

b x

∞

=

+

=

∑

+

And 0 0

( ) ( )

n

n i n i

n i

f x g x

a b

x

∞

−

= =





=









∑ ∑

. (2)

GENERATING FUNCTION FOR

P

n_:

First, we develop a generating function

g x

( )

for

P

n. So, let 0

( )

_n n

n

g x

P x

∞

=

∑

;

2 1

0 1 2 1

( )

....

_n n _n n

....

g x

= +

P

P x

+

P x

+

P x

+

P x

₊ +

+

2 1

0 1 1

2

xg x

( )

=

2

P x

+

2

P x

+ +

.... 2

P x

_n₋ n

+

2

P x

_n n+

+

....

2 2 1

0 2 1

( )

....

n n n n

....

x g x

=

P x

+

P

₋

x

+

P x

₋ +

+

2

0 1 0

(1 2

x

x g x

) ( )

P

P x

2

P x

2

x

∴

−

= +

−

=

(Since

P

n+1

=

2

P

n

+

P

n−1)

Using

P

0

=

0

and P

1

=

1,

we get

2

0

(1 2

)

n n n

x

P x

x

∞

=

−

∑

. (3)

Therefore,

2

(1 2

)

x

−

_{generates the sequence of Pell numbers.}

P P

n+1 n+2_:

Let

1 2 0

( )

_n _n n

n

g x

P P

x

∞

+ +

=

∑

; so that

2 3 1

1 2 2 3 3 4 4 5 1 2 2 3

( )

....

n n n n n n

....

g x

=

P P

+

P P x

+

P P x

+

P P x

+

P P

₊ ₊

x

+

P

₊

P x

₊ +

+

2 3 1

1 2 2 3 3 4 1 1 2

5

xg x

( )

=

5

P P x

+

5

P P x

+

5

P P x

+

....

+

5

P P x

_n _n₊ n

+

5

P P

_n₊ _n₊

x

n+

+

....

2 2 3 1

1 2 2 3 1 1

5

x g x

( )

=

5

P P x

+

5

P P x

+

....

+

5

P P x

_n₋ _n n

+

5

P P x

_n _n₊ n+

+

....

3 3 1

1 2 2 1 1

( )

....

_n _n n _n _n n

....

x g x

=

P P x

+

P

₋

P x

₋

+

P P x

₋ +

+

(

) (

)

2 3 2 2 2

1 2 2 3 3 4 1 2 2 3 1 2

(1 5

x

5

x

x g x

) ( )

P P

P P x

5

P P x

5

P P x

5

P P x

∴

−

+

=

+

−

+

−

P

n+2

P

n+3

P P

n−1 n

5

(

P P

n+1 n+2

P P

n n+1

)



+

=

+





Q



(

) (

) ( )

2 2 2

2 10

x

60

x

10

x

50

x

10

x

(8)

Thus,

1 2 2 3

0

2

(1 5

5

)

n

n n

n

P P

x

∞

+ +

=

−

+

∑

.

GENERATING FUNCTION FOR

2

n

P

:

Let

2 0

( )

n n

n

g x

P x

∞

=

∑

; so that

2 2 2 2 2 3 2 4 2 2 1

0 1 2 3 4 1

( )

....

_n n _n n

....

g x

=

P

+

P x

+

P x

+

P x

+

P x

+

P x

+

P x

₊ +

+

2 2 2 2 3 2 4 2 2 1

0 1 2 3 1

5

xg x

( )

=

5

P x

+

5

P x

+

5

P x

+

5

P x

+

....

+

5

P x

_n₋ n

+

5

P x

_n n+

+

....

2 2 2 2 3 2 4 2 2 1

0 1 2 2 1

5

x g x

( )

=

5

P x

+

5

P x

+

5

P x

+

....

+

5

P

_n₋

x

n

+

5

P x

_n₋ n+

+

....

3 2 3 2 4 2 2 1

0 1 3 2

( )

....

_n n _n n

....

x g x

=

P x

+

P x

+

P x

₋

+

P

₋

x

+

(

) (

)

2 3 2 2 2 2 2 2 2 2 2

0 1 2 0 1 0

(1 5

x

5

x

x g x

) ( )

P

P x

5

P x

5

P x

5

P x

∴

−

+

=

+

−

+

−

(

)

2 2 2 2

1 2

5

1

n n n n

P

₊

P

₊

P

₊



+

=

+





Q



(

) (

) ( )

(

)

2 2 2

0

x

4

x

0

x

5

x

0

x

1

x

= + +

−

+

−

=

−

Thus,

(

)

2

2 3 0

1

(1 5

5

)

n n n

x

P x

x

∞

=

−

=

−

+

∑

.

3

n

P

:

Let

3 0

( )

_n n

n

g x

P x

∞

=

∑

; so that

3 3 3 2 3 3 3 4 3 5 3 3 1

0 1 2 3 4 5 1

( )

....

n n n n

....

g x

=

P

+

P x

+

P x

+

P x

+

P x

+

P x

+

P x

+

P x

₊ +

+

3 3 2 3 3 3 4 3 5 3 3 1

0 1 2 3 4 1

12

xg x

( )

=

12

P x

+

12

P x

+

12

P x

+

12

P x

+

12

P x

+

....

+

12

P x

_n₋ n

+

12

P x

_n n+

+

....

2 3 2 3 3 3 4 3 5 3 3 1

0 1 2 3 2 1

30

x g x

( )

=

30

P x

+

30

P x

+

30

P x

+

30

P x

+

....

+

30

P

_n₋

x

n

+

30

P x

_n₋ n+

+

....

3 3 3 3 4 3 5 3 3 1

0 1 2 3 2

12

x g x

( )

=

12

P x

+

12

P x

+

12

P x

+

....

+

12

P x

_n₋ n

+

12

P

_n₋

x

n+

+

....

4 3 4 3 5 3 3 1

0 1 4 3

( )

....

_n n _n n

....

x g x

=

P x

+

P x

+

P

₋

x

+

P x

₋ +

+

(

) (

)

(

) (

)

2 3 4 3 3 3 2 3 3 3 3 2 3 2

0 1 2 3 0 1 2

3 2 3 3 3 3

0 1 0

(1 12

30

12

) ( )

12

30

12

x

x g x

P

P x

∴

−

+

=

+

−

+

−

+

3 3 3 3 3

4

12

3

30

2

12

1

0

n n n n n

P

₊

P

₊

P

₊

P

₊

P



−

+

=





Q



(

) (

) ( )

2 3 2 3 2 3 3

0

x

8

x

125

x

0

x

12

x

96

x

0

x

30

x

0

x

= + +

+

−

+

−

+

2 3

4

x

= −

−

Thus,

(

2

)

3

2 3 4

0

1 4

(1 12

30

12

)

n n n

x

P x

x

∞

=

−

=

−

+

∑

.

A LIST OF GENERATING FUNCTIONS

Using the above technique and Pell Identities, we can also derive the following generating functions:

(i)

1 2

0

1 2

(1 2

)

n n n

x

P x

x

∞ − =

−

=

−

∑

₍₄₎

(ii)

1 2

0

1

(1 2

)

n n n

P x

x

∞ + =

=

(9)

(iii)

(

)

2

0

2 1

(1 2

)

n n n

x

Q x

x

∞ =

−

=

−

∑

₍₆₎

(iv)

1 2

0

2(1

)

(1 2

)

n n n

x

Q

x

∞ + =

+

=

−

∑

₍₇₎

(v)

1 2

0

2 6

(1 2

)

n n n

x

Q

x

∞ − =

− +

=

−

∑

(vi) 2 2 0

2

(1 6

)

n n n

x

P x

x

∞ =

=

−

+

∑

(

Using

P

2n+2

+

P

2n−2

=

6

P

2n

,

n

≥

1

)

(vii)

2 1 2

0

1

(1 6

)

n n n

x

P

x

∞ + =

−

=

−

+

∑

(

Using

P

2n+3

+

P

2n−1

=

6

P

2n+1

,

n

≥

1

)

(viii)

(

)

2 2

0

2 1 3

(1 6

)

n n n

x

Q x

x

∞ =

−

=

−

+

∑

(

Using Q

2n+2

+

Q

2n−2

=

6

Q

2n

,

n

≥

1

)

(ix)

(

)

2 1 2 0

2 1

(1 6

)

n n n

x

Q

x

∞ + =

+

=

−

+

∑

(

Using Q

2n+3

+

Q

2n−1

=

6

Q

2n+1

,

n

≥

1

)

(x)

(

)

2 2 2 0

2 3

(1 6

)

n n n

x

Q

x

∞ + =

−

=

−

+

∑

(xi) 2 2 2 0

2

(1 6

)

n n n

P

x

∞ + =

=

−

+

∑

(xii) 3 2 0

5

(1 14

)

n n n

x

P x

x

∞ =

=

−

∑

(

Using

P

3n+3

−

P

3n−3

=

14

P

3n

,

n

≥

1

)

(xiii)

3 2

0

2 14

(1 14

)

n n n

x

Q x

x

∞ =

−

=

−

∑

(

Using

Q

3n+3

−

Q

3n−3

=

14

Q

3n

,

n

≥

1

)

(xiv) 2 1 2 3 0

1

(1 5

5

)

n n n

x

P x

x

∞ + =

−

=

−

+

∑

(xv) 2 2 2 2 3 0

4 5

(1 5

5

)

n n n

x

P

x

∞ + =

+

−

=

−

+

∑

(xvi) 2 2 2 3 0

4 16

4

(1 5

5

)

n n n

x

Q x

x

∞ =

−

=

− −

+

∑

(xvii) 2 2 1 2 3 0

4 16

4

(1 5

5

)

n n n

x

Q

x

∞ + =

+

−

=

−

+

∑

(xviii) 2 2 2 2 3 0

36 16

4

(1 5

5

)

n n n

x

Q

x

∞ + =

+

−

=

−

+

∑

(xix) 2 3 3

2 3 4

0

8 88

120

8

(1 12

30

12

)

n n n

x

Q x

x

∞ =

−

+

=

−

+

∑

(xx) 2 3 1

2 3 4

0

1 4

(1 12

30

12

)

n n n

x

P x

x

∞ + =

−

=

−

+

∑

(xxi) 2 3 3 1

2 3 4

0

8 120

88

8

(1 12

30

12

)

n n n

x

Q

x

∞ + =

+

−

=

(10)

(xxii)

2 3

3 2

2 3 4

0

8 29

12

(1 12

30

12

)

n n n

x

P

x

∞ + =

+

−

=

−

+

∑

(xxiii) 2 3 3 2

2 3 4

0

216 152

104

8

(1 12

30

12

)

n n n

x

Q

x

∞ + =

+

−

=

−

+

∑

(xxiv) 1 2

2 3 4

0

10

(1 12

30

12

)

n

n n n

n

x

P P P

x

∞ + + =

=

−

+

∑

(xxv) 2 3 1 2

2 3 4

0

24 120

120

8

(1 12

30

12

)

n

n n n

n

x

Q Q Q

x

∞ + + =

−

+

=

−

+

∑

(xxvi) 2 0

1

( 1)

n k nk k n k

P x

Q x

x

∞ =

=

−

+ −

∑

(xxvii) 2 0

( 1)

1

( 1)

r

n

r k r

nk r k

n k

P

P x

P

x

Q x

x

∞ − + =

+ −

=

−

+ −

∑

(xxviii) 2 0

2 3

1

( 1)

n k nk k n k

Q x

x

∞ =

+

=

−

+ −

∑

(xxvii) 2 0

( 1)

1

( 1)

r

n

r k r

nk r k

n k

Q

x

Q

x

Q x

x

∞ − + =

+ −

=

−

+ −

∑

GENERATING FUNCTIONS FOR

s

n_:

Let 1

n n i i

s

P

=

∑

and 0

( )

_n n

n

g x

s x

∞

=

∑

; then

(

1

)

0 0

1

2

n n

n n n

n n

s x

P

x

∞ ∞ + = =

=

+

−

∑

[Since

(

1

)

1

2

n

i n n

i

P

₊

=

+

−

∑

]

1

0 0 0

1

2

n n n

n n

n n n

P x

x

∞ ∞ ∞ + = = =





=



+

−





∑



2 2

1

2 (1 2

)

(1 2

)

1

x





=



−

+

−







_{[Using (1), (3) and (5)]}

2 3

1

2

2 (1 3

)

x





=





− + +





`Thus, 2 3

0

(1 3

)

n n n

x

s x

x

∞ =

=

− + +

∑

.

GENERATING FUNCTIONS FOR

P

m n+ _and

Q

m n+ _:

Using Binet formula, generating function for

P

m n+ can be derived as follows:

0 0

m n m n

n n

m n

n n

P

x

γ

δ

x

γ δ

+ + ∞ ∞ + = =



−



=





−





∑

0 0

1

m n n m n n

n n

x

γ

δ

γ δ

∞ ∞ = =





=



−



−



∑



1

m m

x

γ

δ

γ δ

γ

δ

(11)

(

)

(

)

(

)(

)

1

m m

x

γ

δ

γ

γ δ

γ

δ



−



=





−





−





(

)

(

)

(

)

1 1

2

1

m m m m

x

γ

δ

γδ γ

δ

γ δ

γδ

− −



−







=

−





− +

+





(

) (

)

(

)

1 1

2

1

m m m m

x

γ

δ

γ

δ

γ δ

γδ

− −



−

+

−







=

−





− +

+





1 2

m m

P

x

−

+

=

−

Thus,

1 2

0

1 2

n m m

m n n

P

x

P

x

∞

− +

=

+





=





−





∑

.

Likewise, it can be shown that

1 2

0

1 2

n m m

m n n

Q

x

Q

x

∞

− +

=

+





=





−





∑

.

EXPONENTIAL GENERATING FUNCTIONS

Since 0

!

n t

n

t

e

n

∞

=

∑

, it follows that 0

!

n n

x

n

x

e

n

γ ∞

γ

=

∑

and 0

!

n n

x

n

x

e

n

δ ∞

δ

=

∑

0

!

0

!

x x n n n n

n

n n

e

x

P

n

γ δ

γ

δ

γ δ

∞ ∞

= =





−

∴

=





=

−

∑



−



∑

. (8)

Thus, the exponential function

x x

e

γ

e

δ

γ δ

−

_{generates the numbers}

!

n

P

n

.

Likewise, we can show that

0

!

n

x x

n n

x

e

Q

n

γ δ ∞

=

+

=

∑

; (9)

That is

(

)

x x

e

γ

+

e

δ

generates the numbers

!

n

Q

n

.

More generally, we can show that

0

!

k_x k_x

n nk

n

P

e

x

n

γ δ

∞

=

−

=

−

∑

_. ₍₁₀₎

Likewise,

0

!

k_x k_x _n

nk

n

Q

e

x

n

γ δ ∞

=

+

=

∑

. (11)

Using (8), we can write

(

)

(

)

(

) (

)

1 2 1 2

0

!

1

2

1

2

x x _n

n n

e

x

P

n

+ − _∞

=

−

=

+

− −

∑

Or

(

1 2

)

(

1 2

)

0

2 2

!

n

x x

n n

x

e

P

n

∞

+ −

=

−

=

∑

Or

2 2

0

2 2

!

n

x x x

n n

x

e

P

n

∞ −

=



−



=





∑

2

sinh 2

2 2

n x

n

x

e

x

P

∞

(12)

Thus,

0

sinh 2

2

!

n x

n n

x

e

x

P

n

∞

=

∑

Similarly,

0

2

cosh 2

!

n x

n n

x

e

x

Q

n

∞

=

∑

.

IDENTITIES USING GENERATING FUNCTIONS

We have 1 0

1

n n n

P x

D

∞

+ =

=

∑

,

1 0

1 2

n n n

x

P x

D

∞

− =

−

=

∑

and

(

)

0

2 1

n n n

x

Q x

D

∞

=

−

=

∑

where

2

1 2

D

= −

x

−

x

_. _Also,

(

)

2 1

x

1

1 2

x

D

−

=

+

−

.

It follows that

(

)

0 0 0

2 1

1

1 2

n n n

x

D

∞ ∞ ∞

= = =

−

=

+

−

∑

1 1

0 0 0

n n n

Q x

P x

∞ ∞ ∞

+ −

= = =

⇒

∑

=

∑

+

∑

(

1 1

)

0

n

n n

n

P

x

∞

+ −

=

∑

+

∴

Q

n

=

P

n+1

+

P

n−1

.

Next we shall prove that

P Q

m n

+

P Q

m−1 n−1

=

Q

m n+ −1

.

(

1 1

)

1 1

0 0 0

m m m

m n m n n m n m

m m m

P Q

x

Q

P x

Q

P

x

∞ ∞ ∞

− − − −

= = =

+

=

+

∑

1

1 2

n n

x

Q

D

−

D

−

=

+

(

)

1

2

1 1 2

n n n n n

Q

x

Q

x

D

−

+

−

− −

+

−

=

1 0

m m n m

Q

x

∞ + − =

=

∑

1 1 1

.

m n m n m n

P Q

₋ ₋

Q

_{+ −}

∴

+

=

Similarly, we can prove other identities like

P P

m n

+

P

m−1

P

n−1

=

P

m n+ −1

,

Q Q

m n

+

Q

m−1

Q

n−1

=

8

P

m n+ −1 etc.

COMBINATORIAL IDENTITIES USING GENERATING FUNCTIONS

Let 0

( )

!

n

n n

t

A t

a

n

∞

=

∑

and 0

( )

!

n

n n

t

B t

b

n

∞

=

∑

; then

0 0

( ) ( )

!

n n

k n k

n k

n

t

A t B t

a b

k

n

∞

−

= =



 



=



 



 





∑ ∑

, (12)

0 0

( ) (

)

( 1)

!

n n

n k

k n k

n k

n

t

A t B

t

a b

k

n

∞

−

= =



 



− =



−

 



 





∑ ∑

(13)

and 0 0

(2 ) ( )

2

!

n n

k

k n k

n k

n

t

A t B t

a b

k

n

∞

−

= =



 



=



 



 





∑ ∑

(13)

In particular,

( )

t t

e

A t

γ δ

−

=

−

_and

B t

( )

=

e

t_;_then

2 2

(2 ) ( )

t t

t

e

A t B t

e

γ δ



−



=





−





2 2 0 0

2

!

n t t

n

k t

k n k

n k

n

t

e

a b

e

k

n

γ δ

∞ − = =



 





−



=



 







−

 









∑ ∑

By (14)

(1 2 ) (1 2 )

0 0

2

!

n t t

n k

k

n k

n

t

e

P

k

n

γ δ

+ + ∞ = =



 





−



=



 







−

 









∑ ∑

By (8)

2_t 2_t

e

γ

e

δ

γ δ



−



=





−





_(Since 2

2

1

0

γ

−

γ

− =

₎

2 0

!

n n n

t

P

n

∞ =

=

∑

By (10) Thus, 2

0 0 0

2

!

n n n k k n

n k n

n

t

P

k

n

∞ ∞ = = =



 



=



 



 





∑ ∑

∑

Equating the coefficients of

!

n

t

n

yields the combinatorial identity

2 0

2

n k k n k

n

P

k

=

 

=

 

 

∑

Now taking,

( )

t t

e

A t

γ δ

−

=

−

_and

B t

( )

=

e

−t_;

2

( ) (2 )

t t

t

e

A t B t

e

γ δ

−



−



=





−





2 0 0

( 2)

!

n t t

n

n k t

k n k

n k

n

t

e

a b

e

k

n

γ δ

∞ − − − = =



 





−



−

=



 







−

 









∑ ∑

( 2 ) ( 2 )

0 0

( 2)

!

n t t

n

n k k

n k

n

t

e

P

k

n

γ δ

− + − + ∞ − = =



 





−



−

=



 







−

 









∑ ∑

t t

e

δ

e

γ

γ δ

− −



−



=





−





_(Since

γ δ

+ =

2

₎

0

(

)

( 1)

!

n n n

t

P

n

∞ =

−

=

∑

−

1

0

( 1)

!

n n n n

t

P

n

∞ − =

=

∑

−

Thus, 1

0 0 0

( 2)

( 1)

!

n n

n

n k n

k n

n k n

n

t

P

k

n

∞ ∞ − − = = =



 



−

=

−



 



 





∑ ∑

∑

!

n

t

n

_{yields another combinatorial identity}

1 0

( 2)

( 1)

n

n k n

(14)

Obviously, by selecting

A t

( )

and

B t

( )

as suitable exponential functions, we can apply this method to derive other Pell and Pell-Lucas combinatorial identities.

For example, choosing

( )

t t

A t

=

e

γ

+

e

δ

and

( )

t

B t

=

e

, we get

2 0

2

n k k n k

n

Q

k

=

 

=

 

 

∑

; and taking

( )

t t

A t

=

e

γ

+

e

δ

and

( )

t

B t

=

e

−

, we

get 0

( 2)

( 1)

n

n k n

k n k

n

Q

k

− =

 

−

 

= −

 

∑

.

COMBINATORIAL IDENTITIES USING THE DIFFERENTIAL OPERATOR

Since 0

( )

!

n n n

t

A t

a

n

∞

=

∑

, it follows that 0

( )

!

r n n r r n

d

t

A t

a

dt

n

∞ + =

=

∑

(15)

In particular, taking

2 2

( )

t t

e

A t

α β

−

=

−

_and

B t

( )

=

e

t_;_{we get}

2 2

( ) ( )

t t

t

e

A t B t

e

α β



−



=





−





0 0

( ) ( )

2

!

n n k k n k

n

t

A t B t

P

k

n

∞ = =



 



=



 



 





∑ ∑

0 0

( ) ( )

!

n n

k n k

n k

n

t

A t B t

a b

k

n

∞ − = =





 





=





 







 









Q

∑ ∑



Using (15) we can write

0 0

( )

2

!

r n n

k r k r t n k

n

d

t

B t

A t

P

k

dt

n

∞ + + = =



 



=



 



 





∑ ∑

Or 2 2 0 0

2

( )

!

n r t t

n k r

k r t

n k

n

t

d

e

P

B t

k

n

dt

α β

∞ + + = =



 





−



=



 







−

 









∑ ∑

( )

2

( )

2

.

r _t r _t

t

e

α β

α

β

α β



−



=





−









( )

(2 1)

( )

(2 1)

2

α

r

e

α t

2

β

r

e

β t

α β

+

−

+

=

−

2 2

2

r t r t

r

e

α β

α

β

α β



−



=





−





_(Since 2

2

1

0

α

−

α

− =

)

2 0

2

!

n r n r n

t

P

n

∞ + =

=

∑

Thus, 2

0 0 0

2

!

n n

n

k r r

k r n r

n k n

n

t

P

k

n

∞ ∞ + + + = = =



 



=



 



 





∑ ∑

∑

.

!

n

t

n

yields the combinatorial identity

2 0

2

.

n

k r r

k r n r

k

n

P

k

+ + + =

 

=

 

 

∑

Or 2 0

2

.

n k

k r n r

k

n

P

k

+ +

(15)

HYBRID IDENTITIES

Here we derive some hybrid identities involving both Pell and Pell-Lucas numbers. If we choose

( )

t t

e

A t

γ δ

−

=

−

_and

( )

t t

B t

=

e

γ

+

e

δ

; then

(

)

( ) ( )

t t

e

A t B t

e

γ δ γ δ

γ δ



−



=





+

−





Or 2 2

0 0

!

n t t

n

k n k

n k

n

t

e

P Q

k

n

γ δ

∞ − = =



 





−



=



 







−

 









∑ ∑

[By (8) and (12)]

Or 0 0 0

(2 )

!

n n

n

k n k n

n k n

n

t

P Q

P

k

n

∞ ∞ − = = =



 



=



 



 





∑ ∑

∑

Or 0 0 0

2

!

n n

n

k n k n

GENERATING FUNCTIONS FOR PELL AND PELL-LUCAS NUMBERS

CONTENTS

Page No.

M. GURUSAMY

CHIEF PATRON

PATRON

ADVISORS

PROF. R. K. SHARMA

DR. AMBIKA ZUTSHI

PROF. ANIL K. SAINI

MOHITA

PROF. A. SURYANARAYANA

DR. PARDEEP AHLAWAT

MOHITA

LEGAL ADVISORS

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INTRODUCTION

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GENERATING FUNCTIONS

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EQUALITY OF GENERATING FUNCTIONS

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GENERATING FUNCTION FOR

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GENERATING FUNCTIONS FOR

IDENTITIES USING GENERATING FUNCTIONS

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COMBINATORIAL IDENTITIES USING THE DIFFERENTIAL OPERATOR

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