Milovanovic

7/26/2019 Milovanovic

1/5

ON SOME PROPERTIES OF H UMBERT S POLYNOMIALS

G .

V .

M I LOV A N OV I C

University of Nis, P.O. Box 73, 18000 Nis, Yugoslavia

G, DJORDJEViC

University of Nis, 16000 Leskovac, Yugoslavia

(Submitted February 1986)

SNTRODUCTiON

I n 1 9 2 1 , H u m b e rt [8 ] d e f i n e d a c l a s s o f p o l y n o m i a l s

{ I I

n ? m

}

=0

b y t h e g e n -

e r a t i n g f u n c t i o n

( 1 -

mx t

+

t

m

)~

x

= f H

r

A

m

{x)t

n

.

(1 )

n = 0

These satisfy the recurrence relation

(n

+

l)n

+

sm

(a;)

-

mx(n

+

X)Tl^

m

(x) - (n

+

mX

-

m

+

l)Iin-

m+ 1

,

m

(x)

= 0.

Particular cases of these polynomials are Gegenbauer polynomials [1]

and Pincherle polynomials (see [8])

Later,Gould [2] studied a class of generalized Humbert polynomials

P

n

(m, x, y

s

p

5

C)

defined

by

(C - mxt +

yt

m

)

p

= P

n

( ^ , x

9

y

9

p, C)t

n

, (2)

H=0

where TTZ

1 is an

integer

and the

other p aramete rs

a r e

unrestricted

in

gener al.

The recurrence relation

for the

generalized Humbert po lynomial s

is

Cn P

n

- m( n - 1 - p)xP

n

_

1

+ in - m- mp)yP

n

_

m

==

0 n > m > 1, (3)

where we put P

n

=

P

n

(m

9

x

9

y

9

p

9

C).

In

[6]j

Horadam

and

Pethe investigated

t he

polynom ials associated with

the

Gegenbauer polynomials

[n/2]

(X)

where (A)

0

= 1,

( X )

n

=

X(X

+ 1 )

o e e

(A +

n

- 1 ) , n = 1, 2, ... . Listing the

polynomials of (4) horizontally and taking sums along the rising diagonals,

Horadam and Pethe obtained the polynomials denoted by p

x

(x)

.

For these poly-

nomials

,

they proved that the generating function

G

x

(x

9

t)

is given by

G

X

(x

5

t) =E

p

X

(x)t

n

~

1

= (1 - 2x t +

t

3

) "

A

.

(5)

n =1

356 [Nov.


2/5

ON SOME PROPERTIESOF HUMBERT'S POLYNOMIALS

Some special cases

of

these polynomials were considered

in

several papers

(see

[3]

5

[4],and[7],forexample).

Comparing (5) to (1),we seethat their polynomialsareHumbert polynomi-

als

form =3,

with

x

replaced

by

2x/3

9

i.e.,

p

+1

(x)

=II

A

(2#/3).

2. THE

POLYNOMIALS

p

A

x)

In this paper

9

we

consider

the

polynomials

{p* }

defined

by

Pl

m

(x) = Jl

x

m

(2x/m).

Their generating function

is

given

by

G

x

(x ,

t) = (1 - 2xt + t

m

)-

x

=f) p\

m

(x)t

n

.

(6)

Yl =0 '

Note that

p

x

(x) = C^(x)

(Gegenbauer polynomials)

and

Pn 3^)

=

Pn+

i(

x

^

(Horadam-Pethe polynomials).

For m = 1, we have

G

x

(x, t) = 1 - (2x - l ) t r

A

= E p i Ax)t

n

n

=

0

an d

^ .

P^G =

-ir ~

n

x

) 2x - vf -~zr 2x - D .

These polynomials

can be

obtained from descending diagonals

in the

Pascal-type

array

for

Gegenbauer polynomials

(see

Horadam

[5]).

Expanding

the

left-hand side

of

(6),

we

obtain

the

explicit formula

[n/m]

(A) ,

_,.

7

^w.m

^

0

Ac

(n

- mk)

These polynomials can be obtained from(2) byputting

C = y =

1, p = -A,

and

x : = 2x/m.

Thenwehave

p

x

(re)= P

n

(jn

9

2x/m

9

1, -A, 1).

Also,if we put C =z/= m/2 and p = -A, weobtain

p

A

(x) = (-) P

n

Qn

9

x

5

m/2

9

-A, mil).

Then,

from (3),

we get the

following recurrence relation

np ^

m

(x ) =(A+n - l)2xp

n

_

lim

(x ) - (n +m(X-

1)

)p

n

_^

m

(*), (8)

for

n

> m > 1.

The starting polynomials

are

p

A

Or) =

r-

(2x)

n

s

n = 0, 1, . . .

s

m - l .

1987]

357


3/5

ON SOME PROPERTIESOF HUMBERT'S POLYNOMIALS

Remark: Forcorresponding monic polynomialsp

A

, wehave

p

x

(x) = xp

x

(x) - b p

x

(x )

,

n

^

m

> 1,

V

X

(x) = X

n

9

0 E

k

f

r

= f

r+k

-

Theorem

2: The

polynomial

x

H-

p

x

(x) is a

particular solution

of the

follow-

ing w-order differential equation'

m

y

m)

+ E

a

s

x

s

y

{s)

= 0, 13)

=

0

where

the

coefficients

a

s

are

given

by

a

s

= ~ A

s

f

0

(s = 0,1,...,m). (14)

Proof: Let n = pm+ q,

where

p =

[n/w]

and 0 < q g.

If

we

substitute these expressions

in the

differential equation

(13) and

compare

the

corresponding coefficients,

we

obtain

the

following relations:

E (

n

~

mk

)sla

s

=2

m

k(X +

n

- (rn- l)k)

m

15)

(k

= 0

9

1,

...,

p - 1)

and

E (";

mP

)s a

s

=2

m

p(X+ -(m- D p ) ^ -

s

=

0

b

Firstsweconsiderthesecond equality,i.e.,

q

(H\l

A

V

=

2

m

^ ^ l \ + a + - , .

m

lm-i

=

Asl

m

J

o

m \

H

m In

This equalityiscorrect, becauseit isequivalentto

(1+ A)V

0

=E

q

f

Q

= f

q

= f(q).

Equality

(15) can be

written

in the

form

(16)

1987] 359


5/5

ON SOME PROPERTIES OF HUMBERT'S POLYNOMIALS

Since

t

->

f{t)

is a polynomial of degree

m,

the last equalities are correct;

(16) is a forward-difference formula for / at the point t = n - mk>

Thus,the proof is completed.

From (14 ), we have

2

m

n/n + mX\ 2

m

n

m

' \ ^ ,. ^ . ...

a

0

^{)

m

^ . ^ ( n + m t t + i - 1)),

=

{

_ /^A|

1

77Z l V 77? /

m

- i \ 777

777

- 1

Since

/() = -( j t

m

+ terms of lower degree,

we f ind

2

m

(m - If'

1

m m \ m

For m =1, 2, 3, we have the following differential equations:

(1 - 2x)y

f

+ 2ny = 0,

(1 - x

2

)y - (2X + l)xy

f

+ n(n + 2X)y = 0,

(l - | | X

3

)T/'" - -^(2A + 3)^2/

- Jy(3n(n + 2X +1) - (3A +

2)

(3A +5))xz/'

+ - ~

n{n

+

3A)

(n + 3(A +

l))y

= 0.

Note that the second equation is the Gegenbauer equation.

REFERENCES

1. L. Gegenbauer. "Zur Theorie der Functionen

C^(x)."

Osterreichische Aka-

demie der Wissenschaften Mathem atisch N aturwissen Schaftliche Klasse Den k-

scriften 48

(1884):293-316.

2. H. W. Gould.

ff

Inverse Series Relations and Other Expansions Involving Hum-

bert Polynomials." Duke Math. J. 32 , no. 4 (1965):697-712.

3. A. F. Horadam. "Polynomials Associated with Chebyshev Polynomials of the

First Kind." The Fibonacci Quarterly 15, no. 3 (1977):255-257.

4.

A. F. Horadam. "Chebyshev and Fermat Polynomials for Diagonal Functions."

The Fibonacci Quarterly 17, no. 4 (1979):328-333.

5. A. F. Horadam. "Gegenbauer Polynomials Revisited."

The Fibonacci Quarter-

ly 23,

no. 4 (1985):294-299, 307.

6. A. F. Horadam & S. Pethe. "Polynomials Associated with Gegenbauer Polyno-

mials."

The Fibonacci Quarterly 19, no. 5 (1981):393-398.

7. D. V. Jaiswal. "On Polynomials Related to Tchebichef Polynomials of the

Second Kind." The Fibonacci Quarterly 12, no. 3 (1974):263-265.

8. P. Humbert. "Some Extensions of Pinch erle

s Polynomials." Proc* Edinburgh

Math.

Soc (1) 39 (1921):21-24.

360

^o*^

[Nov.

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