Internat. J. Math. & Math. Sci.
VOL. II NO. 2 (1988) 405-412
405
SOME APPLICATIONS OF LEGENDRE NUMBERS
PAUL W. HAGGARD
Department of Mathematics, East Carolina University
Greenville, North Carolina 27858 U.S.A.
(Received August 26, 1986)
The associated Legendre functions are defined using the Legendre numbers.
ABSTRACT.
From these the associated Legendre polynomials are obtained and the derivatives of
these polynomials at x
0 are derived by using properties of the Legendre numbers.
These derivatives are then used to expand the associated Legendre polynomials and
in series of Legendre polynomials.
x
n
Other applications include evaluating certain in-
tegrals, expressing polynomials as linear combinations of Legendre polynomials, and
A connection
expressing linear combinations of Legendre polynomials as polynomials.
between Legendre and Pascal numbers is also given.
KEY WORDS ANY PHRASES. Aoed Legenre functions and polynomials, Legenre polyand rntegr of Legennomials, divatives of associated Legende polynomi,
dre poyno, Legene and Pasc numbers.
1980 MATHAIICS SUBJECT CLASSIFICATION CODES. IOA40, 26C99, 53A45.
1.
INTRODUCTION.
The Legendre numbers were introduced and many of their elementary properties were
We apply these pzoperties to a variety of problems and the use of
developed in [i].
Legendre numbers may provide somewhat simpler solutions to the problems.
2.
DERIVATIVES OF ASSOCIATED LEGENDRE POLYNOMIALS AT
For
n
and
s
x
0.
non-negative integers the associated Legendre functions are
defined as usual by
pS(x)
n
where
see [i]
P (x)
n
(i
x2)S/2Dspn (x),
D
is a Legendre polynomial and
by using the Legendre numbers
pi,
n
n
P (x)=
n
Equation (2.1) becomes
i=O
s
-
(2.1)
ds
Ox
s
(where
Since
P (x)
pi(x)
n
n
x=0
can be expressed,
)’ as
pixi
n
i
(2.2)
P.W. HAGGARD
406
pixi
n
pS(x)
n
It is clear that
pS(x)
n
(I-
x2)S/2Ds I
i=0
n
il
(2 3)
is a polynomial of degree
and recall, see [1], that
pi
for
0
and
n
for
n
Thus, let
even.
s
2m
s
Oitting these
of different parity.
i
null terms from (2.3), one has
n__
xZ)mDZ
(I
p2i x2i
21
m
n
n-I
2
x2 )mD2m
(I
even
n
(2i)
i= 0
(2.4)
p2i+l x2i+i
n
odd.
n
(2i+i)
i= 0
Taking the indicated derivatives in (2.4) gives
n-2m
2
2)m I
i= 0
(i
p2m+2 ix2i
n
x
P2nm(x)
n-2m-I
2
x2) m
(I
(2.5)
p2m+2iix21+1
n
DkF2nm(x),
Leibniz’ Theorem is used.
odd.
n
(2i+i)
i= 0
To obtain
even
n
(2i)
For
one has from
0 < k < n,
(2.5)
Dk-i(l_
il
Dkp2nm (x)
i
odd,
Dkp2nm(X)]x=
n-2m
2
Di I
if 0
n
(2.6)
for
for
0
even,
n
0
0
odd.
for
O
even and the second is
k
for
odd, since
i
odd.
even, since in each term the first factor is
k
for
i
n
i
0
even.
even,
n
p2m+2ix2i
n
(2i)
IJ
x=0
0,
i
0
i>n-2m
p2m+i
n
odd
i
even,
(2 7)
i
_<
n
2m,
odd,
Di
n-2m-I
2
i=0
Further,
0
0
odd and the second is
One observes that for
and for
,
’i
Dkp2m (x)] x=O
n
i=0
in each term the first factor is
n
even
[[ki]Dk_i(l_x2)mDi l Pn2m+21+Ix21+l]
From this equation one sees that for
for
n
n-2m-i
2
i=0
For
n
x2)mDii.0=
O=
p2m+2i+l x2i+i
n
(2i+i)
0, i
even
O, i>n-2m
p2m+i,
n
i
odd,
(2.8)
i < n
2m
407
SOME APPLICATIONS OF LEGENDRE ERS
odd
O, k- i
D
k-i
(1
x
2)mlx=0
(2.9)
O, k- i > 2m
m
k-i
(-i)
D
j--O
j()x2j]x=0
i even, k
k
i < 2m.
The derivative on the right in (2.9) can be expressed using factorial notation as
k-i
(-l)J(j)(2j)(k-i)x2j-k+
k- i even, k- i < 2m,
x=0
j=--which reduces to the first term
(-I)
for
(k- i)!
k-i
even,
k- i
m,
2
Now (2.9) can be expressed as
0.
x
k-i
2
Dk-i(l-
i odd
0, k
O, k- i > 2m
x2)m]x=O
k-i
[]
m
(_i)2
(k- i)!, k- i even,
k-i
(2.10)
< m.
Using Equations (2.7), (2.8), and (2.10) in (2.5) and the observations following
(2.6), one sees that
Dkp2m(x)
n
x=0
0, n
even,
0, n
odd,
k___!
2
k (-i)
[
i=O i,
recall that
and
n
0
for
odd
(2.11)
even
k!m!p2m+i
n
0
if
i > n- 2m,
k
for
i
i
k
and
n
() (m-)!
where a term in the series above is
p2m+i
k
k
of the same parity,
m <
is odd or if
odd and
n
even,
-4k_:
Also,
and for
i
even
odd.
n
Equation (2.11) provides a formula
for evaluating
Dm(x) ]x=0.
answers obtained by (2.11) agree with those obtained by other methods
Of course, the
and can be
easily verified for small integers
k, m, and n.
n
ASSOCIATED LEGENDRE POLYNOMIALS AND x
AS SERIES OF LEGENDRE POLYNOMIALS.
It is known that an associated Legendre polynomial can be expressed as a series of
3.
Legendre polynomials.
Equation (2.11) and a table of Legendre numbers, see Table I,
can be used to provide a formula for the coefficients in the series.
method, let
p2m(x)
n
Take
derivatives to obtain
n
In these
n
+
1
n
+
i
identities let
identities and recall that
n
[ AiPi (x).
(3. i)
i=0
other identities,
n
Dkp2m(x)n
-.
To outline the
n
i=0
x
AiDiPi(x),
0.
k
i, 2, 3
,n.
Use (2.11) on the left sides of the resulting
DkPi(X)]x= 0 pk.,-i
see [i].
The right sides can be
P.N. IGGARD
408
simplified by using
k
0
Pi
values of the
k > i
for
Ai’s
unknowns can be solved for the
The system of
Dnp 2m (x)
n
+ 1
identities in
An, An_l, An_ 2
can be obtained in the order
Ai’s
n
(3.1) gives the desired expansion.
and
A0,
n
+ I
The
by
x--O
pn
n
n-ip2m (x)
x=O
n
k=n-l+l
(3.2)
,l<i<n
pn-i
n-i
,l<i<n,
pn-i
n-i
for
where
p
i
and
even
pO
n
p2
pl
n
n
0
n
3
0
3
8
0
0
15
8
0
o
105
8
15
-4-
15
2
105
0
o
4-o
105
8-W
945
48
TABLE i.
n
odd.
i
p4
p3
n
p6n
p5
n
n
p8n
p7
n
3
0
0
x
for
1
2
Now,
i 1
15
-- 0
105
105
2
0
945
o
945
o
945
o
0
10,395
0
8-
LEGENDRE NIR4BERS
lO,395
I0,395
0
135 135
135,135
0
2,027,025
pm
n
can be expanded in a series of Legendre polynomials in a similar way.
Let
n
n
x
[ AlP i
(3.3)
i=0
and proceed as in the derivation of (3.2) to obtain in order
An, nA-l’ znA-"’
uA’
as
n!
n
pnn
i
Ai
1
p.i
n
k, k
n-i
--i
j
odd
[IPi+2j Ai+2j
(3.4)
i
n
k, k
even.
SOME APPLICATIONS OF LEGENDRE NUMBERS
Ai’s
With these values of the
409
(3.3) gives the desired expansion, which agrees with the
known expansion
n
[] (2n-4k+1) Pn_2k (x)
2
4.
n
()n-k
k!
k=0
SOME INTEGRALS INVOLVING LEGENDRE NUMBERS.
In [i], the result
ip
0
for
Pn+l
n(X)dx
Here, two other important integrals are ex-
any positive integer is given.
n
It is known, see [2], that if
pressed in terms of Legendre numbers.
IPn
More generally, if
m
(x)
(4. i)
n
Pm’(X)
and
m
n,
then
1
2n+l
dx
(4.2)
are different non-negative integers, Rainville,
n
[2], gives the result
With
a
even and
0, b
P
n
i,
P
(1-x2)[p(x)Pn(X)-Pm(X)P(x)]
Pn(X)Pm(X)dx;
(n-m)(n-+l)
m
0
for
m
Iip (X)Pm(X)
0 n
and
dm(pn
pmn
and the results
dx
(x))]
(4.3)
Pln= p1
m
x=O
0
for
and
m
n
odd from [i], Equation (4.3) becomes
n
pip
_p pl
ran_ran
(m-n) (m+n+l)
0,
mland
n
of the same parity
(4.4)
PP
m n
m
n) ((m+n+l)
(m
and
n
of different parity.
A third integral can be evaluated as shown by the following computation.
With
n > l,
ffl
joXnPn-2k(X)dx JO xn
pm
n-2k
i
n-2k
l
m= 0
]
.I
-dx
x
m!
J
(by 2.2)
xm+n.)dx
m=O
n-2k -i pm
xm+n dx
n-2k
.
i
[0
J
n-2k
pm
[
m=O
J
n-2k
pm
n-2k
m!
m!
0
n-2k
xm+n+l].I
m=O
n-2k
o
Therefore,
fxnPn_2
(x) dx
k
i
0
pm
n-2k
m.’ (m+n+l)
pm
n-2k
n-2k
v
L
m! (m+n+l)
m=0
Since the value of this integral is known one can use this value for the series.
(4.5)
410
P.W. HAGGARD
Thus,
m
n-2k
Pn-2k
"
5.
n
m! (m+n/l)
m=0
(4.6)
3
"n’k! ()n-k
POLYNOMIALS AS LINEAR COMBINATIONS OF LEGENDRE POLYNOMIALS.
Since the Legendre polynomials form a simple set, any polynomial of a single vari-
able can be expressed as a finite series of Legendre polynomials.
Using Table 2, this
can be done in much the same way any polynomial can be expressed in terms of factorials.
Consider the problem of expressing
5x3
H(x)
2
4x- 3
/
By continued subtraction of Legendre polynomials we
in terms of Legendre polynomials.
obtain a zero remainder.
3x
Detaching coefficients,
2P
H(x)
5
3(x)
5
-3
Diff.
-2P2 (x)
-4
Dif.
7Pl(X
Diff.
-4Po (x)
O.
Diff.
Thus
H(x)
2P
H(x)
2P3(x
3(x)
+
2P
7P l(x) + 4P
2(x)
O(x)
0,
from which
1
Qn
0
Qn
0
1
0
3
8
0
2
Q
0
+
2P2(x
3
Qn
4
Qn
7Pl(X
4P0(x
5
Qn
Q6n
7
Qn
0
15
8
0
70
8
0
63
8
I-
0
105
16
0
315
16
0
231
16
0
35
16
0
315
16
0
693
16
0
35
128
0
1260
128
0
6930
128
0
35
pm
TABLE 2.
8
Qn
2
0
3O
8
5
(5.1)
12,012
128
429
16
0
6435
128
SOME APPLICATIONS OF LEGENDRE NUMBERS
411
For a second method, set
5x
3
3x
2
+ 4x
=- AP3(x)
3
+ BP 2(x) +
identities
pm (0)
and use
pm
DRo(X).
in each of the four
0
x
to obtain the system
n
n
let
x,
Take the first three derivatives with respect to
+
CPI(X)
AP3 + BP2 + CPI + DPo
-3
4= AP 3
AP 3 +
30
3
(5.2)
2
2
-6
gp2
3
7, and
-2, C
Next, use Table i and solve for A
2, B
listed to obtain (5.1) again.
More generally, if V(x) is a polynomial of degree n
D
-4
in the order
in
x,
write
n
[. AiPi(x)’
V(x)
(5.3)
iffiO
take
derivatives, let
n
’obtain the
Ai’s
x
in the
0
in the order
n
i
v
A
1
identities, and use
0
i
to
(n)
+
n
p(m)(o)
n
pm
to
n
as
(0)
pn
n
n
V (i) (0)
(5.4)
n-i
[
AjPj
j=n-i+l
n-l,n-2
i
pn-i
,l,0
n-i
Since
pm
n
0
for
m
+
n odd,
the second equation of
(5.4) can be expressed as
n-i
2
v(i) (o)
IAI+2J Pi+2j
A
i
i
i
(5.5)
n-l, n-2,.,,,l,0
Pi
Table 2 can be used to evaluate a finite series of Legendre polynomials as a
polynomial in
As an example, we evaluate
S(x)
(x)
P7(x)
x.
Detaching coefficients,
P7(x)
Then,
S(x)
6.
-
5
4
5
185
-x
0
105
0
o
-sP5 (x)
-i-
5
-4P 6 (x)
Sum
35
0
75
o
185
16
105
4
105 2
4P6
5P
5(x).
693
315
16
0
0
315
4
0
350
o
315
8
315
4
1323
16
1015
16
3
315 4
x +
+ 1015
16
i--
1323 5
x
16
0
429
16
231
4
231
4
429
16
213 6
429 7
+-x
----x
LEGENDRE AND PASCAL NU}ERS.
Consider Table 3, which gives values for
result that can be easily proved.
m.
L
The entries shown are integers, a
The alternate diagonals have entries of the form
412
P.W. HAGGARD
2
n+l n-i
Pn+i
0
i
(n-i)
reading from upper right to lower left.
1
0
Lm
2
L3
L
L
Ln
n
n
n
If
(6.1)
n
to
2np/n!
is factored from
4
L
L5
L
n
n
6
L7
n
n
0
2
-2
0
6
0
-12
0
20
6
0
-6O
0
7O
0
60
0
-280
0
252
-20
0
420
0
-1260
0
924
0
-2B0
0
2520
0
-5544
0
3432
70
0
-2520
0
13,860
0 -24,024
0
Lmn
TABLE 3.
L
8
12,870
2npm
n
2n_m
n
m!
each entry on such a diagonal, the remaining factors are
(-1)iC(n,i).
In notation,
one has
2n+ipn-in+i
(-1)
i2npncn (n, i)
i:0
n!
(n-i)
to
(6.2)
n.
Equation (6.2) can be simplified to
(_l)in
(n-i)
-i n-i
2
Pn+i
pnn
C(n,i), i
0
to
(6.3)
n,
which shows a connection between Legendre numbers and Pascal numbers.
be easily proved using the general form of the Legendre numbers
pmn’
This result can
given in [i].
REFERENCES
i.
HAGGARD, P.W.
2.
RAINVILLE, E.D. Special Functions, The Macmillan Company, New York, 1960.
COPSON, E.T. An Introduction to the Theory of Functions of a Complex Variable,
Oxford University Press, London, 1935.
RICHARDSON, C.H. An Introduction to the Calculus of Finite Differences, C. Van
Nostrand Company, Inc., New York, 1954.
3.
4.
On Legendre Numbers, International Journal of Mathematics and Mathematical Sciences, Volume 8, Number 2, 1985, 407-411