407
Internat. J. Math. & Math. Scl.
Vol. 8 No. 2 (1985) 407-411
RESEARCH NOTES
ON LEGENDRE NUMBERS
PAUL W. HAGGARO
Department of Mathematics, East Carolina University
Greenville, North Carolina 27834 U.S.A.
(Received February 7, 1984)
The Legendre numbers, an infinite set of rational
ABSTRACT.
numbers are defined
from the associated Legendre functions and several elementary properties are pre-
A general formula for the Legendre numbers is given.
sented.
Applications include
summing certain series of Legendre numbers and evaluating certain integrals.
Le-
gendre numbers are used to obtain the derivatives of all orders of the Legendre
polynomials at x
i.
KEY WORDS AND PHRASES.
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of Legnee polyno
THEMATI SUBJECT CIFICATION CES.
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INTRODUCTION.
For example, Stlrllng
Many sets of numbers are associated with polynomials.
numbers of the first and second kinds, Bernoulli
defined from certain polynomials.
2.
numbers
We follow this pattern by defining the Legendre
numbers from the associated Legendre functions.
ropertles
and Euler numbers are
These Legnedre numbers have many
and applications and our purpose is to examine some of these.
THE LEGENDRE NUERS.
The associated Legnedre functions are defined by
m
m
pm(x)
(l
(x)),
n
x2) (Pn
where
P (x)
is the nth order Legendre polynomial and
n
integers.
these
m
and
n
are non-negatlve
Using Rodriques’ formula, one has
P(x)
With
(2.1)
x2.)2 Dm+n(x2
.(I.
n[2
n
I)
n.
(2.2)
the legendre numbers can be defined as follows.
Definition i.
The Legendre number
pm
n’
are the values of
Pro(x)
for
From (2.1) and the definition, it is clear that
pmn
where
p(m) (0)
is the mth derivation of
p(m)(o),
n
P (x),
(2.3)
evaluated at
x
O.
From (2.2),
P.W. HAGGARD
408
one sees that
pmn
From (2.4) it is clear that
m
+
n
even and
m < n,
i
n!2 n
pm
for
0
n
Dm+n (x2
+
m
n
I )n
(2.4)
Ix=O.
odd and also for
there is exactly one term (2.4) void of
m > n.
x
For
(before taking
x-O) and this term simplifies to the third part of the explicit formula
0, m + n odd
pm
O, m > n
n-m
n
(-I) 2 (n+m)!
m
2n() ()
This gives all Legendre numbers with
[1]
gives all values of
m
and
n even, m < n
non-negatlve integers.
(2.5)
Ralnville,
pO
P
1
P0
P2n+l
0
(-l)n(1/2)
P2n
which agrees with
n
+
(2.6)
n
n!
O.
(2.5) for m
The following table gives some of the Legendre numbers.
Note from (2.5) that all
Legendre numbers are rational.
TABLE I. LEGENDRE NUMBERS
3.
SOME BASIC PROPERTIES.
The following list of simple properties, observable from the table, can be
easily proved using (2.5).
pn
n
pm
1-3-5---(2n- I), n
n-i
n-m
_nm
(m +n
pm=
-(n
n
(3.l)
pm+l
n
p
i.
m
m
n, n > i.
l)pm-i
n-i
+ 2)(n +
m
m
(3.2)
n > 1
I)Pm-2
n
(3.3)
m > 2.
(3.4)
LEGENDRE NUMBERS
(n + m- 1)
pm_.__
n
n
pm
(n + m
n
pm
n-2
m
1)(n + m
409
3)---(n
(3.5)
2.
n > 2, m < n
+ 3)(n
m
+ 1)Pn-m’
m
m > 1.
(3.6)
Equation (3.1) gives the value on the "main" diagonal of the table. Equation (3.2)
gives each entry, except the last, in a row of the table from the entry just above
and to the right, while (3.3) gives each entry from the one just above and to the
left. Equation (3.4) allows one to fill in the entries of a row from left to right.
Equation (3.5) shows the connection between entries in the same column but two rows
apart.
Finally, Equation (3.6) gives each Legendre number in terms of a Legendre
number in the first column of the table.
4.
EXPANSIONS OF LEGENDRE POLYNOMIALS AND ASSOCIATED LEGENDRE FUNCTIONS.
P (x),
By Maclaurln’s expansion, the Legendre polynomials,
P (x)
.
n
n
p (x)
n
With the table
(4.1)
pmxm
n
(4.2)
m!
m= 0
(4.2) gives a simple way of writing out P n
105 3
63 5
15
Ps(X)
is easily obtained using
can be expressed as
m!
0
mffi
Using (2.3), one has
n
p(m)
(0)xm
n
n
-x --x m+-x
PS’
(4.2) and the entries
for
m
For example
(4.3)
zero through five, from
the table.
from (4.2) into (2.1) one has
m
n
m
n
pm(x)
D
(I
[ m!
n
m= 0
P (x)
Substituting
n
pmxm
x2)
If
m
and
n
(4 4)
are not too large, (4.4) is easy to use to obtain
Pnm(X).
table provides the summation entries and the mth derivative is then evaluated,
example,
P(x)
Pxm
x2)D 2
(i
m=0
(i-
For
(4.5)
m!
x2)(P + Px)
x2)(O +
15x(l- x2).
(I-
5.
The
15x)
SOME SERIES AND AN INTEGRAL INVOLVING LEGENDRE NUMBERS.
Taking
x
0
and
t
in the known generating relation for the Legendre
1
polynomials, see Rainville, [I],
(i
2xt
+
t
2)
1
2
[
n=0
p (x)t n
n
gives
1
(5.2)
the sum of the non-zero terms in the first column of the table.
In (5.1) take
k
410
P.W. HAGGARD
then let
x,
derivative with respect to
pk
n= k
and t
0
x
1 to obtain
2k+l.
1-3-5.7--.(2k- i)2
n
(5.3)
This gives the sum of the non-zero terms in the kth column of the table.
A well known series involves the Legendre numbers.
Let
in (5.1) to
0
x
obtain
+
(i
Next, let
tan 0
t
for
I01
1
2
2)
t
n
P t
Z
n
n 0
(5.4)
and use the appropriate trigonometric identities
<
to obtain
cos
P0 + P2 tan20 + P4 tan4 + P6 tan6
1
tan20 + 3 tan40 5 tan60
1
+’’"
(5.5)
+.--
where the coefficients in the series are the Legendre numbers,
Next
I Pmxm, "I
1
n
for
n
>_ O.
Using (4.2), one has
an integral is evaluated.
lIP0
P2n’
n
_---)dx
(x)dx
(5.6)
0 m=O
n
_
pmxm
n
I
m=O
dx
0
m=0 [.(-i)J 0
n
m-0
Therefore, for
n
pm
n
(-l)
any non-negative integer, one sees that
pm
n
ipn (x)dx
I0
(m+l)n
m--0
(5.7)
A better formula for this integral will be obtained in that the summation will be
evaluated.
Recall that the Legendre polynomials form an orthogonal set for
positive integer.
fl
-i
For
n
P (x)dx
0
P (x)dx
fo
n
n
P (x)dx
-i
2
0
P (x)
n
is an even function for
0
are thus seen to have the value
for
any positive integer.
n
n
Pn(X)dx
+
Pn(X)dx
0
Pn
(5.9)
(x) dx,
The integral and summation in (5.7)
positive and even.
pm
0
n
even.
n
More generally,
for
a
positive and even,
-1
since
n
Thus,
n
(re+l)’
m--0
Pn+l
(5.1o)
n
Now, (5.10) certainly holds for
n
even since
LEGENDRE NUMBERS
411
O, by (2.5). To prove (5.10) for n odd an inductive type argument omitted
here, can be used.
6. DERIVATIVE OF LEGENDRE POLYNOMIALS AT x
i.
First the Oth derivative, that is, Pn (x), can be evaluated at x
i. From
Pn+l
(4.2), one has
pm
n
(6.1)
n
I
can be shown that
ment can be used.
PI(1)
Pk+l(1)
P (i)
n
First if
1
by evaluating the series
i, then since
n
Also, recall that
i.
m
I.
PO(1)
P (x)
n
Next, if
Pk(1)
it is clear that
i,
we can argue that
The proof can be completed by inducting on k twice once for k even
k odd. ,erefore, for all positive integers n,
i.
and once for
pm
n
n
P (i)
n
Since
An inductive type argux,
i.
(6.2)
0
P (x)
I for n
0
(6 2) than holds for all non-negatlve integers
n
From (4.2), the ith derivative of P (x) evaluated for x
1 is
pm
n
e(1)(1)
n
n.
n
(6.3)
m=l(mi)!
It can be shown that
(n-i+l)
p(i) (i)
n
i!2
where the numerator is in factorial notation.
here also.
The induction is on
not be given here.
k
0
for
i
for
i
0,
k
# 0,
i.
2i
(6.4)
i
An inductive type proof can be used
Since the argument is long and involved it will
Nith the usual agreements,
P(O)(x)
n
P (x)
n
(6.4) holds for all non-negative integers
(6.4) reduces to (6.2).
and the factorial
i
and
n.
Also,
Equation (6.4) can easily be shown to have
the forms
p(i) (i)
n
C(n+i,2i)
(n+i)
i! (n_l) t2
i
(6.5)
Pi
REFERENCES
i.
2.
3.
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.