Internat. J. Math. & Math. Sci.
VOL. ii NO. 3 (1988) 619-623
619
SOME FURTHER RESULTS ON LEGENDRE NUMBERS
PAUL W. HAGGARD
Department of Mathematics, Last Carolina University
Greenville, North Carolina 27858 U.S.A.
(Received February 12, 1987)
ABSTRACT.
pmn
The Legendre numbers
in the previous column down to
Pmn
are expressed in terms of those numbers
and in terms of those,
m
Pk
above but in the
Other results are given for numbers close to a given number.
same column.
m-i
Pk
The limit
of the quotient of two consecutive non-zero numbers in any one column is shown to be
Bounds for the Legendre numbers are described by circles centered at the origin.
-i.
A connection between Legendre numbers and Pascal numbers is exhibited by expressing
the Legendre numbers in terms of combinations.
KEY WORDS AND PHARASES. Associated Legendre funns, bounds for the Legendre numbs,
Legendre numbers, Legendre lynomi, limits of ratios of Legendre numbers.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES. 10A40, 26C99, 33A45.
i.
INTRODUCT ION.
The Legendre numbers were introduced in [i] and several elementary properties were
given.
In [2], some applications of the numbers were presented.
Further applications
In this note some relationships between the numbers are shown, bounds are
given for the numbers, and the numbers are described in terms of combinations. For
reference, we give (from [i]) the definition that we use, a general formula for the
are needed.
numbers, and a partial table of them.
Definition i.
functions
pm
The Legendre numbers,
pm(x)
n
for
x
0
n’
and
m, n
are the values of the associated Legendre
non-negative integers.
A general formula for the Legendre numbers is
[’0,
pro=
m
+
n
odd
O, m>n
n
(-i)
n-m
2
.(n
(i.i)
.+ m)!
L 2n(-) (-)
m
+
n
even, m < n
Another result needed is that
emn
where
at
2.
p(m) (0)
n
x=0.
p(m)
n
(1.2)
(0),
is the ruth derivative of the Legendre polynomial,
P (x),
n
evaluated
SOME RELATIONSHIPS BETWEEN LEGENDRE NUMBERS.
Many relationships between Legendre numbers have been shown in [i], and [2].
620
P.W. HAGGARD
the preterms of the non-zero entries in
Here, each Legendre number is expressed in
is expressed
entry in two ways. Further, each
vious colum (see Table 1) dow to this
entry.
the
same column but above
in terms of the non-zero entries in the
pn
’. n
p
n
=p0
n
n
p2
pl
p4
p3
p5
n
n
n
n
p7n
p6
n
n
p8
n
0
- -3
0
15
15
0
15
8
0
105
0
48
0
_i0.__.5
105
384
0
105
0
105
2
0
945
945
0
0
945
0
-9454- -
0
10,8_395
48
TABLE i.
8
10,395
0
0 135 135
2
0
_135,1352
02,027,025
LEGENDRE NUMBERS
From the known result, see [3],
Z
P’n(X)
where
n
is
n
(2n- 4k +
3)Pn_2k+l(X
k=l
is even and
n+l
n
---if
(2,1)
odd, it follows that by taking m- i
is
derivatives then using (1.2), one has
pm
n-___l
=n
This gives each Legendre number,
(2n- 4k +
k=l
m-i
3)Pn_2k+l,
m, n
_>
i
(2.2)
m
Pn’
as a sum of products involving the Legendre
numbers in the preceeding column of Table i and above
Other such formulas are
n
possible. Four that can be proved (mathematical induction, inducting on n is one way)
are:
2n
p2m+l
2n+l
p2m
2n
2n+l
pm.
(-l)n-m(4m-l)
(2.3)
V2k-i
k=m
n
Z P2k2m
(-I )n-m (4m + i)
(2.4)
k--m
(-i)n-.m.. (4m+l.)n. i
2 (n-m)
k"m
2(n-m)
2m
P2k
k=m
n > m
Y2k+l
n > m.
(2.5)
(2.6)
Noe that (2.3) and (2.4) give each Legendre number as a product that involves
the sum of the absolute values of the entries in the previous column and above the
entry of Table I specified. Similarly, (2.5) and (2.6) involve the entries in the same
621
RESULTS ON LEGENDRE NUMBERS
Equations (2.4) and (2.6) can be obtained from (2.3) and
i. In fact,
(2.5), repectively, by replacing 2n with 2n + i and 2m with 2m +
(2.3) and (2.4) can be comblned as
column but above the entry.
pm
(-i)
n
2(2m-i)
while (2.5) and (2.6) can be combined
n-___m
(-i)
Pmn
for
m
and
n
Pn-2k+l
k= I
2
n-m
as
n-m
(2ml)T lPm
(2.8)
n-2k
k=l
of the same parity.
There are several results concerning entries in Table 1 that are near each other.
These can be easily proved using properties of Legendre numbers or by using (1.1).
For
example,
pm
pm+l
n-i
m+l
+ Pn+l
n > m
2m+ i
n
+ 2
(2 9)
gives each entry in terms of the entries in the next column and just above and below.
Each entry in terms of the entries in the previous column and just above and below is
given by
pm
(n+m-l) (n-m+2)
pmn
Considering
m-i
(pm-i
+ Pn+l
)’
n-i
2m- 3
n
(2.10)
n, m > i.
and the nearest entries on slant lines through
pmn
leads to a deter-
minant type result,
2 (2n+l)
m+l m-i
m-i m+l
Pn-iPn+l Pn-iPn+l
(n+m-l) (n-m+2)
(Pm)n 2
(2.11)
Next, if we look at a particular non-zero entry and consider the first four nonzero entries above, below, to the left, and to the right, we have
m m
n
n
nP-2Pn+2- pm-2pm+2
left
below
which one can express as above
_> 4,
n
0,
2
_< m_<
(2.12)
n- 2
right.
In [i], it was shown that the sum of the non-zero entries in any column of Table i
converges.
The limit of the ratio of consecutive entries is somewhat surprising.
Choose the mth column of Table I.
pmn+2
pmn
For
n
+
even and using (i.i), we have
m
i
n+m+l
n-m+2
Therefore,
lim
n-
pm
n+2
pm
+ _mn +-ni
i
ratios is
i.
m > i.
(2.13)
+2
n
(2.14)
i
n
From (2.13) it is clear that the limit approaches
from the left for
m
n
-i
from the right for
m
0
It is clear that the limit of the absolute value of the
and
P.W. HAGGARD
622
3.
BOUNDS FOR THE LEGENDRE NUMBERS.
From the known bound from [3],
i
P (x)
<
n
for the Legendre polynomials,
Pn(X),
C
with
D
/’
2n
D
0
x
(3.2)
,/C
is the circumference of a circle of radius
2n
2n
one has, for
/2n
n
where
(3.1)
.2n(l-x)
n
centered at the origin
the diameter of the circle.
In [i], the relationship
pmn
was given where
P
n-m
the more general result
Iemln
<
Using (3.2) in (3.3) we have
is in the first column of Table i.
(n+m-l) (n+m-3)-." (n-m+3)
(n-re+l)/
-,
m > i, n
2(n-m) is the circumference of a circle of radius
2(n-m) the diameter of the circle.
origin with D
4. LEGENDRE NUMBERS IN TERMS OF COMBINATIONS.
where
(3.3)
(n+m-l) (n+m-3)-.. (n-m+3) (n-m+l)Pn m’ m > 1
C
n
(3.4)
>m,
m
centered at the
In [2], combinations were expressed in terms of the Legendre numbers.
express the Legendre numbers as combinations.
C(q,i)
Here, we
The equation
i
0
(4.1)
q
to
i)!Pqq
(q
from [2] becomes
(-i)
n+m
n-m
2
m!
pm=
- after solving for
n+m
n-m
i and
en+m2 (, __nm)
C
2
n
(-)1
p
+
q.
then letting
Since
pn
2
q
n
(4.2)
n-m
2
+
i
and
1.3.5.-.(2n-i),
n
m
q
i.
Notice that
see [i],
n+m
p 2
n+m
1-3"5’’’ (n+m-l)
2
(n + m)
(4.3)
n+m
2
2
n+m
(---/--)
Substituting (4.3) into (4.2) gives
pm
n
(-!)
n-m
2
ml(n+m)!
2n[(-)!]2
C
(.nm, nm)
(4.4)
RESULTS ON LEGENDRE NUMBERS
for
n > 0,
m
and
are given in [i] as
n
0
P0
of the same parity, and
m < n.
pmn
and
i
and
0
for
m
623
The remaining values of
n
pmn
of different parity.
REFERENCES
i.
HAGGARD, P.W.
2.
HAGGARD, P.W.
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.
4.
5.
On Legendre Numbers, International Journal of Mathematics and Mathematical Sciences, Volume 8, No. 2 (1985) 407-411.
Some Applications of Legendre Numbers, International Journal of
Mathematics and Mathematical Sciences, (to appear).
RICHARDSON, C.H. An Introduction to the Calculus of Finite Differences, C. Van
Nostrand Company, Inc., New York, 1954.