ORTHOGONAL POLYNOMIALS
by
DAVID POMEROY
A
THESIS
submitted to the
OkEGON STATE COLLEGE
in partial fulfillment of
the requirements for the
degree of
MASTR
OF ARTS
October 1944
jUrOVED:
Head of Department of Mathematics
In Charge of Major
Chairman of School Graduate Coimiijttee
Chairman
ot
State College Graduate Council
AC KNOWLED GEMEN T
I
wish to express my great appreciation of the
help and guidance
I
received from Dr.
stage of my work on this thesis.
Mime
at
every
TABLE OF CONTENTS
Chapter
Introduction.
..................
Classical Orthogonal Polynomials
i.
Fourier Series
2.
Tchebicheff Polynomials
.
3.
Properties of Orthogonal Polynomials
..............
......
II
i
.
.
.
4
5
.
.
.
5
.................
.........
.............
..............
...........
............
Continuous Variable
i. Construction
of Orthogonal Polynomials
2.
Example
3.
Evaluating Determinants by Pivotal
Condensation .
Roots of P (x)
Tables of
4.
4.
5.
III
Page
Graph of
.
..................
P(x)
B
9
li
12
13
14
Orthogonal Polynomials for Discrete Points
J_.
2.
Factorial Polynomials
Example.
Function
15
Representation of Given
.
.
.
........
()()
16
3.
Persymmetric Determinants
4.
Proof.
5.
Laguerre Po1ynomir1s for inite Diff ere noes
Partial Difference Equation. .
Tables of L(x).
Laguerre Polynomials for
Finite Differences
Difference Equation. . . .
Recurrence Formula
P(X). ModjfjCtio of Formula
Evaluation of ,I
LÇ(X) 2-x
28
29
30
30
33
Expansion of an Arbitrary Function in
Series of L(x)
Biorthogonal Functions
Tables f(x)
L.(x)
Graph of L(x) for n
0,1, 2, 3, 4 .
35
38
40
41
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
L(x)
'(_j)J
=
.......
.....
..................
.......
...........
............
......
Bibliography
............
.........
-2b
.....
.
.
.
.
......
.
.
.
.....
.
.
20
22
25
26
43
ORTHOGONAL POLYNOMIALS
INTHODUC TION
The subject of orthogonal polynomials finds a place in
diverse branches of study in mathematics, pure and applied.
Their properties make their use in typically modern problems
in quantum mechanics, for example, fairly prevalent, while
in the field of analysis they have proved to be of inestimable value in the study of differential equations with
boundary conditions.
By definition, if we have a system
P(x)
nO
:iii
aX
with the property
b
f
Pm(X)
P(x)
m
O
4 n,
(m,n
0,
1,
2 -
-
then any two of the polynomials of different degrees are
said to be mutually orthogonal over the interval (a,b).
If we are granted the further hypotheses:
(i)
a
weight function
w(x)
>
O
in (a,b).
2
(ii)
all moments
b
w(x) xidx exist;
mj
f
>
(iii) m
(j
0,
1,
2
-
-
-)
O,
then
fb
w(x) Pm(X)
defines
a
P(x) dx
(m + n)
O
unique system of polynomials which are mutually
orthogonal with respect to
If,
further, the system be
norma1izedwe have
¡b
Eq.1
/
w(x) Pm(X)
P(x) dx
=
S
m,n
'a
where
g
is the Kronecker delta;
i.e.
m,n
O
l
mn
(mn.
By analogy with the expansion of an arbitrary function
f(x) in a Fourier Series, f(x) may be expanded in the form
fP(x)
where
(b
=
f
w(x) f(x) Pn(X) dx.
Ja
It is clear that the type of polynomials set up will
depend on the choice of w(x) and on whether (a,b) is finite
or infinite.
The reader should bear in mind,
too,
that we
3
have here assumed for the independent variable the property
of continuity.
This is by no means necessary, and in many
important applications is not even true.
In problems of
statistics, or probability, involving distribution aggregates, for instance,
of discrete values,
the independent variable assumes a set
so that
the defining equation, instead
of being an integral proper, is a sum of a number of dis-
crete terms.
Thile this thesis is the outcome of an examination of
orthogonal polynomials with certain chosen weight functions,
no attempt has been made to unify the treatment of x as a
continuous variable or as
a
set of discrete points by
utilizing the convenient properties of Stieltjes integrals.
For such a treatment the reader is referred to Szegd's
"Orthogonal Polynomials."
CHAPTER
I
CLASSICAL ORTHOGONAL POLYNOMIALS
1.
The paper by Fourier,
Fourier Series.
/
n Theorie
analytique de chaleur" (1822) wherein the assertion was
first put forward that an arbitrary function, given in a
fixed interval, could be expressed in a certain trigonometric
series, may be taken as the starting point of the present
study.
Let
1ir
Sin mx Sin nx dx
and let
I'
111fl
,'m,n
denote the corresponding integrals with Gos mx Sin mc and
We find
Gos mx Gos ax respectively.
O;
'm,n
m
4'
n.
Similarly,
=
O,
.
the former equation holding also for
m
-
n.
These equations express the fact that the system of trigono-
metric functions
1,
Gos x, Sin x, Gos 2x, Sin 2x
orthogonal over the interval (7T
7T).
-
-
-
is
5
An arbitrary function f(x) may be expanded in the form
a
ZakCoskx+bkSiflkx
where ak
f(x) Ces kx dx,
Sin kx dx
f(x)
bk
are the Fourier coefficients.
Tehebichef polynomials.
2.
In an analogous manner,
if
we substitute
X
Cos e,
then
Gos n O
Tn(x)
/
i
=
+
ì4i
i
(x)
Sin (n
=
4
e,
(nO,1,2---)
are polynomials of degree n in x.
Gos n
n
2
=
-
i
Co8
e
-
(Tchebichef polynomiaiE
n
n
2
-
n-2
Gos
e
Tr
+
ri(n -
3)
2
2
n
-
5
Ccs
n
-
4
e
e7Thine2mCosme_ n-1
U
Sin (n
4 1)
2
n-2
Cos
n-2
O
C3)
3.
T(x)
Properties of Orthogonal Polynomials.
and
U(x),
of
Properties of
heuristic value when sets of orthogonal
6
polynomials are being developed, are
(a)
-x2)*
T(x) dx
Tm(x)
O
=
m4n
¡
ie.
Tm(X),
-
x2)
T(x)
Um(X)
U(x)
o
=
are mutually orthogonal (-1,
to the weight function (1
Um(x),
U(x) dx
-x2).
if we substitute
(1
with respect
1)
The same is true for
for (1
x2)
-
x2)4
-
for
the weight function.
(b)
The zeros of
T(x) and iJ(x)
and lie within the interval (-1,
(e)
are all real, distinct,
1).
Between any three successive polynomials
a
relation
U
-
i(x).
-
1,
of recurrence exists.
j(x)
=
xT(x)
(1 - x2)
-
.
Un(x)
(d)
=
xTJ
1(x)
T(x).
+
n
2,
3 - - -.
The polynomials satisfy a second order linear differ-
ential equation.
(1 - x2)
(1
If,
-
x2)
T(x) -xT(x)
U1.'
(x)
-3xIJ(x)
n2T(x)
f
n (n
in equation 1, we put (a,b)
w(x) =
(1 - x)
(1 +
x),
we have the Jacobi polynomials,
(e',
4
O
2)
(-1,
U(x)
O.
1),
1),
of which the Tehebichef
7
polynomials form a subclass;
(O'
Ç3
--*)
while, if
0,
the Legendre polynomials are formed.
infinite at one end,
(0,00
).
w(x)
If the interval be
x0e_X,
(
> -1),
we have the Laguerre polynomials; if the irterval be infinite
at both ends,
(-cC ,oO), w(x)
e2,
the Hermite polynomials
result.
These four kinds of orthogonal polynomials constitute
the Classical Orthogonal Polynomials, and a study of them
should precede any investigation in the construction of a set
of orthogonal polynomials.
CHAPTER II
CONTINUOUS VARIABLE
1.
Construction of Orthogonal Polynomials.
We have
th moment;
defined the
mj ...fbW(x) xi dx
(j
-
0,
2 - - -)
1,
m1--n2---
If we form
m1
m1
m
[I
m1
and define
P(x)
ma--- m2fl2
i
X
mO
ml
n1
n2
X2---
-
X
-
LI
m1
in___
then
'b
i
Eq. 2
(a)
D
+
/
w(x)
P(x)
m21
b
i
(b)
D
D
w(x) Pm(X)
(
"a
For a detailed proof,
P(x) dx
m,n.
the reader is referred to "Numerical
Calculus" by W. E. Mime, p. 60, where the subject is treated
in a theory of least squares, or to notes by the same author
The properties of the polynomials
on Orthogonal Polynomials.
so constructed will be analogous with those of the Tchebichef
polynomials discussed in the first chapter.
We will now consider
Example.
2.
11w(X)
=
xdx
with the given weight function defined as
w(x)
=
x<O
i+x
l-x
x>O.
1
¡'o
Then m
/
(1 + x)
xdx
'-1
/
f
xdx.
(1 - x)
'0
Whence
Eq.
3
m
(''
O
n
odd.
1
n
even.
+
i)(n
+ 2)
'i
It
is
o
apparent that the interval of orthogonality is symmetric
with repect to the origin, as is w(x).
It follows that
I.e.,
(_1)T1
P(-x)
=
(odd) powe±s of x for n
P(x) contains only even
even (odd).
By definItion,
P(x)
1
x
m0
ni1
jm1
-
-
-
-
-
-.
-.
-
-
-
and any polynomial of this group, e.g. Pk(x) would, with the
aid of Eq. 3, have the form
=
i
x
i
0
1/6
0
1/15
o
i/6
0
1/15
0
1/6
0
0
1/28
o
1/15
1/15
o
1/28
o
11
Pivotal Condensation.
Evaluating Determinants
3.
The evaluation of each determinant becomes increasingly
laborious.
Much time can be saved, however, by use of the
theorem.
(6)O)
fi,jI
-
i
Hai,i ai,d
I
a1,3
a1
lai,i ai,nl
fi,i)r-21
la2
a2
-
'
I
I
I
21
a2,1 a2,31
-
1a2,l
a2,nI
a1,1
al,flI
a1,1 al,2I
a1,1
a3,i a3,2
a3,1 g33i
a3,1 a3,n\
a1,1 a1,2
a1,1 al,3'
a1,1 a1,
a,1
a,1 a,3
a,1
an,2
a,
rzJ
Each operation of this kind reduces the order of the determinant by one.
Work can be simplified further by moving columns or
rows,
if necessary,
so as
to bring a simpler term (e.&. 1)
to the leading element position.
By these methods we find
P0C)
= i
-x
1/36(6x2
P2(x)
P3(x)
- -
P4(x)
= 19
=
1)
7/5400 (5x3
4(6300)2
P5(x)
-
-
(490x4
2x)
-
310x2
-683
(798x5
735(180 X 420)2
+
-
19)
700x3
+
109x)
12
In working this problem,
had been hoped that, at
it
least, a recursion formula might be found so that values of
Unlike the Legendre
P(x) could be tabulated therefrom.
polynomials, whose coefficients are convenient to handle,
those of the present problem became unmanageable.
Nor was
any luck experienced in finding a function which would gen-
erate the polynomials.
P(x),
12. Roots of
The graph of the polynomials given
up to P5(x) should be compared with that of the Legendre
polynomials.
zeros of
F(x)
It will be observed that between every two
a
zero of P
+
1(x) occurs.
This is a prop-
erty shared by all orthogonal polynomials in addition to
those cited in the first chapter.
Roots of P(x) up to P5(x) obtained algebraically are
listed below.
P(x)
Roots
P1(x)
o
P2(x)
+
P3(x)
0,
P4(x)
4
P5(x)
0,
.408
+
.632
.262,
+
+
.820,
.751
+
.451
4. Tables of
x
-
P1(x)
i.0
1.0000
.9
.9000
.8000
.7000
.60o0
.5000
.4000
.3000
.2000
.1000
.8
.7
.6
.5
.4
.3
.2
.1
0.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
P(x).
0
-
.1000
.2000
.3000
.4000
.5000
.6000
.7000
.8000
.9000
-1.0000
36 P2(x)
5400
po
5.0000
3.8600
2.8400
1.9400
1.1600
-3.0000
-1.8450
.5000
.0400
.4600
.7600
.9400
.3750
.4800
-1
-
- .9600
- .3150
.1200
.4.650
.3600
.1950
0
.9400
.7600
.4600
.0400
.5000
1.1600
1.9400
2.8400
3.8600
5.0000
-
.1950
.3600
.4650
.4800
.3750
.1200
.3150
.9600
1.8450
3.0000
(630O)2
p(x)
9.9500
4.8435
1.0655
L18O.420)2.7
683
10.3500
2 .9600
-
- .9028
- .2466
.3692
.7974
.4855
- 1.4800
- 1.2874
.4032
.3 485
.6650
.8228
.5104
.9500
.7974
.3692
- .2466
- .9028
-
- .7625
-1.4548
-1.4062
-1.4062
-1.4548
- .7625
1.0655
4.8435
9.9500
o
.5104
C'-'
.Ö.
.6650
.3485
.4032
1.2874
1.4800
.4855
- 2.9600
-10.3500
P5(x
CHAPTER III
ORTHOGONAL POLYNOMIALS FOR DISCRETE POINTS
Mention was made in the
Factorial x-olynomials.
1.
introduction to this thesis that there is no need to treat
the two cases separalely, provided that we use Stieltjes
integrals, for the underlying principles and the resulting
Since we are not using that method,
formulae are identical.
we propose instead to develop the case for discrete points.
It will be convenient to recall some properties of
factorial polynomials and some equations from the finite
calculus.
Eq.
4
(a)
x
(b)LX(n)
x(x
-
(x
4
a
1)
-
-
(x - n + 1) - nt(n)
-
)(n)
x
(n)
nx
(n
1)
-
n-1
___
dx
X
(c)
(n)
(x
=
)(fl 4
n+l
s:0
"X
/
c.f.
i
xn4
snlds
n+1
IO
Eq
1)
4.
5
LxAy
f(x,y)
f(x
=
4'
1,
y)
f(x
4'
1,
y
4
f(x,y)
-L
y
y
1)
4.
f(x
f(x,y).
4.
1,
y)
-
f(x, y
4.
1)
16
The analogy of factorial polynomials and the binomial co-
efficients with permutations and combinations respectively,
is noteworthy;
e.. from
3(a)
3(b)
2.
Example:
The series
()
;
xPn
x4lxX
Representation of Given Function.
Consider
__
I110zx
is generated by the function (1
-z).
We obtain, after n
successive differentiations,
n
(l-z)1
:xzx_n
whence
fl
a series
Zn (1
-
Z)
-n--1
cO
/xO
which is convergent for all
Z
x(zx,
in the unit circle.
17
Hence, put
Z =
and obtain
(')
-
x.O
or
Eq.
6
(n)
2-x
=
We now construct
2n.
(1)
Pk(x)
in0
in1
in1
In2
(2)
-
- -
-
- -
(k)
mk+1
I
I
inkl
-
ink
- -
where
Iflj
=
2j!
x.J
2
(from Eq. 6)
m2k_1
i
Thus, Pk(x)
i
(l)
,j2)
-
-
-
2
2.11
2.2L
-
-
-
2.k!
2.l
2.2
2.3.1
-
-
-
2.(k4.l)!
2.(k-l)
2.kl
a(kfl)
-
-
-
2.(2k.-l)
If we factor out all the elements of the first row and
column and, in addition, divide each column by the factorial
of the exponent of n heading that column, we will have
n
Pk(X)
-
a
lT
j=O
i
2j(j
f
1)i
1
íx\Ix\
kl)12) -
-
-
Ix
Lk
111---1
i
'
_f'kfl
3
\.1
1
i
3
6
-
-
_(kf2
(2k-i
For example,
P4(x)
2
2.1!
2.2! 2.3!
2.4!
2.1!
2.2!
2.3!
2.4!
2.5!
2.2!
2.3'
2.41
2.5!
2.6!
2.3!
2.4!
2.5!
2.6! 2.7!
(x\
kj)
i
=
2j(j
(4)
(2)
i
+
1)
11
12
(x\
(x\
(x
.2)
.3)
4
1
1
1
3
4
5
1
3
6
10
15
1
4
10
20
35
We proceed to show that
P(x)
2j
(j
L11
(x)
4
1)1
j-0
or
Eq. 7
P(x)
211
n!
where L11(x) is the
(1)2
o
normalizedtt
Eq. 8
L
(
x
_L1
)
-
X
(-l)
ji0
(j)(i)
Before doing so, we discuss
type of determinant.
sorne
properties of a special
20
3.
Persymmetric Determinants
There the m
If we write
m0-
-
-
Inn_
-
-
m2n
q.
3,
and factor out the
are given by
terms of the first row and column, we may write
=
2
+
)2
'
(n
n-1
In..'
K) ()
-
-
(o)
(n41)
K
The ttnormalizedtt determinant is persymmetric;
i.e.,
all
elements in any diagonal at right angles to aLt are alike.
If we adopt the notation
a
0,0
---a n,0
then aj,j
aj
.
-
-
)
C.)
Persynimetric determinants in general have the property of
lending themselves to rapid evaluation by the successive
operations
row
1
=
row i
row
2
-
row
1
row
3
-
row
2
row
+
and so on.
1,
In practice, we subtract row 1 from row 2, row 2 from
row
3,
row
from row 4
3
-
-,
-
row (n
repeat the operation omitting row
row 2, then omitting row
-
i)
from row n, then
repeat again omitting
i,
and so on down the line.
3,
We see that since the first row (column) is a poly-
nomial
inn
degree
2,
O,()is
of degree
-
4'
or in general
it follows,
of degree 1,
of
is of degree k in n,
from the equations
A
(n
4.
k k
(n
i')
\k/)
4
k
('n
\kJ
J
(n+k_i
(n4'k
etc.,
1)
(2
-
4.
k
k
k-1,,J
k
vanish, while
1;
i.e.,
ajj
si,j
i
so that the determinant has the value 1.
_>
1
(nfk-1
that all the elements below the leading diagonal
au
-
j,
Ç (
r
In fact, persymmetric determinants whose corresponding
elements in each row (column) reduce
to binomial coefficients
of successive differences, may be shown to have the value i
by premultiplying with a determinant whose elements are the
binomial coefficients of successive order.
For example,
if
we premultiply
il
12
i
i
3
4
with
i
o
o
0'
-1
1
0
0
-2
1
0
1
3
6
10
1
1
4
10
20
1
we obtain
i
o
o
c
o
i
o
c
o
0
1
C
o
o
o
i
331
By either method, we find
Eq. 9
2nl
(n!)2
We are now ready to prove that
n
4.
"-1)
L(x)
j
(n\i'x
o
=
.
If we subject the deter-
minant L(x) to the operation, column
column
3 -
-
-
i
-(y) column
n
l)i() column
i
n
we obtair., for exaiple,
ï
5.
(1 +
j),
2
23
(x'\
(\
1)
(x)
3)
4)
o
i
i
i
i
i
o
2
3
4
5
6
o
3
6
10
15
21
0
4
10
20
35
56
0
5
15
35
70
126
15\íx")
4.
- 3A3)
5!5J
()
Consistent with the notation which we have adopted for
j
if we write
L(x)
=
a0,
-1
a0,
O
a1,
-1
-
-
-
-
+
f'
a0,
n
1
-
(j
we observe that the element a1
)
,
.
-
a,
-
a,
-
a,
-1
-
The cofactor
of the leading element is composed of rows of differences
of successive order.
a1,
1
1
1
1
1
1
012345
.7
L¡
L)
a1
i,
a1,
o
O
O
i
O
O
O
O
0
0
1
0
0
o
o
o
o
i
o
000001
:1
Thus, the value of L5(x) Is
5
a0,
-1
A0,
-1
=
and, in general,
n
L(x)
(-1)
1(Î)
if we can show that our operation
(i)i(n)
column (1
+
j)
-
-
gives
a0,
k
=
O
1,
(k = 0,
2,
-
n
- 1).
Since
ai,
j
(i+j
a1
=
j
i.
/
it suffices to show that
n
I
)
= 0.
Let the generating function
(t)
(1 -
t)
1)
n
j
(i.'
25
Differentiating
times,
i
t)7
______
dLtL(___
(-1)
'J
dt1
fl\%íj
(ç
4.
J
t1
Apply Leibniz rule for the 1th derivative of a product and
obtain
ti)U
+
/i\
l)
+---(
where the
suffxs
Putting
entiatirig,
t
i
=
ti
(i
I
-
u
1)
-
Ui
(1)
-
4.
1
(1
u
-
2)
(2)
u
indicate the order of the derivative.
in the above expression,
after differ-
we get
u
)fl
(1
Q
21(1)
=
5.
i
+i)
(n)i
J
3=0
Laguerre Polynomials for Finite Differences.
It
should be observed that since
n
2x,
X
the weight function
2X
chosen is analogous to the weight
function e"X of the Laguerre polynomials (defined on page
with
c=
7
0).
We accordingly defIne the set of polynomials
Laguerre Polynomials for finite differences.
L(x)
as
26
From the equations
Partial Difference Equation.
6.
L(x
L](x)
/
+
1)
-L(x)
nfl
),
we get
[L(X
I(x)
-L(x)7
+ 1)
n41
i)i
n
=_(-i)
::
kfl(fl\Ix'\
¼k)kJ
=
-L(x).
Thus
Eq. 10
L(x)
,L\
L(x)
f
= 0.
This equation affords a rapid means of tabulating the values
L(n, x).
Thus,
Ln(X)
-
L
L(x)
f
1(x)
f
2
Ln(x
4
1)
.
L(x)
1(x
L
-L(x
f
f
1)
1)
O
or
Eq.
11
Ln
1
1(x)
f
- 2
L(x)
1(x
L
.
f
1).
27
S inc e
L0(x)
La(0)
1,
=
we tabulate values
in accordance with Eq. 11; i.e.,
a
b
C
d
b
in any block
+
c
-
2a
d,
a relationship which must hold for every block of 4 cells.
Eq.
12
n
L(x)
-
x [L(x)
-L(x
follows from the property of symmetry in x and n.
-
l)J
Laguerre Polynomials for Finite Differences.
7.
X
o
1
2
3
4.
5
6
7
8
9
10
II
12
13
14
15
16
17
18
19
20
Lp(x)
x)
i
1
1
1
1
1
i
0
L2(x)
113(x)
-4
i
-1
-2
-2
-1
1
i
-2
-2
0
3
6
1
1
1
1
1
-5
4
-6
8
13
8
8
1
1
1
1
1
-10
1
1
1
1
1
-15
-16
-17
-18
-19
-1
-2
-3
-7
-8
-9
-U
-12
-13
-14
19
26
34
43
53
64
76
89
103
118
134
151
Lj(x)
L(x)
i
i
-3
-1
-4
1
3
6
6
6
6
0
2
-6
-17
-29
-39
-32
-57
-90
-132
-184
-247
-322
-410
-512
-629
237
409
643
951
1346
5
-2
-14
Weight function
2X
Çx)
c)
i
i
1
-5
4
-6
-7
2
-10
-6
-20
-17
-25
1
-C
19
-2
-29
-20
-10
-20
-25
-20
-1
-20
-20
-5
-20
0
-5
35
25
70
35
70
84
70
70
14
70
-126
-43
34
-36
-12
84
144
204
248
100
116
92
-152
56
-28
-168
-344
-512
-98
-238
-350
-358
-182
-395
-713
-1076
-1424
-1658
-603
-526
-176
552
1742
239
919
1795
2699
116
253
188
13
-322
-878
1.3
5
25
64
8
3337
1
-9
26
1
2
4
-14
-39
-1
32
64
84
14
-126
-252
64
128
256
512
1024
-252
-294
-168
174
708
-252
-42
378
2048
4096
8192
16384
1311
1752
1709
818
-1243
1142
0
888
1248
273
-1522
-4122
-7001
8
16
32768
65536
131072
262144
524288
1048576
29
8.
series
During the discussion of the
quation.
Difference
as Tchebichef poly-
T(x) and IJ(x), (also known
nomials of the first and second kinds respectively) we found
that a relation of recurrence existed between any three suc-
cessive polynomials; also that the polynomials satisfied a
2nd order linear differential equation.
if we set up a table for
j
2
1,
= 0,
With this in mind,
from the relations
already established, viz.,
L(x)
_
n
1)3(fl\1X\
i)
n
L(x)
X
(i)
=
-
3=0
L(x
-
1)
1)
=y_'(l)3(
2),
-
j-=
o
we will have
¿L(x)
i
fl
fl(
X
2
\2/ 2)
2
LxLn(x1)
O
(n(x
[n
2/l/
L(x)
from which we find that the coefficients of L(x),/.
andÌ
L(x
-
1)
which will make the sum vanish are
and 2x respectively.
That is,
we have established
n,
the
(1 -x)
30
difference equation
Eq. 13
2x
4
L(x
1)
-
(1 - x)
f
+
n
L(x)
0.
The relation of symmetry,
Recurrence Formula.
9.
Ln(x)
L(x, n)
L(n, x)
=
conveniently establishes the recurrence formula as well.
Thus, from equation 12,
L(x
2x
+
4
-2 Ln(x)
1)
(1 -
x)
[Ln(x
Ln(x
+
1)
f
-
i)]
+
n
L(x)
L(x)
+
2x Ln( X
Lfl(x)]
-
0.
Simplifying, we get
(x
1)
+
L(x
+
1)
+
(n
-
3x
-
1)
-
1)
and using the symmetry relation mentioned, we have
Ea.
14
(n +
1)
L
+
10.
P(x).
4
(x - 3n
2n L
1(x)
1(x)
-1) L(x)
0.
Llodification of Formula.
of the proof of equations 2(a)
An essential part
and 2(b) is the fact that
when the first row of the determinant Pn(X) is multiplied by
dx and each element integrated from
become identical for
:'rting, however,
m<n.
a to b,
two rows
For the Pn(x) which we are inte-
while no difficulty is experienced on
account of a summation replacing the integration, we are
31
dealing by the nature of the case with a factorial polynorni
therefore, we multiply
Then,
O
to,
P(x) by
(n)
2
and sum from
whih corresponds
we do not obtain a row
to the last
and our equations 2 no longer hold.
one of
We make use of the relation
(x + rn)m x
(x +
m)(m
4
n)
and replace
2_x
(k)
-
ink
x-O
by
xO
If we let x
+
(x+m) (k
s,
in
k
then we obtain
-7
2
4
n
(k)
s
5(k)
2m
-
2
2s
m
5(k)
2j
But the last expression vanishes for
S = 0,
1,
2 -
-
k
-
-
1,
whence
Eq.
15
2_=
2m
2k12m41k1
k>
For
m.
We accordin1y redefine
P(x)
(2)
(1)
i
co
co
o
1
Ci
n-1
(ra)
-
-
-
-
-
-
-
-
0n-1
n
-
-
co
n
n-1
_C2n_1
where
2m41k1
This relation,
equation
then,
-7
=
(x+m) (k)
2-x
supersedes the one established by
5.
By factoring out elements in the first row and column,
proceeding as for equation
6,
that equation is now
superseded by
Eq. 16
¡n
L(x) ¡-j-
a
P(x)
=
2''
_2X L(x).
11. Evaluation of
We proceed to
obtain the formal equivalent of equation 2(b).
If we multiply
L(x)
=
3e0
L(x), and sum on x from
by 2
x0
L(x)
2
O toco,
we get
(i)(n)
=
x0
i=o
(V2
L(x),
and this, by virtue of the orthogonal property, vanishes
for
j
4
n.
We obtain
L(x)
(_1)fl
L(X),
___
which for convenience we express ss
(-i)
+
n) 2
L(x),
which again we can do, because of the orthogonality of the
L(x) polynomials.
This permits us to write
00
x0
___
L
-
00
2
(x)
(-')'
x0
j0
(-1)
-
n
t'(-1)" (i),)
r,)2_X
+
211
n
(_1)i
211(%x n+ n\fx\ 2-x
(fl)
x0
i
n
(i)i
()2I
Ii
xO
(x+ n)
(n4j)
n
i!
With the aid of equation 14, we write
(_1)n
L(x)
2
j
=
(-i)
2'
+
2
(_1)i
i
(n+j)
n!j
=0
i
n
i)
1)i(n)(n
From the formu1a
n
F(x)
4'
i)
F(-
- 1).
j
i-0
If we suppose
F(x)
we have
n
(X)
n
i
=
i
J
n
-
j) =
-
n
(-1)
-
j
X
(7)(
4
j
2
35
by virtue of the relations9
/fl\
(
(-1)
ísfk-1
k
n-k);
Let x
n,
k
)
then we will have
n
(-1)
n-j
(fl)(n4i"
1'
j
j=o
whence
(1)fl
-
n
n
=
n/nfj
jX
Y;
j
50 that
2
x:O
L(x)
(_1)fl
2n
i
(_1)fl,
and we finally emerge with
Eq.
17
4
L(x)
As will presently be seen,
1
this equation is of great impor-
tance when we seek to expand an arbitrary function in the
00
form
ak Lk(x).
12. Expansion of
L(x).
If we can assume that we may write the function
1(x)
then,
an Arbitrary Function in Series of
=
a0 L0(x)
if we multiply by 2
a,
L,
Lk(x)
(x)
4
-
-
-,
and sum on x from O to 00,
36
we will have
00
c'o
L(x)
f(x)
A
(x)
x0
x=O
This is true because each sum on the right is zero except the
2
one containing Lk(x), by virtue of the orthogonality property.
Thus
Eq. 18
Ak
x=O
f(x)
+
2X
Lk(x)
i
We first give a few simple examples to illustrate and con-
firm the results arrived at.
Ex. i
f(x) = X
a0
From eq. 17.
2:-_x
2
-
x)
X2
-
f(x)
a
=
X
-
L0(x)
-
L1(x);
result which is readily verified.
=
-1;
37
f(x)
Ex. 2
We have
a0
Th_x X/
=
22_x
a1
a2 =
x2(l
{_x
x2
-
22_x [(2)
=
x(')]/2
_2_x
+
x,ì,/
2x
[i -
X(
(2
-
1)
-
J
2
2ï
x3J/
3
-
To convert the expression in parentheses on the right to
factorial form, we make use of
Eq. 19(10) (7)
f(x)
f(0)
4
(x)2
f(0)
4.
-
-
-+()Am
and set up the table of differences
x
f(x)
o
o
1
-1
¿f(x)
df(x)
Lf(x)
-1
-6
-7
2
-8
3
-3
-10
3
-18
12
2
4
-16
12
15
Ç(0);
38
From equation 18 we now have
f(x)
=
_(1)
DX
-
(2)
+
4
i.e.
2II_x (_(1)
a2
i
(
-
2
-
12
(2)
-Óx
24)
6 4
4
(3)4
1
=
1(4))
2,
8
and
f(x)
=
x2 = 3 L0(x)
-
L1(x)
5
4
2
L2(x).
As before, this result is easily substantiated.
13.
Biorthogonal ?unctions.
The problem of expanding
an arbitrary function Is much more complex when, instead
of a polynomial, we have to deal with a nonterminating
series.
We do not propose to enter into the problem of
exemining the series for convergence, and we leave open the
question of the validity of the series as
a
representation
of the function,
since it would take us outside the scope
of this thesis.
That we do have bo take into account is the
concept of biorthogonal functions,
a
new se
since we obtain thereby
of coefficients for our Ln(x)
which give us
rapidly convergent series for large values of x.
By definition,
if two
sets of functions
ui(x), Vj(X),
(i
0,
=
bear the relation
Uj Vj dx
=
0,
+
1,
2
-
-
-)
a
more
39
then the two systems of functions are said to be biorthogonal
over the interval (a, b).
L0(x)
Li(x)
,
- -
L(x)
L(x)
2
the case of the functions
In
-
-L(x)
Ln(x),
-
since we may write
c'O
x0
____
the L1(x),
L(x) L(x)
= O
m
n,
are two sets of mutually biorthogonal
L1(x)
2
2x
functions, for the case of discrete
points.
The aj given by equation 17 is now replaced by
b1 L1(x),
where the bk are given by
Eq. 20
2(x) Lk(X)
2k41
bk
Example.
f(x) = Sech x.
The expansion is of the form
Sech x
22bk Lk(x).
To calculate the coefficients
5ech X L,(x)
4
1
We find it sufficient to take values for Xup to 10.
(Sech
lOa
.0001.)
We then compute
k
L
Sech x
Sedi X
2X
Lk(X)
0
2.0711
1.0355
1.000
1.002
1
.3286
.0822
.648
.640
2
-.3515
-.0438
.2b6
.273
3
-.5809
-.0363
.099
.099
4
-.6340
-.0198
.0366
.0360
5
-.6188
-.0097
.0135
.0100
6
-.4728
-.0037
.0050
.0051
7
-.5105
-.0C20
.0018
.0030
close an approximation to
have obtained.
The accompanying graph shows how
the given function
we
BIBLIOGRAPHY
1.
Aitken, A. C. Determinants and matrices.
Publishers, New York, p. 45, 1939.
2.
Comm. by Muir.
On some ersymmetric
Geronimus, J.
determinants. Proc. Roy. Soc. Edin. 50:304.
3.
Hobson, E. W.
A treatise on plane trigonometry.
bridge, p. 104, 1939.
4.
Fourier series and orthogonal polynomials.
Jackson, D.
Carus Mathematical Monographs. 6:151-153.
5.
Jordan, C.
Calculus of finite differences.
p. 78, 1939.
6.
Ivlilne,
7.
Mime
8.
Po"lya, G., u. G. Szeg6.
J.
Numerical calculus, p. 13.
E.
unpublished.)
Interscience
Cam-
Budapest,
(As yet
and Thompson.
Calculus of finite differences.
MacMillan, London, p. 28, 1933.
Aufgaben u.
Lehrstze
aus der
Analysis. 2:75.
9.
10.
Shohat, J. A.
Szeg6 on Jacobi polynomials.
Am. Math. Soc.
Mar., 1935.
Bull. of
Stephens, E.
Elementary theory of operational mathematics. McGraw-I-uil, New York, p. 79, 1937.