Aequationes Mathematicae 46 (1993) 174-198
University of Waterloo
0001-9054/93/010174 25 $1.50+0.20/0
© 1993 Birkh/iuser Verlag, Basel
A set of orthogonal polynomials induced by a given orthogonal polynomial*
W A L T E R GAUTSCHI AND SHIKANG LI
Dedicated to the memory o f Alexander M. Ostrowski on the occasion o f the lOOth
anniversary o f his birth.
Summary. Given an integer n/> 1, and the orthogonal polynomials n,( - ; dcr) of degree n relative to
some positive measure da, the polynomial system "induced" by n, is the system of orthogonal
polynomials {r~k.~ } corresponding to the modified measure d~, = n~ d~. Our interest here is in the
problem of determining the coefficients in the three-term recurrence relation for the polynomials ~k,,
from the recursion coefficients of the orthogonal polynomials belonging to the measure do. A stable
computational algorithm is proposed, which uses a sequence of QR steps with shifts. For all four
Chebyshev measures da, the desired coefficients can be obtained analytically in closed form. For
Chebyshev measures of the first two kinds this was shown by AI-Salam, Allaway and Askey, who used
sieved orthogonal polynomials, and by Van Assche and Magnus via polynomial transformations. Here,
analogous results are obtained by elementary methods for Chebyshev measures of the third and fourth
kinds. (The same methods are also applicable to the other two Chebyshev measures.) Interlacing
properties involving the zeros of n, and those of ~, + ~,, are studied for Gegenbauer measures, as well as
the orthogonality--or lack thereof--of the polynomial sequence {~. . . . i }.
1. Introduction
L e t da b e a p o s i t i v e m e a s u r e
h a v i n g finite m o m e n t s
orthogonal
on R (with infinitely many points of increase)
o f all o r d e r s , a n d let {rim } b e t h e c o r r e s p o n d i n g
(monic)
polynomials,
n , , ( t ) = rim(t; dtr),
(l.1)
m = 0, 1, 2 . . . . .
*Work supported in part by the National Science Foundation under grant DMS-9023403.
AMS (1991) subject classification number 33C45.
Manuscript receivedAugust 26, 1991 and, in final form, November 18, 1992.
174
Vol. 46, 1993
A set of orthogonal polynomials induced by a given orthogonal polynomial
175
They are known to satisfy a three-term recurrence relation
~ +, (0 = (t - ~ ) ~ k ( t ) - / ~ k ~ k - ,
7Z0(t) = 1,
(t),
k=0,1,2 .....
(1.2)
n_l(t) =0,
where ct~ = :tk(do-) e •, flk = flk(da) > 0, and by convention, flo =
For a fixed n >i 1, we define
d8 = dS~ .-= [zc.(t; da)] 2 da(t).
flo(da) = ~R dtr(t).
(1.3)
The set of orthogonal polynomials we wish to study is
7~k(t) = ~k.,(t) =
nk(t; d~,),
k = 0, 1, 2 . . . . .
The existence of {72k } is assured, since d~ is a positive measure satisfying the same
assumptions as those made for dtr. We consider the following problem:
Given the recursion coefficients c~,(da), flk(da) for
coefficients ~g.. = ctg(d3.), ilk,. = i l k ( d r 7 . ) for d~..
da, determine the recursion
Modification of a measure d(r by multiplying it with the square of a polynomial
occurs, for example, in constrained polynomial least squares approximation, where
the approximant is required to agree at certain given points exactly with the
function to be approximated (cf. [4, §8]). Other modifications, involving more
general multipliers, are discussed in [12], where the class of Bernstein-Szeg6
weights is significantly generalized, and in [13], [11] in connection with the
asymptotic t h e o r y - - a n extension of Szeg6's t h e o r y - - o f orthogonal polynomials.
The particular modification in (1.3) has been proposed by Bellen [2] to provide a
source of additional interpolation p o i n t s - - t h e zeros of ~, + 1,,--in the process of
extending Lagrange interpolation at the zeros of ft,. When dtr is a Chebyshev
measure of the first or second kind, the polynomials Z2k also arise in the work of
A1-Salam, Allaway and Askey [1] on sieved ultraspherical polynomials, and in the
related work of Van Assche and Magnus [14] on new orthogonal polynomials
generated by a polynomial mapping (see also [9]). The problem stated above, in
these special cases, has been completely solved in [14]. For polynomials n, other
than those of Chebyshev, the measure d~ in (1.3) appears to define a new set of
orthogonal polynomials, which we term "induced orthogonal polynomials", or
more precisely, "orthogonal polynomials induced by the (fixed) polynomial n , " .
176
W A L T E R G A U T S C H I A N D S H I K A N G LI
AEQ. M A T H .
In §2, we give an algorithm based on successive Q R decompositions with shifts,
which, for any given N i> 1, constructs fij.n, flj.,, J = 0, 1. . . . . N - 1. In §3 we treat
the Chebyshev measures da of all four kinds, and for those of the third and fourth
kind obtain the desired recursion coefficients by elementary methods (which could
also be applied to rederive the results in [14]). Our results, unlike those in [ 14], are
phrased in terms of monic, rather than normalized, polynomials. The zeros of
lr~ ( • ) = n n ( . ; dtr) and those of ~n + ~,~( • ) = rc~+ i ( " ; d ~ ) , in particular their interlacing properties, will be the topic of §4. In §5 we will show that, for any Chebyshev
measure, {z2. . . . 1} is a sequence of polynomials orthogonal relative to a measure of
Bernstein-Szeg6 type, but give reasons to believe that for other Gegenbauer or
Jacobi measures, they do not form an orthogonal system.
2. Computational method
Let J = J ( d a ) be the Jacobi matrix for the measure da, i.e., the infinite
symmetric tridiagonal matrix
J = J(da) = tri(ao, a,, a2 . . . . . x/~J, x ~ 2 . . . . )
(2.1)
with the recursion coefficients a, = ak(da ) (cf. 1.2)) down the main diagonal, and
the x ~ k = X/flk(da) down the two side diagonals. Similarly, let
J =
= tri(
o, a , ,
....
,
....
)
(2.2)
be the Jacobi matrix for the measure d~ (cf. (1.3)). By arm = J,,(da) and a*m = J,,(d~)
we denote the leading principal minor matrices of order m of J(da) and J(d~),
respectively.
Our computational method is based on the following observation, due to
Kautsky and Golub [10]. Given the Jacobi matrix J,,+ ](da) and a number z e ~,
one can obtain the Jacobi matrix of order m, J , , ( ( . - ~ ) 2 da), for the measure
(t - z) 2 da(t) by one step o f the Q R algorithm with shift z:
Let Q (orthogonal) and R (upper triangular with diag R i> 0) be the matrices of
order m + 1 such that
J,,,+ ,(da) - zI,,,+ , = Q R ;
(2.3)
then
J,,,((. - z) z da) = ( R Q + ZIm+ ]),,, × ,,,,
(2.4)
Vol. 46, 1993
A set of orthogonal polynomials induced by a given orthogonal polynomial
177
where, as indicated by the subscript, the last row and last column of the transformed matrix are discarded.
We define
]
d ~ ( t ; j ) = ]-I (t - rv) 2 da(t),
j = O, 1, 2 . . . . .
n,
(2.5)
*'=1
where r, are the zeros of n . ( • ; da) and the empty product is defined to be 1. It
follows immediately from (2.5) that
d~(t;j)=(t-zj)2
d~(t;j-1),
j=l,2
.....
n;
dd( . ; O ) = d a ,
dd( . ; n ) = d 6 ,
(2.6)
and hence, from (2.3), (2.4), that ]tv = J u ( d 6 ) can be obtained by the following
algorithm:
ALGORITHM
For j = 1,2. . . . . n do
let
Jold = J,+N ,+a(dS(';j-1))
do QR step with shift ~j
Jold - xjI = QR
J~e~=(RQ+xjI)(n+N j)×I.+N ~/
end
end
Upon exit, Jnew will be JJv. For the individual Q R steps, one can use the algorithm
in [l 5, p. 567].
Comparison of double-precision with single-precision results suggests that the
algorithm is remarkably stable, numerically. This is also confirmed by comparing the
computed coefficients with the exact ones, given below in Theorems 3.2, 3.3, 3.5 and
3.7, for the three Chebyshev measures da ['], i = 1,2,3 (cf. (3.1)). Indeed, if
E~!~ = maxo ,<k .< N --I [~'!n - - (~?n I' where the starred quantities are those computed (in
single precision on the Cyber 205; machine precision ~ 7 x 10- ~5) and ~ ! , = ~k(d~ ])
(cf. (3.2)), and if E~!a similarly denotes the maximum error in the first N of the
/?-coefficients, then, for N = 320, we have observed the results given in Table 2.1.
We furthermore observed by computation (in double precision) that perturbations of the zeros rv in (2.5) by random amounts of order e = ~1 x 10- Jo, 51 x 1 0 - 5 and
1
× 10- ~ lead to comparable perturbations rt in the coefficients 02~!,,/~],, 0 ~<k ~< N,
the ratio q/e, for N = 320, never exceeding 30 if n ~< 160, and 110 if n ~< 320.
178
WALTER GAUTSCHI A N D SI-tlKANG LI
AEQ. MATH.
Table 2.1. Accuracy of Chebyshev recursion coefficients
n
1
2
5
10
20
40
80
160
320
Ell,~
3.6(--15)
1.7(--12)
5.4(--12)
2.8( - 11)
2.6(-11)
1.8( - 10)
7.6(-10)
4.3(-9)
5.2(-9)
EI2~
E TM
E TM
3.6(--15)
1.1(-12)
1.1(--11)
2.4( -- 11)
4.8(-11)
9.6( - 11)
1.5(-10)
8.2( - 10)
8.4( - I0)
1.2(-11)
5.3( - 11)
8.3(--11)
1.2( -- 10)
1.5(--10)
4.3( - 10)
1.1(-8)
1.9(-8)
5.7(-9)
5.6(--11)
2.7(--11)
4.2(--ll)
5.4( -- 11)
9.3(--11)
3.7( -- 10)
8.6(--9)
9.7(-9)
2.1(-9)
3. Recurrence coefficients for Chebyshev measures
In general, there are no analytic expressions known for the coefficients ~j, fl:,
and one has to resort to the algorithm of §2 to compute them numerically. As so
often is the case in the constructive theory of orthogonal polynomials, the four
Chebyshev measures on ( - 1, 1),
d~rtll(t)=(1--t 2) l/2 dt,
da~2l(t)=(1-t2)'/2
dt,
(3.1)
do'D1(/) -- ( l -- t) -,/2( l + f) 1/2 dl,
do'j41(/)= ( 1 - t) 1/2( 1 q- i) -~/2 dr,
are exceptional in that they permit closed-form expressions for the desired coefficients. In the case of &r I21, they have already been obtained in [ 1] from the theory
of sieved ultraspherical polynomials, and for both d o "Ill and da E21in [ 14] via a theory
of polynomial transformation of measures and orthogonal polynomials (cf. [9]).
Here we derive them from first principles for d~rI31 (and hence for dcr141)and observe
that the same elementary methods can also be applied to da 111 and da 121.
We shall denote the n th modified measures by
d~tAl(t) = [n~l(t)]2 dat'l(t),
i = 1, 2, 3, 4,
(3.2)
where n~1 is the nth-degree monic Chebyshev polynomial of the ith kind. We will
suppress the superscript i when there is no danger of confusion.
One of the reasons for the existence of analytic formulae is the fact that the
polynomials nk( • ; dd~J), i = 1, 2, 3, 4, of degree k ~<n (k ~<n + 1 if i = 2) are pure
Chebyshev polynomials of the first kind (cf. Theorems 3.1, 3.4 and 3.6). Another is
the special algebraic structure enj oyed by Chebyshev polynomials (for example, (3.3)).
Vol. 46, 1993
A set of orthogonal polynomials induced by a given orthogonal polynomial
179
3.1. Chebyshev measure of the first kind
Here, and throughout this subsection, do(t) = dolU(t) and rt,,(t) = 2 ('' ~T,,(t) =
2k,(t), the monic Chebyshev polynomial of the first kind. Heavy use will be made o f
the well-known identity
1
(3.3)
Tp(t)T~(t) = ~ (T; . ql(t) + Tp+q(t)),
valid for arbitrary nonnegative integers p, q.
THEOREM 3.1. For any integer n >i 1, there holds
,L,,,(t)
=
L,(t),
o ~< k ~< ,,,
(3.4)
where ~k,n( " ) = ~zk( " ; d ~ f ) .
Proof It suffices to prove that
f_
Tk(t)TAt)T~(t ) do(t) : O,
0 <~r < k <~n.
(3.5)
1
We use (3.3) to write T] = ½(1 + T2.) and then use Tk Tr ~ P z k , c P2,
with orthogonality to obtain for the integral on the left in (3.5)
-
Ira
2 _~
together
[]
T~(t)Tr(t) do(t) = o.
For completeness, we state the results of [14] for do m (in terms of monic
orthogonal polynomials) as Theorems 3.2 and 3.3. For elementary proofs, see [6, pp.
7-14].
THEOREM 3.2. I f n = 1, then
(" k/2
o
J : ~ o ( - 1)s4
ITk 2j(t)'
k (even) >12,
(3.6)
~k"(t)=r(~)2(-l)Je-jk
(, j=0
~2JT k
2j(l),
k(odd)>>.l.
n~
Moreover, the recurrence relation for the polynomials ~k
r2k+ l(t) = tT~k(t) --flkT~k_l(l),
r20(t) = 1,
r~_, (t) = O,
k=0,
= ~k,1
is given by
1,2 . . . . .
(3.7)
180
WALTER GAUTSCHI AND SHIKANG LI
AEQ. MATH.
where
k (even) >>.2,
L= 7,
(3.8)
L=
k (odd) >>.I.
THEOREM 3.3. For
m = O, 1, 2 . . . . , satisfy
any
n >>-2,
the
polynomials
7~m,n(
"
) : ~rn( " "~ d ~ [ 1 ] ) ,
k=0,1,2 .... ,
~_,..(t)
~o..(t) = 1,
(3.9)
= O,
where
7~
22n
~(1
tfk = 0,
1
-(~-- 1)kin "~
tfk = 0 mod n (k ~ 0),
1 + (k/n)}
( _ 1)(k-1)/,
1( l + 1+(k-1)/n]
1
(3.10)
tfk = 1 mod n,
otherwise.
We remark that, for k ¢ 0, the result (3.10) can be written as
~flZkl~,2
if k = 0 mod n,
flk'~ = l ~-l + 2(k- l)/n2'
otherwise,
ifk = 1 modn,
(3.10')
A
It is also of some interest to note that the polynomials {rck.
. }k=0 are special
symmetric random walk polynomials, since (cf. [3, §7]), as is readily verified,
f l k , , = D k ( 1 - - D k - ~ ) for k = 1 , 2 , 3 . . . . . with D k = ½ ( 1 - ( - 1 ) k t n / ( l + k / n ) )
if
k = 0 mod n, and Dk = ½otherwise.
3.2. Chebyshev measure o f the second kind
We now consider d a ( t ) = do'121(t), for which f t , ( t ) = 2 - " U n ( t ) = U,(t) is the
monic Chebyshev polynomial of the second kind. It satisfies the well-known identity
Ue(t) _ 1 1 -- T2,+ z(t )
2
1 -t 2
We have the following analogue of Theorem 3.1.
(3.11)
Vol. 46, 1993
A set of orthogonal polynomials induced by a given orthogonal polynomial
I8l
THEOREM 3.4. For any integer n >>.1, there holds
~k.~(t) = 7~k(t),
0 ~< k ~< n + 1,
where ~k.,( " ) -- nk( • ; dat,21) and 7"k is the monic Chebyshev polynomial o f the first
kind.
Proof. The basic orthogonality to be shown, in view of (3.11), is
Tk(t)T~(t)
dat21(t) = O,
0 <~r < k <~n + 1,
l
which, since dat21(t)/(1
f
-t
2) ----do'Ill(t),
' T~(t)Zr(t) dot'J(t) = O,
is equivalent to
O<~r < k <~n + l.
1
This is clearly true.
[]
The result in Theorem 3.4 has also been noted in [1, Eq. (2.16)]. The following
theorem paraphrases results of [1], [14].
THEOREM 3.5. For
m = 0, 1,2 . . . . . satisfy
any
n >>.1,
~k + 1,.(t) = t~k..(t) - l~k,.~k_ l..(t),
~o.,,(t) = 1,
~_,,,,(t)
the
polynomials
k=0,1,2
7~m,n( ' )
= gm(
" ; d~[2]),
.....
(3.12)
= O,
where
22n + 1
°
=
,(
1
t f k = 0,
1)
1 + k/(n +
1)
ifk =Omodn
+ l, k ¢ 0 ,
(3.13)
1 + ( k - l)/(n + 1)
i f k = 1 m o d n + 1,
otherwise.
For an elementary proof, see [6, pp. 15-16].
182
WALTER GAUTSCHI AND SHIKANG LI
AEQ. MATH.
We may write (3.13), for k 4 0, alternatively in the form
~2k/(n+ I),1
~k,n =
f
i f k = 0 mod n + 1,
~1 +2(k-- 1)/(n+ 1),1 i f k = 1 mod n + 1,
1
4
(3.13')
otherwise.
Again, {rtk,,, } are symmetric random walk polynomials with parameters
D.=½(l-1/(l+k/(n+l)))
if k = 0
modn+l,
D k = 31 otherwise, in ilk..=
D ~ ( 1 - D k 1), k > ~ l .
3.3. Chebyshev measure of the third and fourth kind
To the measure dtr t31 (cf. (3.1)) there corresponds nt~31(t) = 2-"V,(t), the monic
Chebyshev polynomial of the third kind, and to d o "[4] the one o f the fourth kind,
rt~l(t) = 2-"W.(t). It is well known that
V,(cos 0)
cos(n + ½)0
cos½0 '
W,(cos 0)
-
sin(n + 1)0
sin½0
(3.14)
Replacing 0 by 0 +~z in the second formula yields W , ( - t ) = ( - 1 ) " V n ( t ) , so that
f
~k(t)p(t) V2(t) dat31(t) = 0,
P e Pk l,
1
upon changing t ~ - t implies
f
~k( -- t)q(t) W](t) dat41(t) = 0,
qe~k_l.
I
Consequently,
~,(t; d,~J) = ( - 1 ) % ( - t ; d,~E.~l),
and it suffices to consider the case o f dtr E31.
(3.15)
Vol. 46, 1993
A set of orthogonal polynomials induced by a given orthogonal polynomial
183
THEOREM 3.6. For any integer n >! 1, there holds
7~k,~ (t) = ]h (t),
0 ~< k ~< n,
where ~k,~( ' ) = nk( ' ; d # ~ 1) and Tk is the monic Chebyshev polynomial of the first
kind.
Proof. W e need to s h o w t h a t
f
l Tk(t)T,(t)V~(t)d~rt31(t) =0,
O<.r < k <~n.
--1
This follows at once f r o m the first r e l a t i o n in (3.14) b y n o t i n g that, with t = cos 0,
V~(t) da13J(1) -
cos2(n + ½ ) 0 c l
cos 2 ½0 ~. - t)
1/2(1 + t) j'2 dt
1 + c o s ( 2 n + 1)0 (1 -- t)-1;2(1 + t) 1/2 dt
1 + cos 0
(3.16)
= (1 + T2o+ l(l)) d°'[ll(t) •
[]
THEOREM
m = 0, 1, 2 . . . .
3.7. For
satisfy
any
n >1
1,
the
~k + ,,.(t) = (t -- ~k..)~k,n(t) --/~k,.7~_ l,n(t),
polynomials
k=0,
~m,.(' ) = z~( '; d ~ l ) ,
1,2,...,
(3.17)
~_ l,.(t) = 0,
"~o,.(t) = 1,
where
4 1 + ( k - n ) / ( 2 n + 1)
~k,tt
_m
I
4 1 + (k - n
0
1
- 1)/(2n + 1)
t f k = n m o d 2n + 1,
/ f k = n + 1 m o d 2 n + 1,
otherwise
(3.18)
184
WALTER GAUTSCHI AND SHIKANG LI
AEQ. MATH.
and
7I
tfk = 0 ,
22n
,
1 + 2k/(2n + 1)
,
1(
~ l+
)
ifk =Omod2n
)
tfk = I mod 2n + 1,
l + 2(fc - 1 ) / ( 2 n + 1)
1 -4(1
+(k
-n
+ l, k ¢ : 0 ,
- 1)/(2n + l)) z
1
(3.19)
tf k = n + l mod 2n + l,
otherwise.
Proof The first relation in (3.19) is a simple consequence of (3.16):
flO,n =
d ~ TM = ~
-I
(1 -{-
T2n+l(t))d~[1](t) = ~ .
-I
We thus assume k/> I.
In view of (3.16) and (3.3), the polynomial
~ ( t ) = 2 k - "~k,,(t) = Tk(t) + a , _ , Tk_ ,(t) + ak 2Tk_ z(t) + ' ' '
+ aoTo(t)
(3.20)
needs to be determined so that
(Tk(t) + ak_ l Tk_ l(t) + ak_ 2Tk 2 ( t ) + ' " + a o T o ( t ) )
f_
1
(
,
,
× T,(t) + ~ TI,_ 2,,- ,1(0 + ~ 1", + 2, +, (t)
) aatq(t) = 0
for 0~<r < k .
(3.21)
We first determine ak_ 1 and ak_ 2 (if k ~>2). This will be done by setting up
appropriate systems of linear equations, one for ak_ 1, and another for a~ 2- (For
the proof of Theorems 3.3 and 3.5, only one system, for a~_l, needs to be
considered.)
Vol. 46, 1993
A set of orthogonal polynomials induced by a given orthogonal polynomial
The former involves the unknowns a,
I --s(2n + 1)I, where
~,
185
alk--1--(2n+ 1)I, alk-1 2(2~+1 ) 1 , ' ' ' ,
a]k--
2
-1
i f k = 0 m o d 2n + 1,
i f k = 1 mod 2n + 1,
(3.22)
2L I
S ~-~
2
ifl<k<n+2mod2n+l,
+1
otherwise,
and is obtained by setting r = l k - l - ( j - 1 ) ( 2 n + l ) ] ,
j=l,2 ..... s+l,
in
(3.21). All these values of r are strictly less than k, as required in (3.21). This is
easily checked by examining each of the four cases in (3.22).
We shall first assume k > 2n + 2. Note that this implies s/> 2.
To determine the desired equations, it is necessary to examine the three indices
r, ]r - 2n - 11 and r + 2n + 1 in the second factor of the integrand in (3.21), in
particular, whether they are zero, positive but ~ k , or >k.
For j = l ,
we have r = k - l > 0 ,
]r-2n-ll=k-l-(2n+l)>O
by assumption, and r + 2n + 1 = k + 2n > k. Accordingly, Eq. (3.21) becomes
2ak
l+alk
(3.231)
1 ~2n+J)l=0.
When 2 ~<j ~< s, we again have r > 0 (and r < k, as already observed), since r = 0
would imply k - - l m o d 2 n + l ,
hence by the second relation in (3.22),
k = 1 + s(2n + 1). This in turn implies j = s + 1, which is presently excluded. The
other two indices compute to
Ik - 1 - ( j - 2 ) ( 2 n
lr-2n-I
l=
ik
1-j(2n+l)]
+ l)l
ifk < 1 + ( j otherwise
1)(2n + 1),
(3.24)
and
r+2n+l=
Ik --1 -- ( j - - 2)(2n + 1)1 i f k >t 1 + ( j otherwise.
Ik 1 - j ( 2 n + 1) I
1)(2n + 1),
(3.25)
186
WALTER
GAUTSCHI
AND
SHIKANG
LI
AEQ.
MATH,
Both o f them (as observed previously) are < k , and the second is clearly positive. The first is nonnegafive, and equal to zero if and only if k = 1 rood 2n + 1
and j = s .
Indeed, r = 2 n + l
implies I k - l - ( j - 1 ) ( 2 n + l) I = 2 n + 1, hence
k=l mod2n+l,
which, by (3.22), again implies k = 1 + s ( 2 n + 1). Therefore,
Is(2n + 1) - ( j - l)(2n + 1)1= 2n + 1, that is, (s - j + 1)(2n + 1) = 2n + 1. Since
j ~< s, this implies j = s. The converse is obvious. Note also that for j = s and
k = 1 m o d 2n + 1, we are in the second case for (3.24), and in the first for (3.25).
Altogether, it follows that Eq. (3.21) takes the form
alk-
1 --
(j
--
2)(2n + 1) I J r 2 a l ~ _ I -- ( j
1)(2n + 1)1 -I- al,~ _ ~ - j ( 2 .
j=2,3
.....
+ l) I = O,
s-1
(ifs~>3),
(3.23j)
and for j = s the form
alk - 1- -
2)(2n+ 1)I + 2al, - 1 - -
(s --
(s --
(3.23,)
1)(2n+ I)I -F Talk _ ~ _ ,(2~ + ~)1= O,
where
{~
7=
ifk¢lmod2~+l,
otherwise.
(3.26)
Finally, for j = s + 1, we have for the first index
r = Ik -
1 - s(2n +
1) I
=0
> 0
if k = l m o d 2 n + l ,
otherwise.
F o r the others, we note, first o f all, that
k-
1 -s(2n
+
1) ~ 0 .
This is easily verified by examining separately the four cases in (3.22). It then
follows that
[r - - 2 n --
l I = Ilk
- 1 -s(2n
= Is(2n + !) - k
+
l) I - - ( 2 n
+ 1 -(2n
--- ]k - 1 - (s - 1)(2n + 1)l.
+
1) I
+ 1) I = I(s - 1)(2n + 1) - k
+ 11
Vol. 46, 1993
A set of orthogonal polynomials induced by a given orthogonal polynomial
187
F u r t h e r m o r e , for the same reason,
r + 2n + l = lk -
l - s ( 2 n + l)l + 2n + l = s ( 2 n + l) - k + l + 2n + l
= (s + l)(2n + 1) - k + 1.
By a g a i n e x a m i n i n g the four cases in (3.22), one finds t h a t r + 2n + 1 > k, unless
either k = 1 m o d 2 n + 1, in which case r + 2 n + 1 = ]k - 1 - ( s 1)(2n + 1)], o r
k = n + 1 m o d 2n + 1, in which case r + 2n + 1 -- k. A s a result, the e q u a t i o n (3.21)
gives
al~ ~ {~ -
+ 1)1-'~ 2alk
l)(2n
~ ~(2.+ ~)1= c5,
(3.23~+ 1)
where
0
6 =
ifk#n+lmod2n+l,
-1
In s u m m a r y , letting ~j = alk
given b y
2al
(3.27)
otherwise.
+a2
= 0,
a/ l + 2 ~ l + a r + l
=0,
t-([-
I)(2~a+ 1)[, the system for d e t e r m i n i n g a k _ 1 is
2~<j~<s-l,
(3.28)
~., + 2~X, + 1 = 6 ,
where
7 and
6 are
defined
in (3.26)
and
(3.27),
respectively.
If
k ¢ n + 1 m o d 2n + 1, then 6 = 0 a n d (3.28) represents a t r i d i a g o n a l , n o n s i n g u l a r
h o m o g e n e o u s s y s t e m o f s + 1 e q u a t i o n s in the s + 1 u n k n o w n s ~ , e2 . . . . . es+ ~. It
follows t h a t all ej = 0. In p a r t i c u l a r , ~ = 0, t h a t is, ak I = 0. If, on the o t h e r h a n d ,
k=n+lmod2n+l,
then6=-l,
andsincekg:lmod2n+l,
wehaveT=l
by
(3.26). S o l v i n g (3.28) as a s e c o n d - o r d e r linear difference e q u a t i o n with two
b o u n d a r y c o n d i t i o n s yields
( _ 1).~'+ t
~1
=
ak
1 --
-
-
s+2
188
W A L T E R G A U T S C H I AND S H I K A N G LI
Since,
by
the
k =m(2n + 1) + n
third relation in (3.22), s is even, say
+ 1, we get ak_~ = --1/(2(m + 1)). Thus,
O
AEQ. M A T H .
s=2m,
i f k ¢ n + 1 m o d 2n + 1,
ak-i =
where
(3.29)
1
2(1 + ( k - - n
-- 1)/(2n + 1))
otherwise.
We have shown (3.29) under the assumption k > 2n + 2. One can verify directly
that (3.29) holds also for 1 < ~ k ~ < 2 n + 2 by considering separately the cases
l~<k~<n+l,
n+l <k<2n+l,
k=2n+l
and k = 2 n + 2 ,
putting in (3.21)
r = k - 1 in the first case, and r = k - 1 and r = [k - 2 - 2n I in the others.
We now turn our attention to ak 2. Here, we set up a system o f linear equations
for
ak 2, aik
2 (2n+ 1)4' • " • '
2
-1
alk
2-s(Zn+ I)1' where
ifO ~<k ~< i m o d 2n + 1,
i f k = 2 m o d 2 n + 1,
(3.30)
if2 < k < n + 2 m o d
2 2-777
2
+1
2n + 1, n ~>2,
otherwise, n 1> 2,
by letting r = Ik - 2 - ( j - 1)(2n + 1)l, j = 1, 2 . . . . . s + 1, in (3.21). By a discussion very similar to the one carried out for the coefficient a~ ~, one obtains for
~j = a l k - 2 - ( ] - 1x2,+ i)l, assuming first k > 2n + 3 (hence s t> 2), the system
2~ +~,
=0,
c~'j_l+20t~+~j+' ~ = 0 ,
~'s-1 + 2 ~
'0t'
+Y
s+l
2~<j~<s-l,
(3.31)
=0,
~'~ + 2oq~+ t = a ' ,
where y' and 6' are defined by
Y'={12
otherwise,ifk¢2m°d2n+l'
6,={ i f k~ # lotherwise.
mod2n+l,_
(3.32)
Vol. 46, 1993
A set of orthoganal polynomials induced by a given orthogonal polynomial
S t r a i g h t f o r w a r d solution yields, for ~ = ak
0i
ak_ 2 =
189
2,
i l k ~ 1 m o d 2n + i,
1
+ 2(k - 1)/(2n + I)
i l k = 1 m o d 2n + 1.
T h e same holds true if 2 ~<k ~<2n + 3 ,
separately the cases 2 ~ < k < n + l , n + l ~ < k < 2 n + 2 ,
R e t u r n i n g n o w to (3.20), we can write
z2,,.(t) = Tk(t) +~a,
1
(3.33)
as can be verified by considering
k=2n+2
andk=2n+3.
,#, , ( O + ' " + ~ o ~ o1( t ) ,
or, in view o f (3.29) a n d (3.33),
1
I
r~k'" = 7~k + 4 I + 2(k - l)/(2n + 1)
~1%k 2 "~- { ~
3, L
4 ....
}
ifk=lmod2n+l,
1
"~k.,=L
(3.34)
1
41+(k_n_l)/(2n+l)Tk
1 -]- {~%k - 3, Tk
4 ....
}
ifk=n+lmod2n+l,
~k,, = 7~ + {7"k-3, Tk 4 . . . . }
otherwise,
(3.35)
(3.36)
where {x, y, z . . . . } means a linear c o m b i n a t i o n of x, y, z . . . . or zero, if the list is
empty.
I f k = 0 m o d 2n + 1 (k # 0), hence k + 1 = 1 m o d 2n + 1, we o b t a i n f r o m (3.34)
and (3.36)
1
1
r~k+ l"n(t) -- tT?k'n(t) = T~+ l(t) + 4 1 + 2k/(2n + 1)
T~_ l(t) -- tTk(t)
L-3 .... }
~(
= -
1
) ~ k - ' ( t ) + { " "' )'
1 - 1 + 2 k f ( 2 n + 1)
190
W A L T E R G A U T S C H I A N D S H I K A N G LI
AEQ. M A T H .
where { . . . }
is a linear combination of 2kk_2, Tk_ 3 . . . . .
hence of
~k - 2.~, ~k - 3. . . . . . if k i> 2, or zero otherwise. Since 7"k- I = ~k -- 1., + { " " " }, there
follows
1(
~k+,,,(t) -- t~k,,(t) = --~
1
1 -- 1 + 2k/(2n + 1)
) ~k_ 1,n(t)'
the linear combination { - . . } left over being necessarily zero.
Therefore,
~k,, = O,
1(
flk.,, = ~ 1
1
)
l + 2 k / ( 2 n + l)
ifk=Omod2n+l
(k#O).
This agrees with the third relation in (3.18) and proves the second in (3.19).
Next, consider k = 1 mod 2n + 1 (and n/> 2). Then, similarly as above, (3.36)
and (3.34) yield
1
1
4 1 + 2(k - 1)/(2n + I)
1(l+
4
1
1 + 2 ( k - 1)/(2n + I)
)
Tk
i(t)--tTk(t)
~k 1,n(t),
(3.37)
hence
~k,~=0,
fl,,,=~
l(
1
l+l+2(k_l)/(2n+l
))
i f k = 1 m o d 2 n + 1.
This agrees with the third relation in (3.18), i f n t> 2, and proves the third in (3.19).
If n = 1, then (3.37) must be slightly modified, since (3.35) and (3.34) are now
relevant, and one finds fi~,~=¼(1 + ( k - 1)/3) -1 and L.1 =¼(1 + ( 1 + 2 ( k - 1)/
3)-1), which is the desired result in the case n = 1.
If k = n m o d 2n + 1 (and n/> 2), then a similar argument, based on (3.35) and
(3.36), gives
1
1
1
Rk + L,(t) -- t~e,,(t) = --~ 1 + (k - n)/(2n + 1) r~k,~(t) - ~ ~k- ~.,(t),
Vol. 46, 1993
A set of orthogonal polynomials induced by a given orthogonal polynomial
191
and thus
1
1
1
~k,~=-~l+(k_n)/(2n+l),
[~k,.=~
ifk=nmod2n+l.
This proves the first formula in (3.18) and agrees with
The case n = 1 again requires the use of (3.34)
~k,l = 1 ( 1 + ( k - 1)/3) -1 , flk.~ =¼(1 + ( 1 + 2 ( k - 1)/3)
mula in (3.18) and the third in (3.19), both for n = 1.
Finally, if k = n + 1 m o d 2n + 1, n >~ 1, then (3.36)
~k+l,n(t)
-- l'~k,n(t)
Z k + l ( t ) "1- {7"k-- 2, ]~'k 3 . . . .
=
1
1
t¢~
the last in (3.19), if n ~> 2.
and (3.35), which gives
1), providing the first forand (3.35) produce
}
~(t) + {¢~_~, ¢~ ~ ....
)
- tea(t) + ~ 1 + (k - n - 1)/(2,, + 1)
o
=-4
1Tk_,(t)+
1
4
1
_
l+(k
n
_
1)/(2n+1)
Tk(t)+{Tk-
~.
Tk-3
.
.
.
}.
.
Applying (3.35) to the middle term then yields
1]~g_
~ + ,,.(t) - t~k,~(t) = --~
+
16 (1 + ( k - n
=-~
,
(t)+l 4
1
1 + (k - n - l)/(2n + 1) ~k'"(t)
- l)/(2n + 1)) 2
-
4(l+(k_n-1)/(2n+l))2
1
~k 1.n(t)
1
+ 4 1 + (k - n - 1)/(2n + 1) r~k'"(t)'
that is,
1
~k,n -m-
1
4 1 + (k-n
,(
~k,~=-~ 1
- 1)/(2n + 1)'
,
4(l+(k_n_l)/(2n+l))2
)
if k = n + 1 m o d 2n + 1.
This proves the second relation in (3.18) and the fourth in (3.19). The p r o o f is
completed by noting that, for all remaining values of k, Eq. (3.36) applies, both as
192
WALTER GAUTSCHI AND SHIKANG
LI
AEQ. MATH.
written, and with k replaced by k + 1. F r o m this, the last relations in (3.18) and
(3.19) follow immediately.
[]
In analogy to (3.10') and (3.1Y), we have, for k ¢ 0 ,
l! 1 + 3(k-- n)/(2n + 1),1
~k,n ~
2+3(k--n--
1)/(2n+ 1),1
i f k = n m o d 2 n + 1,
i f k = n + 1 rood 2n + 1,
otherwise,
(3.18')
and
~3k/(2n + I ), I
i f k = 0 m o d 2 n + 1,
J~l + 3(k
if k = l m o d 2 n + l ,
l)/(2n + I),1
J~2 + 3 ( k - n -- l)/(2n + 1),1
1
i f k = n + 1 m o d 2n + 1,
(3.19')
otherwise.
4. Interlacing properties for zeros
In some applications, for example to extended interpolation [2], [5], one is
interested in the zeros r,, of re,,( • ; da) and the zeros f# o f ~ , + ~.~( • ) = ~z~+ 1( " ; d~,).
In particular, one would like to know whether they interlace, i.e., whether, if
ordered decreasingly a n d contained in ( - 1, 1), they satisfy
-1 <f~+~ <~. <'~. <""
<'g2 <~, < ' ~ < 1.
(4.1)
This is the question to be studied in this section, first analytically for Chebyshev
measures, a n d then numerically for G e g e n b a u e r measures.
4.1. Chebyshev measures
THEOREM 4.1. For the Chebyshev measures d~ v3, i = 1, 2, 3, 4 (cf. (3.1)), and for
any fixed n >>.1, the zeros zv of n , ( ' ; d a H) interlace with the zeros ~ o f
~ . + , . . ( . ) = re.+,(. ; d~l).
Proof W e carry o u t the p r o o f only for da m, that is i = 1, since the p r o o f in the
case i = 3 is similar, a n d in the case i = 2 trivial (since 7tn = ~ , , ~,+ 1,n = T , + ~)- T h e
case i = 4 follows directly from the case i = 3 by virtue of (3.15).
Vol. 46, 1993
A set of orthogonal polynomials induced by a given orthogonal polynomial
193
F o r a °1, one has 7r, = 7~, (the monic Chebyshev polynomial o f the first kind)
and, if n ~> 2,
~. + 1,.(t) = t#,(t) - ~
3 o
T._L,
(4.2)
as follows f r o m Theorems 3.1 and 3.3. One easily calculates
1
7~n+ ,,n(1) -- 2n+1 ,
(--1) n+l
7~n+ 1.n( -- 1) --
2n+l
(4.3)
,
and, since r~ = cos 0,,, 0v = (2v - l)~/2n,
~ , + 1 , , ( r , , ) = 2 3 - - ~ ( - 1 ) " s i n 0 ~,
v=l,2
.....
n.
(4.4)
For n = 1, one has rt I (t) = t, ~2: (t) = t z - 3/4, from which the validity o f (4.3),
(4.4) follows also for n = 1. It is seen that r2, + ~.,(t) takes on values with alternating
signs as t assumes the values 1, rv, v = 1, 2 . . . . . n, and - 1 in this order. This
proves (4.1).
[]
4.2. Gegenbauer measures
These are the measures d~r:'(t)= ( 1 - t 2 ) x - l ! 2 d t on ( - I, 1), where 2 > - 1 / 2 .
They include the Chebyshev measures o f the first and second kind as special cases,
2=0
and 2 = 1 ,
respectively. The zeros *v and f . o f = . ( . ; d a x) and
~.+ ~,.( • ) = ~z.+ 1( " ; dS~) are now functions of 2, and we may study their m o t i o n
induced by letting 2 vary from 2 = 0 to either side o f 0. W h e n 2 = 0, we k n o w from
Theorem 4.1 that interlacing holds. As we increase (or decrease) 2 from the starting
value 2 = 0, the interlacing property remains intact so long as no collision occurs
between a zero r,, a n d a zero {~. Once such a collision occurs, the polynomials re.
and r~.+~,, have a c o m m o n zero, and this is t a n t a m o u n t to the resultant
R(n,, ~, + l,n) o f nn a n d 7~n + l,n vanishing. This resultant, o f course, is also a function
(indeed a rational function) o f 2. Therefore, to determine an interval 2, < 2 < A, in
the parameter 2 on which the interlacing property holds, we need to determine the
smallest positive zero, A,, and the largest negative zero, ,~,, if any, o f R ( n , , ~ + 1,,)The same a p p r o a c h has previously been taken in [7] to study the interlacing
property for G a u s s - K r o n r o d quadrature formulae, and we refer to this work for
computational details. The results obtained are summarized in Table 4.1. Evidently,
194
WALTER GAUTSCHI AND SHIKANG LI
Table 4.1. Intervals 2 ~ < 2 < A .
n
,~,
1
AEO. MATH.
in which the interlacing property (4.1) holds for Gegenbauer
measures do 2
A,
n
2,,
A,,
~
oo
9.094948
5.753360
4.670917
4.134548
3.813842
3.600251
3.447629
3.333022
3.215973
3.100459
14
15
16
17
18
19
20
24
28
32
36
40
-.098522
--.093053
--.088177
-.083804
-.079856
-.076275
-.073010
-.062397
-.054536
-.048469
-.043641
-.039703
3.010405
2.938215
2.879043
2.829653
2.787798
2.751893
2.720697
2.628545
2.561196
2.501838
2.458914
2.426421
-½
2
3
4
5
6
7
8
9
10
11
12
13
--.367035
-.298848
--.251041
--.216429
--.190358
--.170044
--.153770
-.140435
--.129301
-.119861
-.111750
--.104704
the 2, increase, and the An decrease, monotonically with increasing n. It is an
interesting open problem to establish that the limits of 2n and An as n ~ ~ indeed
exist, and to determine their values.
The cases 1 ~< ~ ~< 3 o f Table 4.1 can be verified algebraically. T o do so, one uses
the well-known formula,
f_
t 2m da~(t) -
r(m
+ ½)r( + ½),
F ( m + )' + 1)
m = 0, 1, 2 , . .
"'
t o c o m p u t e the coefficients o f n n and ~, + 1,. from the respective orthogonality
relations. The coefficients turn o u t to be rational functions o f ft. One then computes
the required resultant explicitly as a rational function of fl and determines its zeros.
For n = 1, for example, one finds
n~(t) = t,
~za(t) = t 2
3
2(2 + 2 ) '
(4.5,)
a n d these polynomials obviously have no c o m m o n zeros. Hence, ).1 = - ~
Ai --- oo.
F o r n = 2, one computes
n2(t ) = t2
1
2(,~ + 1)'
$3.2(t)
= t 3 --
3(13). + 7)
2()~ + 4)(52 + 3) t.
1
and
(4.52)
Vol. 46, 1993
A set of orthogonal polynomials induced by a given orthogonal polynomial
195
Since t = 0 is not a c o m m o n zero, we m a y consider the simpler polynomials
p2(t) = n2(w/t)'
/~3(t)
= ~t ~3'2(%//t)'
They have a c o m m o n zero if and only if/~2 and ~3,2 do. Their resultant c o m p u t e s
to
R(p2,/~)
where the
zeros, the
since R >
For n
~3(0
3422 + 37). + 9
=
2(2 + 1)()- + 4)(52 + 3)'
d e n o m i n a t o r is positive for all 2 > - 3 1. The n u m e r a t o r has two negative
larger of which is ( - 3 7 + 1 ~ 5 ) / 6 8 = - . 3 6 7 0 3 5 3 7 8 . . . and defines 22,
0 for all 2 > )-2.
= 3, a more lengthy c o m p u t a t i o n gives
3
2()- + 2)
= t3
~.4,3(t) = t4 -
t,
(4.53)
1 5 ( 1 4 2 3 + 7 1 2 2 + 1582 + 7 2 )
t2
(2 + 6)(2623 + 15322 + 3362 + 160)
45(2225 + 18324+ 64423 + 146022 + 17762 + 640)
+
4(2 + 6)(2 + 5)0~ + 4)(2 + 2)(26,~ 3 + 15322 + 3362 + 1 6 0 )
Again, these polynomials cannot have t = 0 as a c o m m o n zero, since otherwise 7~4,3
would have a double zero at the origin, which is impossible (7~4, 3 is an orthogonal
polynomial!). We therefore compute the resultant of p 3 ( / ) = ( l / N / / t ) / l : 3 ( x / / t ) and
/~4(t) = 7~4.3(w/t) and obtain
9(4)- 6 + 572). 5 +
R(p3,/h)
=
51052 4
+ 1729823 + 27256). 2 + 17440). + 3200)
4(2 + 6)()- + 5)(2 + 4)(2 + 2)2(2623+ 15322+ 3362 + 160)
The cubic polynomial in the d e n o m i n a t o r has a pair of complex zeros and a
negative zero at 2 = -.6447375076 . . . . so that the denominator is positive for all
)- > - ½ . The n u m e r a t o r polynomial, on the other hand, has four negative zeros and
a pair o f complex conjugate zeros, and is therefore positive for all )- >)-3, where
)-3 = - 0 . 2 9 8 8 4 7 9 6 5 4 . . . is the largest negative zero.
196
WALTER GAUTSCEI
AND SHIKANG rl
AEQ. MATH.
5. The polynomial sequence {~.:,_ ~}
The special sequence of induced polynomials {~. . . . ~} (where ~z0,_1 = l) whose
degrees exceed the degrees o f the generating polynomials by one, are of some
interest in extended interpolation, as already mentioned. Here we ask the question
whether they also form an orthogonal sequence, and if so, relative to what measure.
We restrict ourselves to the original measure being a Jacobi measure, and begin
with the case of the Gegenbauer measure
d~(t)=(1-t2)
~ ~/2dt
on(-1,1),
2>-1/2.
(5.1)
Thus,
.... ~(t)=x,(';d#~_,),
d~_,=[Tz~
,(';do;~)]2&r ~.
(5.2)
In the special cases 2 = 0 and 2 = 1, the polynomials (5.2) are indeed orthogonal.
F o r 2 = 1, this is trivial, since ~ . . . . 1 = if', (cf. Theorem 3.4). For ). = 0, it is known
[5, Eq. (2.3)] that r~,,_
, ~ = 2 - ( " - 1)(T, --5T,,_2), from which it easily follows that
~.+Ln(t)=t~...
~(t)-fl.r~.
~,. 2(0,
n>~l,
(5.3)
where
[3/4
fl,=Jl/8
~1/4
i f n = 1,
if n = 2 ,
(2=0)
otherwise.
(5.4)
Since B. > 0 for all n >~ 1, Favard's theorem tells us that there exists a positive
symmetric measure with respect to which the polynomial sequence {r~. . . . ~} is
orthogonal. F r o m [8, Eq. (3.8) and Prop. 2.1] it follows indeed that
( 1 - t 2) - I f2
.... l(')=Tt,(';ds),
n~>0;
ds(t)-
l_St 2
dt
on(-1,1).
(5.5)
We believe that these two values, 2 = 0 and 2 = 1, are the only values o f the
parameter 2 for which the sequence {r2..... ~} in (5.2) defines an orthogonal
sequence. The reason for this is as follows. Suppose there are positive constants ft,
Vol. 46, 1993
A set of orthogonal polynomials induced by a given orthogonal polynomial
197
such that (5.3) holds. Then, putting n = 3 in (5.3), using (4.51_3) , and comparing
coefficients of equal powers on the left and right, gives an equation in 2, namely
a30(,~) -I- [azl (2) - as2(2)]a10(2) = 0,
where %(2) is the coefficient o f t j in +,+L~ o f Eq. (4.5,). After some algebraic
manipulations, this simplifies to
(5.6)
)~(2 -- 1)( 1882 4 q- 71223 -- 74322 -- 32022 -- 1680) = 0.
We again recognize the two known solutions, 2 = 0 and 2 = 1. Since the quartic in
(5.6) is less than 188 • 2 4 + 3203 - 2-+ ~ - 1680 = - 6 7 . 2 5 < 0 for 2 ~ ( -½, 0), it has
no zero in that interval. Indeed, all its zeros are real, less than - I/2, except for one
positive zero, 2o = 2 . 2 2 7 6 1 4 2 8 6 3 . . . One could now in principle repeat the reasoning above with values n ~>4 and arrive at a sequence o f additional algebraic
equations that 2 would have to satisfy. It is very unlikely that the single root )~0
found above is a root o f all these equations, and therefore equally unlikely that a
recurrence relation (5.3) exists for any 2 other than 2 = 0 or 2 = 1. Because o f the
extreme complexity of the algebra involved, however, we have not carried through
the calculations.
For Jacobi measures
da~'t~(t)=(l-t)~(l+t)/Sdt
we recall
from
[5,
~ n +• +- I = 2 (n . l ) ( T n - 3 T
l ~
r'~++,,,(t)=Ct--m,)r~+.+
on ( - 1 ,
!),
c~>-l,fl>
(5.7)
-1,
!
Eq.
(2.17)]
that,
when
m=-+,
,), n ~> 1, from which there follows that
then
,(t)--fl,+,
(5.8)
,,,
z(t),
n>~l,
holds, with
--10/4
%=
ifn = !,
otherwise,
fl~
)+3/8
[1/4
if n = I
otherwise.
(5.9)
Moreover, from the first line of Table 1 in [8, p. 222], with /~ = -½, we find
~,., ,(.)=n+C';ds),
n~>0;
ds(t)=
(I - t 2) t/2
5
+-t
dt
on(-1,1).
(5.10)
198
WALTER (3AUTSCHI AND SHtKANG LI
Similar results, o f course, hold for
in (5.9) and (5.10), respectively.
Gegenbauer case, we conjecture,
measure d a = d o ~'~ cannot be an
AEQ. MATH.
~ = ½, fl = - ½ , with the signs o f aj and t reversed
For reasons a n a l o g o u s to those stated in the
however, that {~,,n t} for the general Jacobi
orthogonal sequence, unless
--1/31 = ~1
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Department of Computer Sciences,
Purdue University,
West Lafayette, Indiana 47907, U.S.A.
Department of Mathematics,
Purdue University,
West Lafayette, Indiana 47907, U.S.A.
current address:
Department of Mathematics,
Southeastern Louisiana University,
Hammond, LA 70402, U.S.A.