GENERATING NEW CLASSES OF ORTHOGONAL POLYNOMIALS

Given a sequence of monic orthogonal polynomials (MOPS), {Pn}, with respect to a quasi-definite linear functional u, we find necessary and sufficient conditions on the parameters an and bn for the sequence P,(x) + a,Pn-l(x) + b,Pr,_2(x), n >_ Po(x) 1, P-1 (X) 0 to be orthogonal. In particular, we can find explicitly the linear functional v such that the new sequence is the corresponding family of orthogonal polynomials. Some applications for Hermite and Tchebychev orthogonal polynomials of second kind are obtained. We also solve a problem of this 'type for orthogonal polynomials with respect to a Hermitian linear functional.

We are interested in some partial converse of the above result when 2 as well as some extensions for more general families of orthogonal polynomials.
If u and u2 denote the linear functionals on the linear space IP of polynomials with complex coefficients, defined respectively by for every p IP, then 1,2 (q(x)u2, p(z)> q(x)p(x)dp2(x) fp(x)d#,(x)= (u,,p(x)) i.e. qu2 u in the space of linear functionals on IP.
If q is a polynomial of degree two and u is a quasi-definite linear functional our first problem is to find necessary and sufficient conditions in order to u2 defined by qu2 u be a quasi-definite linear functional.
Maroni studied this problem when p is a polynomial of degree (see [13]) and in the case when p is a polynomial of degree 2 with Zl z2 (see [14]).In this last case, he does not give the proof of this result.The conditions that we give here are basically different from that ones.
On the other hand, we consider the following inverse problem: Given a MOPS {P,} with respect to a quasi-definite linear functional ul, to characterize the sequences (an), (bn) of real numbers, such that the sequence of monic polynomials Rn Pn + anPn-1 + bnPn-2, n >_ (3) is orthogonal with respect to some quasi-definite linear functional u=.As an immediate consequence, to find the relation between ul and A first attempt to propose an inverse problem was made by J.L.Geronimus in [6].In fact, he obtained information about the quasi-definite linear functionals u and v associated to two MOPS {Pn} and (R,) related by (2).Furthermore, he gave necessary and sumcient conditions on the family {R,} in order to be orthogonal with respect to v.This problem is also connected with positive quadrature formulae (see [19]).
Geronimus gave also in that paper a proof of the Hahn's theorem for the clsical MOPS, i.e. the only MOPS {P} such that the is also a MOPS are the classical ones.He used the fact n+l that the classical MOPS satisfy T()+ (),P0(z)=l.
When ,+,+1 0 n E IN he gave a representation of the measure with respect to {P,,} 7n+s is orthogonal.This is because {P} is a finite linear combination of second kind Tchebychev orthogonal polynomials.
Finally, we study an inverse problem for orthogonal polynomials with respect to Hermitian linear functionals on the linear space of Laurent polynomials.This represents a generalization of the theory of orthogonal polynomials with respect to a measure supported on the unit circle.
The structure of this work is the following: (a) In Section 2 we present the basic tools concerning linear functionals as well as the concept of quasi-orthogonality which will play a central role in our paper.
(b) Section 3 is devoted to the direct problem, i.e. the relation between orthogonal polynomials associated with two quasi-definite linear functionals u, v satisfying a relation u p(x)v where degp 2. Furthermore, we give in Theorem 3 the connection between the corre- sponding parameters of the three term recurrence relation.
(c) In Section 4 we study the inverse problem (3) and we consider some examples.Theorem 4.1 gives the characterization of such sequences by a constructive approach.
(d) In Section 5 we give necessary conditions on the sequence a, in order to n(Z) Cn(Z) -" anCn--l(Z) (4) be a MOPS with respect to an Hermitian linear functional if {.} is a MOPS with respect to an Hermitian linear functional.

QUASI-ORTHOGONALITY
We now introduce the algebraic tools that we use in this paper, (see [5] and [12] for more details).Let {P,} be a MOPS with respect to the quasi-definite (respectively, positive definite) linear functional u defined on IP, i.e.
Since {P,} is an algebraic basis in IP, we can define its dual basis (a,,)in IP* (the algebraic dual space of IP) as If v is an element of IP*, we can express it as v y A,a, where A, (v,P,), IN.As an We can represent the elements of the dual basis in terms of {P,,} and u.In fact, since (cn, Pro) 6n P.u Pra) n, m e IN, we have (u,PZn)' ' <u, p.> " () Let define some linear operators on IP: By transposition, we introduce the following linear operators on IP*" 1. (qa, p) <a, qp) where deg for q(x)-a,x' and o C *; 1--0 2. ((x c)-la, p) (o, Ocp) where 0,ifn-0 c "-1-' (a, x'+n) if n > Given a polynomial p(x) -I(xx,)"u, for every f E IP, we define where, is the Lagrange-Sylvester (Hermite) interpolation polynomial of f with nodes x,, 1,... ,s and the L,k verify the following conditions: { lifu=k-1 andj=i ,k (x) 0 elsewhere (11) for k 1,2,...,m, and 1,...,s.Then: DEFINITION 1 p-lu is the linear functional given by <p DEFINITION 2 ([5]) Let {P,,} be a MOPS with respect to the quasi-definite linear func- tional u.We say {p(1)} is the first associated MOPS for the MOPS {P,} if it satisfies (1) (3) DEFINITION 3 ([4]) The MOPS defined by (14) is called a co-recursive sequence of the MOPS {P,, }.
Notice that these polynomials verify the same three-term recurrence relation as {P,} with initial conditions P0(x; c) and Pl(X; c) Pl(X) c.
Next we introduce a basic definition in our work" DEFINITION 4 ([14]) Let {p,} be a MPS with degp, n and u be a linear functional.We say that {p, is strictly quasi-ovthogonal of order s if (U, pmp.)O, Inra >_ s -4- /r >_ s (u,p,._sp,.)O. ( THEOREM 5 ([16]) Let {P,} be a MOPS with respect to the quasi-definite linear functional u; then, {p,, is strictly quasi-orthogonat of order s with respect to u if and only f -a,,kP, O<_n <_s-1 p.(z) .,P,n>_ k=n-s+l where a ..... : O, Vn >_ s. (16) 3 DIRECT PROBLEM This section is motivated by some modifications considered in the literature of orthogonal polyno- mials.As an example, the Bernstein-Szeg6 polynomials (see [5], [9] and [10]) are orthogonal with respect to the weight functions where p is a positive polynomial in [-1, 1] of fixed degree.These polynomials can be represented in an easy way in terms of the Tchebychev Polynomials of first and second kind as can be deduced from (1) (see also [17, Theorem 2.6]).
Here we solve a more general problem: THEOREM 6 ([2]) Let u be a quasi-definite hnear funct,onal with uo and X X2 E C.
Taking into account theorem 5, the MOPS with respect to v is a strictly qui-orthogonal MPS of order two with respect to u.Now, if we substitute in definition 4 u by (x z,)(z z2)v we obtain that v is quasi-definite (because v is strictly quasi-orthogonal of order two with respect to u, i.e b.+2 # 0, n ), if and only if: i.1 (v, n+l) 0 for n , i.2 (v, xR,+:) 0 for n , i.3 (v, R) # 0.
Expressing these conditions in terms of the data: 1. From i.1 with n 0 we obtain al fl0 Vl.   2. From i.3 and taking into account the lt expression for al we get *=-'(Xl v) + -0 ( v) # 1.

REAL INVERSE PROBLEM
The answer to the main problem is the following: (35) THEOREM 9 Let P be a MOPS wdh respect to u and R,} be a stmctly quasi-orthogonal MPS of order two wth respect to P,}; {Rn} is a MOPS f and only f, the parameters n (16)   an, a and a .... (b} a, a, a3, b3 zf a 0 and b aa.
2. (36) provides an algorithm for the computation of (a., b.).In hct, from the first relation we can calculate the a+4 in terms of (-+3 (b+)= in the second u)k= and and substituting a+4 one we get b+4.
As {Rn} is a MOPS associated with v we have by (7) an (v,m.)v for all n E lN; and so (42) can be rewritten as u + a l R l ( x ) l l + R2(X)rhr/2 <v, 1)" Taking into account Rn for n =-1, 2 we get (38). [-1 EXAMPLES.We will construct two MOPS strictly quasi-orthogonal of order 2 with respect to: (1) Second kind Tchebychev MOPS, {U,,}.
It is well known that in this case /3, 0 n IN (see [5, Exercise 4.9]).
Hence, if we choose as initial conditions al a 2a and b b3 -c # 0, with a,c C, we obtain by induction from (36) and considerations about initial conditions, an+3 2a n lN.bn+4 --c   It is straightforward to prove that (37) holds.The coefficients of the three term recurrence relation (ttrr) verified by {R} are { { sXo=-2a, and r/ = c ,nE]N.
,+ 0 r/,+ This is an example of A1-Salam and A. Verma (see [1]).There they found a measure such that {R,} is the corresponding MOPS.We can say that the condition for the positive definitness of the new linear functional is c >and a E IR.Now, we give the quasi-definite (not positive definite, because ?1 < 0) linear functional, v, in terms of the second kind Tchebychev linear functional, u, in the particular case a 0 and From this representation we get (-a6 v 1) v0, i.e v0 -; hence 3vo (6_ + ,) 8 (2) Hermite MOPS, {H,}.
The coefficients of the ttrr verified by {R,,} are given by -ifn=0 The quasi-definite (not positive definite) functional v is given in terms of the Hermite linear functional, u, by: From this representation we get (-v i.e vo -2; hence v x-2u 2(50.

COMPLEX INVERSE PROBLEM
Before working out the problem (4) we will state some basic definitions.
CO Cl ] Let f14 CO ] be an infinite Toeplitz and Hermitian matrix.A linear functional u on the linear space of Laurent polynomials span {z n G } is said to be Hermitian if ca (u, z"), n E IN and , (u, z-"), n IN.Here 7Z denotes the set of integers {0,-t-1, +2,...}.
The linear functional u is said to be quasi-definite when the principal submatrices of .M are non-singular.
In these conditions, we can define a MOPS {,,} with respect to the matrix .or with respect to where A,,_I means the principal minor of order n for .A/[.This family will be called a MOPS on the umt circle T since the shift operator on IP is unitary with respect to the above inner product.
For a MOPS {q,} on the unit circle, T, Szeg6 recurrence relations follows (see [17, T.