SOME RESULTS ON BIORTHOGONAL POLYNOMIALS

Some biorthogonal polynomials of Hahn and Pastro are derived using a polynomial modification of the Lebesgue measure dθ combined with analytic continuation. A result is given for changing the measures of biorthogonal polynomials on the unit circle by the multiplication of their measures by certain Laurent polynomials.

Throughout this paper we assume that q is real and, for convergence of the infinite products, <nsidering the denominator of (z;q) we also want [q[ < 1 and[qbz- [ 1, that is <l l <l l We also require that [ < 1 and l < .H ote that these restrictions, besides euring convergence and existence, make both sides of equation (1.1) analytic in the parameters a and b. is we will need in ction 3.
In section 2 we state, in determinant form, a pair of polynomial sets which are biorthogonal on the unit circle with respect to the measure dv(O).z-"(z_at)(z-oa)...(z-cq)dO z.e ' assuming that no a is zero and that 0 m h.
In Section 3 we show how these yield Pastro's polynomials in the special case a q2,, b The full result follows by analytic continuation.
Pastro also gave in [9] explicit examples of Laurent orthogonal polynomials, making concrete the earlier work of Jones and Thron [7] in which such "polynomials" were introduced. (They are not actually polynomials, they contain both positive and negative powers of their variable.)More than this, he states an interesting connection between biorthogonal polynomials and orthogonal Laurent polynomials.
There is a well-known formula of Christoffel for modifying the measure da(x) by polynomial multiplication.That is, let o(x (x x) (x x2). .(x x, be a polynomial which is non-negative on [a,b and let {q,(x)} be the polynomials orthogonal with respect to the new measure p(x)dcx(x) on [a,b].Then the polynomials {q,,(x)} can be represented in terms of the polynomials {p,(x)} by p(x)q,(x c,det for suitable constants c,,.

p.(x,) p./(x,)
Both this formula of Christoffel and a related formula of Uvarov carry over to polynomials orthogonal on the unit circle.See Godoy and Marcellan [3] or Ismail and Ruedemann [6].The natural question is, does this formula of Christoffel have an analogue for biorthogonal polynomials on the unit circle?In Section 4 we show how a trivial modification of the result in [6] yields a result for biorthogonal polynomials, at least for certain cases.Unfortunately, we only allow certain modifications and must assume that certain determinants do not vanish.Actually, this assumption of nonzero determinants is common to biorthogonality (see the work of Baxter [2]).
In the remainder of this paper we adopt the following notation.For p,(z) a polynomial ofdegree r we define p(z) z" ,(z-1).For nonzero complex numbers t, ct" denotes 1/.Finally, z denotes e ie in the integrals presented.

A PAIR OF BIORTHOGONAL POLYNOMIAL SETS
In this section we consider a pair of polynomial sets which are biorthogonal on the unit circle with respect to the measure dv(O).z-"(z_ctl)(z_%)...(z_ah)dO z.e i assuming that no ctj is zero and that 0 m -: h.First we need two lemmas.
The problem now is to evaluate A.,t in general We have r + s zeros in our weight function but A.,t is only a (r + 1) by (r + 1) determinant.We "fill out" A.,t and use the Jacobi-Trudi identity in reverse. A Jo ""J,-, L -* /,-2 L +,-1 ""L "".
We could use Lemma 2 to find the other set of polynomials required for biorthogonality but, as noted previously, the conjugation of the weight function w(z) merely switches the roles of r and s and hence those of a and b as well.Thus the polynomials q,,(z,a,b): p,,(z,b,a) satisfy z-e At this point we have the biorthogonality of the polynomial sets {p,(z)} and {q,(z)}.We still must compute the value of p,,(z)q,,(z)w(z)dO.

MODIFICATION OF MEASURES BY LAURENT POLYNOMIALS
In this section we start with a measure dv(O) which is not necessarily positive on z e .F rom Baxter [2] we know that if certain Toeplitz determinants are nonzero then there exists a unique pair of polynomial sets which are biorthogonal on the unit circle.We will call this pair {,,(z)} and {,,(z)}.That is, for any polynomial p,,_l(z) of degree at most n we have What we want to do is multiply the complex measure dv(O) by a Laurent polynomial and get determinant formulas for the new biorthogonal polynomials, {ap,(z)} and {,(z)}, in terms of the old polynomials, {,,(z)} and {,(z)}.Actually, we are going to restrict ourselves to two types of Laurent polynomials, those of the forms R(z) z-'G,,(z) and R l(z) z -(" 1)G (z), where G,(z) and Gz,, (z) are polynomials having precise degrees 2m and 2m + 1 respectively.Further- more, we shall require that neither G,,(z) or G l(z) have z as a factor.We have two eases: the even case and the odd case.

(z).(z)z .(zv(O)O
then the polynomial sets {,(z)} and {.(z)} are biorthogonal on the unit circle with respect to the measure z " ")G. (zv(O).e unusual form of the determinant in (4.4) comes about as we are writing zG* . in the form z " zG dz)] so that the same idea behind Theorem 2 applies in a sense.
However, multiplying both sides of (2.1) by z-" and then expanding the determinant along the first row we find that +h-m .(z)[z'(z-ct)Cz-c)...Cz-ct)]-E c z + E c,,z -1 As each of the terms in the above sums are orthogonal to any p,, (z) the result follows immediately.
PROOF OF (2.2).We want to show that if ,,(z) is defined as in equation (2.2) then for any polynomial p,_ x(z) of degree at most n we have f p._,(z) .(z) [z-'Cz ct,) (z a)...(z ct)]ao o.
Conjugating both sides of this equation we see it is equivalent to showing that ,co._,Cz ,.cztz-"--'c -,)c -)...Cz -,')o o and (2.2) follows as an instance of (2.1) with m replaced by h -m and the at's replaced by ct,'s.
We only have two types of polynomials in the first row of the determinant in (4.1).We consider each separately.Let p,,_ (z) be any polynomial of degree at most n 1.
Equivalently, we wish to show that f Zv.(z) p._ (z) z-'.(z) av(O) o and this we get simply by applying (4.1) to the modification of the measure dv(O) by the Laurent polynomial z-'G,(z).Note that having dr(O) rather than dr(O) simply switches the roles of z) and z)in (4.1).
PROOF OF (4.3).We want to show that if ap,,(z) is defined as in equation ( 4. for -0,1,2, m.However, these statements are equivalent to (i) and (ii) in our proof of (4.1).