Some results on biorthogonal polynomials

Richard W Ruedemann

doi:10.1155/s0161171294000906

Abstract

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.

Highlights

Some biorthogonal polynomials of Hahn and Pastro are derived using a polynomial modification of the Lebesgue measure dO 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
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

Summary

A PAIR OF BIORTHOGONAL POLYNOMIAL SETS

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.ei assuming that no ctj is zero and that 0 m -: h. If p._ x(z) is a polynomial of degree at most n 1 we have. Let the polynomial sets {ap,,(z)} and {,(z)} be defined as in the above two lemmas. That for each n, ap.(z) and ,(z) are of precise degree n. (This is equivalent to assuming certain subdeterminants in equations (2.1) and (2.2) are nonzero.) provided that for each n we have,. [z-(z ct,)(z ,)...(z a,)]a0 ,, 0, these polynomial sets are biorthogonal on the unit circle with respect to the measure dv(O)-z-(z-aO(z-ag...(z-a,)dO, z-e’. Where 0 s m s h and no ct is zero

APPLICATION TO THE POLYNOMIALS OF PASTRO

MODIFICATION OF MEASURES BY LAURENT POLYNOMIALS

REMARKS