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When a measure Ψ (x) on the real line is subjected to the modification d Ψ (t) (x) = e − t x d Ψ (x) , then the coefficients of the recurrence relation of the orthogonal polynomials in x with respect to the measure Ψ (t) (x) are known to satisfy the so-called Toda lattice formulas as functions of t . In this paper we consider a modification of the form e − t (p x + q / x) of measures or, more generally, of moment functionals, associated with orthogonal L-polynomials and show that the coefficients of the recurrence relation of these L-orthogonal polynomials satisfy what we call an extended relativistic Toda lattice. Most importantly, we also establish the so called Lax pair representation associated with this extended relativistic Toda lattice. These results also cover the (ordinary) relativistic Toda lattice formulations considered in the literature by assuming either p = 0 or q = 0 . However, as far as Lax pair representation is concern, no complete Lax pair representations were established before for the respective relativistic Toda lattice formulations. Some explicit examples of extended relativistic Toda lattice and Langmuir lattice are also presented. As further results, the lattice formulas that follow from the three term recurrence relations associated with kernel polynomials on the unit circle are also established. [ABSTRACT FROM AUTHOR]

It was shown recently that associated with a pair of real sequences {{cn}n=1∞,{dn}n=1∞}, with {dn}n=1∞ a positive chain sequence, there exists a unique nontrivial probability measure μ on the unit circle. The Verblunsky coefficients {αn}n=0∞ associated with the orthogonal polynomials with respect to μ are given by the relation αn-1=τ¯n-11-2mn-icn1-icn,n≥1,where τ0=1, τn=∏k=1n(1-ick)/(1+ick), n≥1 and {mn}n=0∞ is the minimal parameter sequence of {dn}n=1∞. In this manuscript, we consider this relation and its consequences by imposing some restrictions of sign and periodicity on the sequences {cn}n=1∞ and {mn}n=1∞. When the sequence {cn}n=1∞ is of alternating sign, we use information about the zeros of associated para-orthogonal polynomials to show that there is a gap in the support of the measure in the neighbourhood of z=-1. Furthermore, we show that it is possible to generate periodic Verblunsky coefficients by choosing periodic sequences {cn}n=1∞ and {mn}n=1∞ with the additional restriction c2n=-c2n-1,n≥1. We also give some results on periodic Verblunsky coefficients from the point of view of positive chain sequences. An example is provided to illustrate the results obtained. [ABSTRACT FROM AUTHOR]

We consider nontrivial probability measures, obtained as simple modifications of the Lebesgue measure, which include mass points at z = 1 and z = i . The orthogonal polynomials, para-orthogonal polynomials and Toeplitz matrices associated with these measures are presented. [ABSTRACT FROM AUTHOR]