Extended Relativistic Toda Lattice, L-Orthogonal Polynomials and Associated Lax Pair

When a measure $\varPsi(x)$ on the real line is subjected to the modification $d\varPsi^{(t)}(x) = e^{-tx} d \varPsi(x)$ , then the coefficients of the recurrence relation of the orthogonal polynomials in $x$ with respect to the measure $\varPsi^{(t)}(x)$ are known to satisfy the so-called Toda lattice formulas as functions of $t$ . In this paper we consider a modification of the form $e^{-t(\mathfrak{p}x+ \mathfrak{q}/x)}$ 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 $\mathfrak{p}=0$ or $\mathfrak{q}=0$ . 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.