Upward extension of the Jacobi matrix for orthogonal polynomials

Orthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We investigate new orthogonal polynomials by adding to the Jacobi matrix $r$ new rows and columns, so that the original Jacobi matrix is shifted downward. The $r$ new rows and columns contain $2r$ new parameters and the newly obtained orthogonal polynomials thus correspond to an upward extension of the Jacobi matrix. We give an explicit expression of the new orthogonal polynomials in terms of the original orthogonal polynomials, their associated polynomials and the $2r$ new parameters, and we give a fourth order differential equation for these new polynomials when the original orthogonal polynomials are classical. Furthermore we show how the orthogonalizing measure for these new orthogonal polynomials can be obtained and work out the details for a one-parameter family of Jacobi polynomials for which the associated polynomials are again Jacobi polynomials

Recurrence relations for connection coefficients between q-orthogonal polynomials of discrete variables in the non-uniform lattice X(s) = q2s

We obtain the structure relations for q-orthogonal polynomials in the exponential lattice q 2s and from that we construct the recurrence relation for the connection coe cients between two families of polynomials belonging to the classical class of discrete q-orthogonal polynomials. An explicit example is also given.Comisión Interministerial de Ciencia y Tecnologí

Stieltjes sums for zeros of orthogonal polynomials

AbstractStieltjes considered sums of reciprocals of differences of zeros of a solution of a homogeneous second-order linear differential equation. Here we re-examine the derivation of these sums with a view to extending the class of differential equations to which the theory applies and including sums involving the zeros of the derivatives as well as those of the polynomials themselves

Landau-Zener-St\"uckelberg interferometry in pair production from counterpropagating lasers

The rate of electron-positron pair production in linearly polarized counter-propagating lasers is evaluated from a recently discovered solution of the time-dependent Dirac equation. The latter is solved in momentum space where it is formally equivalent to the Schr\"odinger equation describing a strongly driven two-level system. The solution is found from a simple transformation of the Dirac equation and is given in compact form in terms of the doubly-confluent Heun's function. By using the analogy with the two-level system, it is shown that for high-intensity lasers, pair production occurs through periodic non-adiabatic transitions when the adiabatic energy gap is minimal. These transitions give rise to an intricate interference pattern in the pair spectrum, reminiscent of the Landau-Zener-St\"uckelberg phenomenon in molecular physics: the accumulated phase result in constructive or destructive interference. The adiabatic-impulse model is used to study this phenomenon and shows an excellent agreement with the exact result.Comment: 22 pages, 7 figure

Orthogonal polynomials-centroid of their zeroes

Let c (n,k) (k=1,...,n) the n zeroes of the monic orthogonal polynomials family P (n) (x). The centroid of these zeroes: controls globally the distribution of the zeroes, and it is relatively easy to obtain information on s (n) , like bounds, inequalities, parameters dependence, ..., from the links between s (n) , the coefficients of the expansion of P (n) (x), and the coefficients beta (n) , gamma (n) in the basic recurrence relation satisfied by P (n) (x). After a review of basic properties of the centroid on polynomials, this work gives some results on the centroid of a large class of orthogonal polynomials