64 research outputs found
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 new rows and columns, so that the original Jacobi matrix
is shifted downward. The new rows and columns contain 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 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
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