  We consider the orthogonal polynomials on $[-1,1]$ with respect to the weight
$$ w_c(x)=h(x)(1-x)^{\alpha}(1+x)^{\beta} \Xi_{c}(x), \quad \alpha, \beta >-1,
$$ where $h$ is real analytic and strictly positive on $[-1, 1]$, and $\Xi_{c}$
is a step-like function: $\Xi_{c}(x)=1$ for $x\in [-1, 0)$ and
$\Xi_{c}(x)=c^2$, $c>0$, for $x\in [0, 1]$. We obtain strong uniform
asymptotics of the monic orthogonal polynomials in $\mathbb{C}$, as well as
first terms of the asymptotic expansion of the main parameters (leading
coefficients of the orthonormal polynomials and the recurrence coefficients) as
$n\to \infty$. In particular, we prove for $w_c$ a conjecture of A. Magnus
regarding the asymptotics of the recurrence coefficients. The main focus is on
the local analysis at the origin. We study the asymptotics of the
Christoffel-Darboux kernel in a neighborhood of the jump and show that the
zeros of the orthogonal polynomials no longer exhibit the clock behavior.
  For the asymptotic analysis we use the steepest descendent method of Deift
and Zhou applied to the non-commutative Riemann-Hilbert problems characterizing
the orthogonal polynomials. The local analysis at $x=0$ is carried out in terms
of the confluent hypergeometric functions. Incidentally, we establish some
properties of these functions that may have an independent interest.
