Orthogonal polynomials for flat-plus-exponential-convolution weights

Determine the monic orthogonal polynomials for the positive weight \(w_B\) that equals the survival function \(S_B(-c_- - t)\) on \((- infty,-c_- )\), equals one on \([-c_-,c_+]\), and vanishes on \((c_+, infty)\); determine their recurrence coefficients and their dependence on \(c_-\), \(c_+\), and the multiset \(B\), characterize the localization and asymptotic distribution of their zeros, establish strong and ratio asymptotics, investigate ladder, Pearson-type, Riemann Hilbert, or spectral relations arising from the piecewise logarithmic derivative of the weight, and analyze limiting transitions as elements of \(B\) tend to zero, coalesce, or diverge under suitable rescaling.

Background

The image theorem shows that Stieltjes-generated moment functionals with no right-hand jumps $A=\emptyset$, a nonempty left-hand jump multiset $B$, and $c_-+c_+>0$ are represented by positive measures whose density is constant on $[-c_-,c_+]$ and whose left tail is the survival function of a sum of independent exponential random variables. This produces the flat-plus-exponential-convolution family $w_B$, including the ordinary exponential tail when $B$ contains one element and hypoexponential-type tails for finite multisets.

The paper establishes positivity and the structural form of these measures but does not derive the associated orthogonal-polynomial systems or their asymptotic theory. The authors note that they have not found a systematic treatment of this particular family and explicitly formulate the listed questions as an open problem.