A reciprocal-Gamma Mehler--Heine limit for Krawtchouk--Sobolev type orthogonal polynomials with an exterior mass point
Abstract
We study the monic polynomials Sobn orthogonal with respect to a Sobolev-type modification of the discrete Krawtchouk inner product in which a single point mass at an exterior point α<0, outside the support {0,1,…,N} of the binomial measure, acts through the j-th forward difference difj (here Nzero={0,1,2,…}, Z+={1,2,…}, R+=(0,∞)). In the joint scaling n,N→∞, n/N→ r with 0<p<r<1 (the \emph{reciprocal-Gamma} regime of Dominici's classification), we prove the Mehler--Heine formula. The exterior mass survives the large-degree limit exactly through the linear factor z−α, with the explicit universal constant C∗, and the limit is independent of both λ>0 and j≥1; the same conclusion holds in the Uvarov case j=0 (Corollary~\ref{cor:uvarov}). The derivation is pointwise and self-contained: it uses only a rank-one connection formula (Theorem~\ref{thm:connection}), the reproducing-kernel recursion, Dominici's pointwise limit and the classical three-term recurrence; the normalised Sobolev correction obeys an exact affine recursion at fixed N, controlled by a triangular contraction argument over two backward windows with an explicit boundary estimate, and the contraction ratio ϱ<1 is exactly the condition r>p. We also discuss the scale of the theorem: in this normalisation the classical term is asymptotically negligible, and the independence of λ concerns each fixed λ>0, not the continuity at λ=0. Multiprecision computations illustrate the convergence and the exact value C∗=2.8 for p=0.3, r=3/5, α=−2.5, j=2, λ=5.
Related
- Sobolev Orthogonal Polynomials on the unit circle with varying inner product
- Bivariate Affine q-Krawtchouk Polynomials and Association Schemes over Galois Rings
- Necessary and sufficient condition for reciprocal polynomials to be monogenic
- Galois groups of twisted reciprocal polynomials: a uniform asymptotic
- Matrix orthogonal polynomials related to diffference operators for L-orthogonal polynomials
BibTeX
@misc{lorente2026reciprocalgamma,
title = {{A reciprocal-Gamma Mehler--Heine limit for Krawtchouk--Sobolev type orthogonal polynomials with an exterior mass point}},
author = {Anier Soria Lorente and Junior Michel},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25060},
eprint = {2609.25060},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25060},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A reciprocal-Gamma Mehler--Heine limit for Krawtchouk--Sobolev type orthogonal polynomials with an exterior mass point
AU - Anier Soria Lorente
AU - Junior Michel
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.25060
UR - https://arxiv.org/abs/2609.25060
AB - We study the monic polynomials Sobn orthogonal with respect to a Sobolev-type modification of the discrete Krawtchouk inner product in which a single point mass at an exterior point α<0, outside the support {0,1,…,N} of the binomial measure, acts through the j-th forward difference dif^j (here Nzero={0,1,2,…}, Z_+={1,2,…}, R_+=(0,∞)). In the joint scaling n,N→∞, n/N→ r with 0<p<r<1 (the \emph{reciprocal-Gamma} regime of Dominici's classification), we prove the Mehler--Heine formula. The exterior mass survives the large-degree limit exactly through the linear factor z−α, with the explicit universal constant C_∗, and the limit is independent of both λ>0 and j≥1; the same conclusion holds in the Uvarov case j=0 (Corollary~\ref{cor:uvarov}). The derivation is pointwise and self-contained: it uses only a rank-one connection formula (Theorem~\ref{thm:connection}), the reproducing-kernel recursion, Dominici's pointwise limit and the classical three-term recurrence; the normalised Sobolev correction obeys an exact affine recursion at fixed N, controlled by a triangular contraction argument over two backward windows with an explicit boundary estimate, and the contraction ratio ϱ<1 is exactly the condition r>p. We also discuss the scale of the theorem: in this normalisation the classical term is asymptotically negligible, and the independence of λ concerns each fixed λ>0, not the continuity at λ=0. Multiprecision computations illustrate the convergence and the exact value C_∗=2.8 for p=0.3, r=3/5, α=−2.5, j=2, λ=5.
ER -