The Marvelous Meixner Polynomials: Second Kind, First Class

Abstract

Meixner's classification asks which exponential generating functions can generate orthogonal polynomials. We follow the continuous Meixner family from this question to its moments, measures, and continued fractions. A beta integral and a bilinear identity establish positivity and orthogonality; Carlitz's Bernoulli and Euler constructions reveal further connections. Turning the parameter into a variable leads to rational orthogonality and Wall's fraction. We derive its Nevanlinna measure and unite the two spectral directions in a joint law with Cauchy marginals. Even finite pencils produce couplings with these exact marginals. Laguerre limits explain the gamma law governing joint extremes. Along the way, Meixner's scientific life, combinatorics, tangent numbers, and even zeta values illuminate the same structure.