Orthogonal polynomials

Physics

Mathematical analysis

Algebra over a field

Exponential function

Hermite polynomials

Pure mathematics

Classical orthogonal polynomials

Mathematics

Wilson polynomials

Discrete orthogonal polynomials

Quantum mechanics

Euler's formula

Degenerate energy levels

Exponential polynomial

Difference polynomials

Polynomial

Advances in Applied Mathematics

The study on degenerate versions of some special numbers and polynomials, which began with Carlitz's pioneering work, has regained recent interests of some mathematicians. Motivated by this, we introduce degenerate Hermite polynomials as a degenerate version of the ordinary Hermite polynomials. Recently, introduced was λ-umbral calculus where the usual exponential function appearing in the generating function of Sheffer sequence is replaced by the degenerate exponential function. Then, among other things, by using the formula of the λ-umbral calculus about expressing one λ-Sheffer polynomial in terms of another λ-Sheffer polynomials we represent the degenerate Hermite polynomials in terms of the higher-order degenerate Bernoulli, Euler, and Frobenius-Euler polynomials and vice versa.

Representations of degenerate Hermite polynomials