Higher Derivatives of Airy Functions and of their Products

TL;DR

This paper derives explicit formulas for higher derivatives of Airy functions and their products, expressing them through Gegenbauer polynomials and hypergeometric series, advancing analytical methods for special functions.

Contribution

It provides new closed-form expressions for derivatives and products of Airy functions using special polynomials and hypergeometric series, which were not previously available.

Findings

Explicit formulas for higher derivatives of Airy functions.

Representation of derivatives in terms of Gegenbauer polynomials.

Solutions for products of Airy functions using hypergeometric series.

Abstract

The problem of evaluation of higher derivatives of Airy functions in a closed form is investigated. General expressions for the polynomials which have arisen in explicit formulae for these derivatives are given in terms of particular values of Gegenbauer polynomials. Similar problem for products of Airy functions is solved in terms of terminating hypergeometric series.