A Basic Analogue of Hermite's Equation

Using basic number and the analogues of differentiation andintegration, a q-analogue of Hermite's equation is introduced.Series solutions are given, and it is shown that polynomialforms of these solutions are orthogonal with respect to basicintegration. By reversing the series representation of thesesolutions, a basic analogue of the Hermite polynomial is obtainedfor which a generating function and a three-term recurrencerelation are deduced. Finally, an orthogonality relation isgiven.     

The Laplace Transform of the Macdonald Function of Argument x3/2 and the Airy Function Ai (x)

The Laplace transform of the Airy function of the first kindAi(x) is obtained conveniently as a special case of the Laplacetransform of the Macdonald function of argument 2x^3/2.