Generating Functions and Rodrigues's Formulas for Special Functions Used in Quantum Mechanics

A generating function is a clothesline on which we hang up a sequence of numbers for display.

—Herbert Wilf

This Demonstration shows generating functions for several special functions of integer order that occur in elementary quantum mechanics. A generating function is a power series in a formal sense, which need not be convergent. Also given are alternative representations of special functions, Rodrigues's formulas, based on multiple derivatives. By selecting the integer index (and , if applicable), you can obtain explicit forms for these special functions. Generating functions are useful in quantum-mechanical computations, particularly for finding general formulas for matrix elements such as .