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

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

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—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 .

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Contributed by: S. M. Blinder (March 2011)
Open content licensed under CC BY-NC-SA


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References:

S. M. Blinder, Chapter 12, Guide to Essential Math, Amsterdam: Elsevier Academic Press, 2008.

H. Wilf, Generatingfunctionology, 2nd ed., Boston: Academic Press, 1994.



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