A Heterosexual Mehler Fromula for the Straight Hermite Polynomials (a la Foata)

by Doron Zeilberger

***Exclusive to the Personal Journal of Zeilberger and Ekhad and the xxx archives***

Written: July 13, 1998.

Refereed by Dominique Foata: this paper was carefully read by him and its formal correctness approved.

In 1978, Dominique Foata revolutionized both Special Functions and Combinatorics in his BOOK-quality proof of the Mehler formula. That this approach is SO NATURAL can be proved that, by ONLY KNOWING THE IDEAS, I was fully able, in a few minutes, to completely reconstruct it from scratch, as well as the rather complicated formula, during a plane ride to Madrid, as I was preparing my talks to be delivered in IWOP '98. I had no references and no computer (besides the one between my shoulders).

My altruistic deed of giving an invited lecture without even once mentioning my own work was rewarded by finding this new, even more Foataesque proof, and even more elegant Mehler-type formula.

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Note Added Sept. 9, 1998: Dick Askey pointed out that what I call `Straight Hermite polynomials' are really so-called Charlier polynomials (with appropriate parameters), and advised me to consult Mizan Rahman whether the main result is new. Rahman pointed out that my `new' polynomials are really Laguerre polynomials in disguise, and found a half-page derivation of the main formula, using `known' formulas from handbooks. Mizan concludes: `So the question you asked: is it new? Yes and no. These formulas are not listed in these exact forms anywhere, but they are contained in other formulas. But, as combinatorially derived formulas they are very interesting, because you have been able to give life to what have so far been lying there as practically dead formulas. In this sense they are new.'

Note added Oct. 30, 1998: Bengt Nagel came up with a Very Interesting Generalization

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