402 research outputs found
Umbral Vade Mecum
In recent years the umbral calculus has emerged from the shadows to provide
an elegant correspondence framework that automatically gives systematic
solutions of ubiquitous difference equations --- discretized versions of the
differential cornerstones appearing in most areas of physics and engineering
--- as maps of well-known continuous functions. This correspondence deftly
sidesteps the use of more traditional methods to solve these difference
equations. The umbral framework is discussed and illustrated here, with special
attention given to umbral counterparts of the Airy, Kummer, and Whittaker
equations, and to umbral maps of solitons for the Sine-Gordon, Korteweg--de
Vries, and Toda systems.Comment: arXiv admin note: text overlap with arXiv:0710.230
On hypergeometric series reductions from integral representations, the Kampe de Feriet function, and elsewhere
Single variable hypergeometric functions pFq arise in connection with the
power series solution of the Schrodinger equation or in the summation of
perturbation expansions in quantum mechanics. For these applications, it is of
interest to obtain analytic expressions, and we present the reduction of a
number of cases of pFp and p+1F_p, mainly for p=2 and p=3. These and related
series have additional applications in quantum and statistical physics and
chemistry.Comment: 17 pages, no figure
Mellin transforms with only critical zeros: generalized Hermite functions
We consider the Mellin transforms of certain generalized Hermite functions
based upon certain generalized Hermite polynomials, characterized by a
parameter . We show that the transforms have polynomial factors whose
zeros lie all on the critical line. The polynomials with zeros only on the
critical line are identified in terms of certain hypergeometric
functions, being certain scaled and shifted Meixner-Pollaczek polynomials.
Other results of special function theory are presented.Comment: 17 pages, no figure
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