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A Short Note on Integral Transformations and Conversion Formulas for Sequence Generating Functions

School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30318, USA
Axioms 2019, 8(2), 62; https://doi.org/10.3390/axioms8020062
Received: 23 April 2019 / Revised: 16 May 2019 / Accepted: 17 May 2019 / Published: 19 May 2019
(This article belongs to the Special Issue Mathematical Analysis and Applications II)
The purpose of this note is to provide an expository introduction to some more curious integral formulas and transformations involving generating functions. We seek to generalize these results and integral representations which effectively provide a mechanism for converting between a sequence’s ordinary and exponential generating function (OGF and EGF, respectively) and vice versa. The Laplace transform provides an integral formula for the EGF-to-OGF transformation, where the reverse OGF-to-EGF operation requires more careful integration techniques. We prove two variants of the OGF-to-EGF transformation integrals from the Hankel loop contour for the reciprocal gamma function and from Fourier series expansions of integral representations for the Hadamard product of two generating functions, respectively. We also suggest several generalizations of these integral formulas and provide new examples along the way. View Full-Text
Keywords: generating function; series transformation; gamma function; Hankel contour generating function; series transformation; gamma function; Hankel contour
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Schmidt, M.D. A Short Note on Integral Transformations and Conversion Formulas for Sequence Generating Functions. Axioms 2019, 8, 62.

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