Sequences related to the differential equation f''= af'f

Stanislav Sykora,

Extra Byte, Via R.Sanzio 22C, 20022 Castano Primo (MI), Italy,
in Stan's Library, Vol.VI. First release 22 June 2017

Permalink:  DOI: 10.3247/SL6Math17.001

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This Note explores a family of sequences related through their exponential generating functions (e.g.f.) to smooth solutions of the nonlinear differential equation f’’(z) = af’(z)f(z) in C. The implied recurrence for the complex expansion coefficients of the solutions leads to a family of sequences uniquely labeled by two complex parameters. Excluding a subset of ‘singular’ cases, each of these sequences has an e.g.f. which can be written as a multiple of the tangent function of a linear form of its complex argument. There exists also a closely related family of singular cases which lead instead to a simple inverse linear form.

A subset of this family contains sequences with exclusively integer elements, one for every pair of integer numbers. Since many of these integer sequences appear on OEIS, often without an explicit recognition of the close relationship between them, it is hoped that the present Note might constitute a useful unifying link and a useful classification. Another interesting subset is that of sequences with complex coefficients which have integer real an imaginary parts. These appear to be novel so far, and a couple of examples are listed.

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Please, cite this online document as:
Sykora S., Sequences related to the differential equation f''= af'f,
Stan's Library, Vol.VI, June 2017, DOI: 10.3247/SL6Math17.001 .


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Copyright ©2017 Stan's Library, ISSN 2421-1230, DOI: 10.3247/SL6Math17.001 Designed by Stan Sýkora