Re: Does This Series Look Familiar?

On Nov 1, 12:37 pm, galathaea <galath...@xxxxxxxxx> wrote:
On Nov 1, 10:31 am, Narcoleptic Insomniac

<i_have_narcoleptic_insom...@xxxxxxxxx> wrote:
My recent musings have lead me to studying the following
general series:

f_{k, m}(z) := \sum_{n = 0}^{oo} (-1)^n z^{nk + m} / (nk + m)!

Does anyone know if these series go by any particular
name?

yes!

this is my generalised trigonometric function!

http://groups.google.com/group/sci.math/msg/5c85912ea563370dhttp://groups.google.com/group/sci.math/msg/c76660860070272ehttp://groups.google.com/group/alt.philosophy/msg/4edc9067b54b6a7c

among others...

i suspect they were known to gauss
but i have yet to see a full exposition outside my own

some properties of the n=4 case were shown by ramanujan
but he apparently never derived they were a part
of the general product and sum properties
i give in the posts above...

if you have any particular questions about these
i have developed many formulae i can share

also
please note i have been looking for references now
since the summer/fall of 2001
and have not found much outside a few specific forms
so if you find any information
i would appreciate them for the paper i have put together on this

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galathaea: prankster, fablist, magician, liar

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