rabbit redux (multisection, modular forms, and lambert series)


for the definition of rabbits see my previous posts
and particularly http://groups.google.com/group/sci.math/msg/1add7ab71dda58c2?hl=en

let me start with one of the simplest appearances of rabbits

look at the function

oo
--- j
\ x
ln(1 - x) = / --
--- j
j=1

multisecting this function
and using the classic wallis/simpson/fourier form (any other names needed here?)
gives

n-1
---
|m 1 \ -mj j
| ln(1 - x) = - / w ln(1 - w x)
|n n --- n n
j=0

-mj
n-1 w
1 / --- j n \
= - ln | | | (1 - w x) |
n \ | | n /
j=0

or the logarithm of a very special rabbit!

by itself it's a very pretty result
but it turns out that it is central to a number of very interesting formulae
relating theta and elliptic functions
with lambert series and other important number theoretic constructions

of course
it all starts with a ramanujan reference
and obviously it will include the number 23

in particular
chapter 16 entry 23 of ramanujan's notebooks
(as presented by berndt)
derives some basic lambert series representations
for the logarithms of some classical theta functions

the approach generalises easily to a beautiful little creature:
theta-rabbits

define

-mj
n-1 w
--- j n
R (x) = | | (w q; q)
m n | | n oo
j=0

where the terms are the classical q-pochhammer (a; q)
k
with the standard lim k -> oo

this family of functions includes several classical forms including

R (x) is euler's variant of the dedekind eta modular form
0 1

R (x) is jacobi's theta-3 (the coolest of the thetas)
1 2

now

oo -mk
--- n-1 w
/ \ \ / --- k j n \
ln | R (x) | = / ln | | | (1 - w q ) |
\ m n / --- \ | | n /
j=1 k=0

oo oo
--- --- * j nk+m
\ \ (q )
= n / / --------
--- --- nk + m
j=1 k=0

oo oo
--- * ---
\ 1 \ nk+m j
= n / ------ / (q )
--- nk + m ---
k=0 j=1

oo
--- * nk+m
\ q
= n / -------------------
--- nk+m
k=0 (nk + m)(1 - q )


a multisected lambert series!

(the asterisk summation means to exclude k=0 if m=0)

the standard relationship with powerseries
brings in the arithmetical function

---
\ 1
sigma (j) = / -
m -1 --- d
n d|j
d=m(mod n)

as the jth term of the expansion

these functions R (x) generalise many of the properties of the classical modular forms
m n
in much the same way as the generalised trigonometry does the classical cos and sin
due the way that multisections
(even here the lambert multisection)
extend product relationships from the base form

and particularly here
several interesting arithmetical results are immediately derivable

the rabbit being the natural algebraic generalisation of fractions
and arising naturally in the iteration
of the generalised polynomials associated with the generalised trigonometrics
now shows it's deep arithmetical form

beware the rabbit's teeth!

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
.

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