functional translations and coefficient extraction (exponential sum decompositions)


one of the fundamental relationships of differentials
is the functional relation between exponentiation and translation

if f(D) is a polynomial in the differential operator
then
c x c x
f(D) e g(x) = e f(D + c) g(x)

now
if the limit of both sides is taken x -> 0
then through application of

lim f(x) = f
x=0 0

(where f_0 is the constant term of a power series expansion)

/ n \
lim | D f(x) | = f
\ /x=0 n

when f is expanded on an exponential basis
ie.
f
2 2
f(x) = f + f x + -- x + ...
0 1 2

now we can use this to extract coefficient relations
for the generalised berneulers that illustrates
the differential structure here

if one looks at

1
f(D) 1 = f(D) g(x) ----
g(x)

with an eye for applying the above
then an exponential sum

n-1
--- b x
\ j
g(x) = / g e
--- j
j=0

can be used
and then coefficients of the inverse can be extracted
using

n-1
---
\ / 1 \
lim (f(D) 1) = / g lim | f(D + b ) ---- |
x=0 --- j x=0 \ j g(x) /
j=0

which is precisely what is needed for the generalised berneulers

here
for instance
we may define

oo gb
--- 0
1 \ n j j
------ = / ---- x
g (x) --- (1)
0 n j=0 j

where the coefficients are nonzero when j equiv= 0 (mod n)

since

n-1 j
--- w x
1 \ n
g (x) = - / e
0 n n ---
j=0

we get


n-1
---
1 \ / j 1 \
lim (f(D) 1) = - / lim | f(D + w ) ---- |
x=0 n --- x=0 \ n g(x) /
j=0

to make this useful for proving simple recursive relations
some useful criteria are
- make f(D) 1 nonzero and normalized to 1
e.g. f(D) has a constant term 1 since D 1 = 0
- f(D + w_n^j) should have a zero constant term
to reduce the size of the recursion
so choose f the polynomial with zeroes w_n^j
iow
n
f(D) = 1 - D

then
n
--- (1)
j \ n n-k jk
f(D + w ) = 1 - / ---------- D w
n --- (1) (1) n
k=0 k n-k
n-1
--- (1)
\ n n-k jk
= - / ---------- D w
--- (1) (1) n
k=0 k n-k


or

n-1 n-1
--- --- (1)
/ 1 \ / \ n n-k jk \ 1 \
1 = lim | - / | - / ---------- D w | ------ |
x=0 \ n --- \ --- (1) (1) n / g (x) /
j=0 k=0 k n-k 0 n

noticing here that

if k =/= 0 (mod n)

then the sum of 1 + w_n^k + w_n^(2k) + ... + w_n^((n-1)k) is 0
then only those terms k equiv= 0 (mod n)
which are left after the others annihilate by symmetry
and each of these contribute n identical copies

/ n 1 \
-1 = lim | D ------ | = gb
x=0 \ g (x) / 0 n
0 n n

this is well known for the n = 2 case
(the classical euler numbers)

where E = -1
2

and shows all generalised eulers have a similar element with that value

we can extract even more relations
by varying the f(D) polynomial

the properties that keep the f(D) with efficient terms above
can be kept by simply raising it to greater powers

n r
the functions p (D) = (1 - D )
r
give higher coefficient relations of the same form

using the nice sum form above for (1 - (D + w_n^j)^n)
the rth power may be extracted using the multinomial theorem

n-1 n-1
--- --- / r \ --- / / n \ n-k jk \ q_k
/ 1 \ \ | | | | | - | | D w | 1 \
lim | - / / \ q0, q1, ..., q_(n-1) / | | \ \ k / n / ------ |
x=0 \ n --- --- k=0 g (x) /
j=0 0 <= q0, q1, ..., q_(n-1) <= r 0 n
q0 + q1 + .. + q_(n-1) = r

the products over k can be simplified
using the sum-to-r property

n-1
---
\
define Q = / k q
--- k
k=0

the

n-1 n-1
--- --- / r \ nr-Q jQ --- / n \ q_k
/ 1 \ \ | | D w | | | | 1 \ r
lim | - / / \ q0, q1, ..., q_(n-1) / n | | \ k / ------ | = (-1)
x=0 \ n --- --- k=0 g (x) /
j=0 0 <= q0, q1, ..., q_(n-1) <= r 0 n
q0 + q1 + .. + q_(n-1) = r

again
the sum over j can be pushed inside
and this time only the terms Q equiv= 0 (mod n) contribute

so
taking limits
this becomes the generalised berneuler coefficient recursions

n-1 r
--- / r \ --- 1 (-1)
\ | | gb | | ------------- = -----
/ \ q0, q1, ..., q_(n-1) / 0 nr-Q | | q_k q_k r
--- n k=0 (1) (1) (1)
0 <= q0, q1, ..., q_(n-1) <= r k n-k n
q0 + q1 + .. + q_(n-1) = r
Q equiv= 0 (mod n)

now
Q = n is the classic selector for partitions of an integer
and selecting the the sum of the q to r
isolates those partitions of r parts

for the other Q (0, 2n, 3n, ...) available
there are similar partition-theoretic interpretations

the case n=2 gives a classical efficient recursions for euler numbers

using similar techniques
relations for coefficients of other inverses of exponential sums may be determined
not only for the other generalised trigonometrics
(giving relations for the rest of the generalised berneulers)
but also for sums of exponentiated roots of other polynomials

similarly
this can be applied to fourier series
allowing the extraction of power series coefficients of an inverse
in terms of the fourier coefficients of the original series

also notice that this can be used on generalised trigonometric polynomials
on which tchebyshef relations may be derived
so the technique provides a powerful language for deriving many important relations

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galathaea: prankster, fablist, magician, liar
.

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