Explicit merit factor formulae for Fekete and Turyn polynomials
Authors: Borwein Peter , Choi Kwok-Kwong 
Published: July 2002

We give explicit formulas for the

L

4

L_{4}

norm (or equivalently for the merit factors) of various sequences of polynomials related to the Fekete polynomials
\[

f

q

(
z
)
:=

∑

k
=
1

q
−
1

(

k
q

)

z

k

f_{q}(z) := \sum ^{q-1}_{k=1} \left (\frac {k}{q}\right ) z^{k}

\]
where

(

⋅
q

)

\left (\frac {\cdot }{q}\right )

is the Legendre symbol. For example for

q
q

an odd prime,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar f Subscript q Baseline double-vertical-bar Subscript 4 Superscript 4 Baseline colon equals StartFraction 5 q squared Over 3 EndFraction minus 3 q plus four thirds minus 12 left-parenthesis h left-parenthesis negative q right-parenthesis right-parenthesis squared">

‖

f

q

‖

4

4

:=

5

q

2

3

−
3
q
+

4
3

−
12
(
h
(
−
q
)

)

2

\|f_{q}\|_{4}^{4} : = \frac {5q^{2}}{3}-3q+ \frac {4}{3} - 12 (h(-q))^{2}

\]
where

h
(
−
q
)

h(-q)

is the class number of

Q

(

−
q

)

\mathbb {Q}(\sqrt {-q})

. Similar explicit formulas are given for various polynomials including an example of Turyn’s that is constructed by cyclically permuting the first quarter of the coefficients of

f

q

f_{q}

. This is the sequence that has the largest known asymptotic merit factor. Explicitly,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Subscript q Baseline left-parenthesis z right-parenthesis colon equals sigma-summation Underscript k equals 0 Overscript q minus 1 Endscripts left-parenthesis StartFraction k plus left-bracket q slash 4 right-bracket Over q EndFraction right-parenthesis z Superscript k">

R

q

(
z
)
:=

∑

k
=
0

q
−
1

(

k
+
[
q

/

4
]

q

)

z

k

R_{q}(z) := \sum ^{q-1}_{k=0} \left (\frac {k+[q/4] }{q}\right ) z^{k}

\]
where

[
⋅
]

[\cdot ]

denotes the nearest integer, satisfies
\[

‖

R

q

‖

4

4

=

7

q

2

6

−

q

−

1
6

−

γ

q

\|R_{q}\|_{4}^{4} = \frac {7q^{2}}{6}- {q} - \frac {1}{6} - \gamma _{q}

\]
where
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma Subscript q Baseline colon equals StartLayout Enlarged left-brace 1st Row 1st Column h left-parenthesis negative q right-parenthesis left-parenthesis h left-parenthesis negative q right-parenthesis minus 4 right-parenthesis 2nd Column a m p semicolon if q identical-to 1 comma 5 left-parenthesis mod 8 right-parenthesis comma 2nd Row 1st Column 12 left-parenthesis h left-parenthesis negative q right-parenthesis right-parenthesis squared 2nd Column a m p semicolon if q identical-to 3 left-parenthesis mod 8 right-parenthesis comma 3rd Row 1st Column 0 2nd Column a m p semicolon if q identical-to 7 left-parenthesis mod 8 right-parenthesis period EndLayout">

γ

q

:=

{

h
(
−
q
)
(
h
(
−
q
)
−
4
)

a
m
p
;

if 

q
≡
1
,
5

(
mod

8
)

,

12
(
h
(
−
q
)

)

2

a
m
p
;

if 

q
≡
3

(
mod

8
)

,

0

a
m
p
;

if 

q
≡
7

(
mod

8
)

.

\gamma _{q}: = \begin {cases} h(-q) (h(-q)-4) & \text {if $q \equiv 1,5 \pmod 8$},\\ 12 (h(-q))^{2} & \text {if $q \equiv 3 \pmod 8$}, \\ 0 & \text {if $q \equiv 7 \pmod 8$}. \end {cases}

\]
Indeed we derive a closed form for the

L

4

L_{4}

norm of all shifted Fekete polynomials
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript q Superscript t Baseline left-parenthesis z right-parenthesis colon equals sigma-summation Underscript k equals 0 Overscript q minus 1 Endscripts left-parenthesis StartFraction k plus t Over q EndFraction right-parenthesis z Superscript k Baseline period">

f

q

t

(
z
)
:=

∑

k
=
0

q
−
1

(

k
+
t

q

)

z

k

.

f_{q}^{t}(z) := \sum ^{q-1}_{k=0} \left (\frac {k+t}{q}\right ) z^{k}.

\]
Namely
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column double-vertical-bar f Subscript q Superscript t Baseline double-vertical-bar Subscript 4 Superscript 4 2nd Column a m p semicolon equals one third left-parenthesis 5 q squared plus 3 q plus 4 right-parenthesis plus 8 t squared minus 4 q t minus 8 t 3rd Column a m p semicolon minus StartFraction 8 Over q squared EndFraction left-parenthesis 1 minus one half left-parenthesis StartFraction negative 1 Over q EndFraction right-parenthesis right-parenthesis StartAbsoluteValue sigma-summation Underscript n equals 1 Overscript q minus 1 Endscripts n left-parenthesis StartFraction n plus t Over q EndFraction right-parenthesis EndAbsoluteValue squared comma EndLayout">

‖

f

q

t

‖

4

4

a
m
p
;
=

1
3

(
5

q

2

+
3
q
+
4
)
+
8

t

2

−
4
q
t
−
8
t

a
m
p
;

−

8

q

2

(
1
−

1
2

(

−
1

q

)

)

|

∑

n
=
1

q
−
1

n

(

n
+
t

q

)

|

2

,

\begin{align*} \| f_{q}^{t} \|_{4}^{4} &= \frac {1}{3}(5q^{2}+3q+4)+8t^{2}-4qt-8t &\quad -\frac {8}{q^{2}}\left ( 1-\frac {1}{2} \left (\frac {-1}{q}\right ) \right ) \left |{\displaystyle \sum _{n=1}^{q-1}n \left (\frac {n+t}{q}\right )} \right |^{2}, \end{align*}

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar f Subscript q Superscript q minus t plus 1 Baseline double-vertical-bar Subscript 4 Superscript 4 Baseline equals double-vertical-bar f Subscript q Superscript t Baseline double-vertical-bar Subscript 4 Superscript 4">

‖

f

q

q
−
t
+
1

‖

4

4

=
‖

f

q

t

‖

4

4

\| f_{q}^{q-t+1} \|_{4}^{4}= \| f_{q}^{t} \|_{4}^{4}

if

1
≤
t
≤
(
q
+
1
)

/

2

1 \le t \le (q+1)/2

.