Identities Rational trigonometry

1 identities

1.1 explicit formulas
1.2 recursion formula
1.3 relation chebyshev polynomials
1.4 composition
1.5 coefficients in finite fields
1.6 orthogonality
1.7 generating functions
1.8 differential equation

identities
explicit formulas






s

n


(
s
)
=
s

∑

k
=
0


n
−
1




n

n
−
k






(



2
n
−
1
−
k

k


)



(
−
4
s

)

n
−
1
−
k


.


{\displaystyle s_{n}(s)=s\sum _{k=0}^{n-1}{\frac {n}{n-k}}{\binom {2n-1-k}{k}}(-4s)^{n-1-k}.}

(michael hirschhorn, shuxiang goh)





s

n


(
s
)
=



1
2



−



1
4





(
1
−
2
s
+
2



s

2


−
s


)


n


−



1
4





(
1
−
2
s
−
2



s

2


−
s


)


n


.


{\displaystyle s_{n}(s)={\tfrac {1}{2}}-{\tfrac {1}{4}}\left(1-2s+2{\sqrt {s^{2}-s}}\right)^{n}-{\tfrac {1}{4}}\left(1-2s-2{\sqrt {s^{2}-s}}\right)^{n}.}

(m. hovdan)





s

n


(
s
)
=
−



1
4





(


(


1
−
s


+
i


s


)


2
n


−
1
)


2




(


1
−
s


−
i


s


)


2
n


.


{\displaystyle s_{n}(s)=-{\tfrac {1}{4}}\left(\left({\sqrt {1-s}}+i{\sqrt {s}}\right)^{2n}-1\right)^{2}\left({\sqrt {1-s}}-i{\sqrt {s}}\right)^{2n}.}

(m. hovdan)

from definition follows that

s

n


(
s
)
=

sin

2


⁡

(
n
arcsin
⁡

(


s


)

)

.


{\displaystyle s_{n}(s)=\sin ^{2}\left(n\arcsin \left({\sqrt {s}}\right)\right).}



recursion formula






s

n
+
1


(
s
)
=
2
(
1
−
2
s
)

s

n


(
s
)
−

s

n
−
1


(
s
)
+
2
s
.


{\displaystyle s_{n+1}(s)=2(1-2s)s_{n}(s)-s_{n-1}(s)+2s.}



relation chebyshev polynomials

the spread polynomials related chebyshev polynomials of first kind, tn, identity

1
−
2

s

n


(
s
)
=

t

n


(
1
−
2
s
)
.


{\displaystyle 1-2s_{n}(s)=t_{n}(1-2s).}

this implies

s

n


(
s
)
=



1
−

t

n


(
1
−
2
s
)

2


=
1
−

t

n


2



(


1
−
s


)

.


{\displaystyle s_{n}(s)={\frac {1-t_{n}(1-2s)}{2}}=1-t_{n}^{2}\left({\sqrt {1-s}}\right).}

the second equality above follows identity

2

t

n


2


(
x
)
−
1
=

t

2
n


(
x
)


{\displaystyle 2t_{n}^{2}(x)-1=t_{2n}(x)}

on chebyshev polynomials.

composition

the spread polynomials satisfy composition identity

s

n




(



s

m


(
s
)


)


=

s

n
m


(
s
)
.


{\displaystyle s_{n}{\bigl (}s_{m}(s){\bigr )}=s_{nm}(s).}



coefficients in finite fields

when coefficients taken members of finite field fp, sequence {sn}n = 0, 1, 2,... of spread polynomials periodic period p − 1/2. in other words, if k = p − 1/2, sn + k = sn, all n.

orthogonality

when coefficients taken real, n ≠ m, have

∫

0


1



(

s

n


(
s
)
−



1
2



)


(

s

m


(
s
)
−



1
2



)




d
s


s
(
1
−
s
)



=
0.


{\displaystyle \int _{0}^{1}\left(s_{n}(s)-{\tfrac {1}{2}}\right)\left(s_{m}(s)-{\tfrac {1}{2}}\right){\frac {ds}{\sqrt {s(1-s)}}}=0.}

for n = m, integral π/8 unless n = m = 0, in case is π/4.

generating functions

the ordinary generating function is

∑

n
=
1


∞



s

n


(
s
)

x

n


=



s
x
(
1
+
x
)


(
1
−
x

)

3


+
4
s
x
(
1
−
x
)



.


{\displaystyle \sum _{n=1}^{\infty }s_{n}(s)x^{n}={\frac {sx(1+x)}{(1-x)^{3}+4sx(1-x)}}.}

(michael hirschhorn)

the exponential generating function is

∑

n
=
1


∞






s

n


(
s
)


n
!




x

n


=



1
2




e

x



(
1
−

e

−
2
s
x


cos
⁡

(
2
x


s
(
1
−
s
)


)

)

.


{\displaystyle \sum _{n=1}^{\infty }{\frac {s_{n}(s)}{n!}}x^{n}={\tfrac {1}{2}}e^{x}\left(1-e^{-2sx}\cos \left(2x{\sqrt {s(1-s)}}\right)\right).}



differential equation

sn(s) satisfies second-order linear nonhomogeneous differential equation

s
(
1
−
s
)

y
″

+

(



1
2



−
s
)


y
′

+

n

2



(
y
−



1
2



)

=
0.


{\displaystyle s(1-s)y +\left({\tfrac {1}{2}}-s\right)y +n^{2}\left(y-{\tfrac {1}{2}}\right)=0.}






^ cite error: named reference wildberger_2005 invoked never defined (see page).