PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 136, Number 1, January 2008, Pages 229–237
S 0002-9939(07)09085-5
Article electronically published on October 18, 2007
PARTIAL FRACTION DECOMPOSITIONS
AND TRIGONOMETRIC SUM IDENTITIES
WENCHANG CHU
(Communicated by Carmen C. Chicone)
Abstract. The partial fraction decomposition method is explored to establish
several interesting trigonometric function identities, which may have applications to the evaluation of classical multiple hypergeometric series, trigonometric approximation and interpolation.
1. Outline and introduction
Recently, in an attempt to prove, through the Cauchy residue method, Dougall’s
theorem (Dougall [6, 1907], see [5] also) on the well-poised bilateral hypergeometric
5 H5 -series, the author found the following trigonometric sum identity:
(1a)
sin a
sin b sin c sin d sin e
=
(1b)
+
(1c)
+
(1d)
+
cot b sin(a − 2b)
sin(c − b) sin(d − b) sin(e − b)
cot c sin(a − 2c)
sin(b − c) sin(d − c) sin(e − c)
cot d sin(a − 2d)
sin(b − d) sin(c − d) sin(e − d)
cot e sin(a − 2e)
.
sin(b − e) sin(c − e) sin(d − e)
Consider the trigonometric fraction defined by
R(z) :=
eiz cot z sin(a + 2z)
.
sin(b + z) sin(c + z) sin(d + z) sin(e + z)
According to the Euler formulae
cos z =
eiz + e−iz
2
and
sin z =
eiz − e−iz
,
2i
this rational function is essentially a fraction in eiz with the degree of the numerator
polynomial less than that of the denominator polynomial by one. Therefore we can
Received by the editors October 25, 2006.
2000 Mathematics Subject Classification. Primary 42A15; Secondary 65T40.
Key words and phrases. Trigonometric interpolation, trigonometric formulae, partial fraction
decomposition.
c
2007
American Mathematical Society
229
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230
WENCHANG CHU
decompose it into the following partial fractions:
(2)
R(z) =
B
C
D
E
A
+
+
+
+
sin z
sin(b + z) sin(c + z) sin(d + z) sin(e + z)
where the coefficients {A, B, C, D, E} are determined respectively by
A =
B
=
C
=
D
=
E
=
sin a
,
sin b sin c sin d sin e
−e−bi cot b sin(a − 2b)
,
lim (b + z)R(z) =
z→−b
sin(c − b) sin(d − b) sin(e − b)
−e−ci cot c sin(a − 2c)
lim (c + z)R(z) =
,
z→−c
sin(b − c) sin(d − c) sin(e − c)
−e−di cot d sin(a − 2d)
,
lim (d + z)R(z) =
z→−d
sin(b − d) sin(c − d) sin(e − d)
−e−ei cot e sin(a − 2e)
lim (e + z)R(z) =
.
z→−e
sin(b − e) sin(c − e) sin(d − e)
lim zR(z) =
z→0
Recall that
sin M i = i sinh M,
cos M i = cosh M
and
lim
M →+∞
sinh M
= 1.
cosh M
For M → +∞, we can check without difficulty the following limiting relations:
sin M i
sin(λ + M i)
i sinh M
→ eλi ,
sin λ cosh M + i cos λ sinh M
e−M cos M i sin(a + 2M i)
(3b) R(M i) sin M i =
sin(b + M i) sin(c + M i) sin(d + M i) sin(e + M i)
2i exp (b + c + d + e − a)i
≈
(3c)
→ 0.
eM sinh M
Multiplying across (2) by sin z, then letting z = M i and M → +∞, we confirm
the trigonometric sum identity displayed in (1).
This trigonometric sum identity is only the tip of the hidden iceberg. In fact,
there exist many more trigonometric formulae of the same nature, which will be
established by means of partial fraction decomposition method. The main theorems
will be shown in the next section through this unified and more accessible approach.
Then the last section will illustrate several interesting examples, including those
due to Calogero [2, §2.4.5.3], Gustafson [8, 9] and Mohlenkamp-Monzón [11]. As
demonstrated in these works just cited, the trigonometric identities presented in
this paper may find further application in the evaluation of classical multiple hypergeometric series, trigonometric approximation and interpolation.
(3a)
=
2. Partial fraction decomposition
Consider the following function:
P (eiz )
,
k=0 sin(z − βk )
R(z) = n
where P (w) is a Laurent polynomial in w consisting of terms wk with |k| ≤ n + 1
and {βk }nk=0 are n + 1 distinct complex numbers.
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PARTIAL DECOMPOSITIONS AND TRIGONOMETRIC SUM IDENTITIES
231
Noting that
sin(z − β) =
e−i(z+β) 2iz
ei(z−β) − ei(β−z)
=
e − e2iβ ,
2i
2i
we can reformulate R(z) explicitly in eiz as follows:
P (eiz ) e(n+1)iz
R(z) = (2i)n+1 eBi n 2iz
− e2iβk
k=0 e
where
B=
n
βk .
k=0
Denote by [wm ]P (w) the coefficient of wm in P (w). Then we have the following
decomposition in partial fractions:
n
P (w)wn+1
P (γk )
γkn
1
n+1
n P (−γk )
n
=
[w
]P
(w)
+
+
(−1)
2
2
2
2
2
w+γk
=k (γk −γ ) w−γk
k=0 (w −γk )
k=0
where w = eiz and γk = eiβk for 0 ≤ k ≤ n.
According to the parity of n and P (w), we can combine the fractions inside the
last braces {· · · } and simplify the result as follows:
(4a)
(4b)
P (w)wn+1
n
= [wn+1 ]P (w)
2
2
k=0 (w −γk )
⎧ n
wP (γk )γkn
⎪
⎪
⎪
,
⎪
⎪
⎪
(w2 −γk2 )
(γk2 −γ2 )
⎪
⎨k=0
=k
+ n
⎪
P
(γ
)γkn+1
k
⎪
⎪
,
⎪
⎪
2
2
⎪
(γk2 −γ2 )
⎪
⎩k=0 (w −γk )
n − even : P (w) = P (−w),
n − odd : P (w) = −P (−w);
n − even : P (w) = −P (−w),
n − odd : P (w) = P (−w).
=k
§2.1. P (w) and n have the same parity. Suppose throughout this subsection
that (−1)n P (−w) = P (w), i.e., the natural number n and the Laurent polynomial
P (w) have the same parity.
Noting that
w2 − γk2
= 2iei(z+βk ) sin(z − βk ),
γk2 − γ2
= 2iei(βk +β ) sin(βk − β ),
we may restate the first case of (4) in terms of a trigonometric sum identity.
Theorem 1 (n-even with P (w) = P (−w) and n-odd with P (w) = −P (−w)). Let
P (w) be a Laurent polynomial consisting of terms wk with |k| ≤ n + 1 and {βk }nk=0
distinct complex numbers with the sum being denoted by B = nk=0 βk . Then there
holds the summation formula:
n
k=0
P (eiz )
P (eiβk )
= n
.
sin(z − βk ) =k sin(βk − β )
k=0 sin(z − βk )
In particular for z = 0, this theorem reduces to the following identity.
Corollary 2 (n-even with P (w) = P (−w) and n-odd with P (w) = −P (−w)). Let
P (w) be a Laurent polynomial consisting of terms wk with |k| ≤ n + 1 and {βk }nk=0
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232
WENCHANG CHU
distinct complex numbers with the sum being denoted by B =
holds the summation formula:
n
k=0
sin βk
n
k=0
βk . Then there
P (eiβk )
(−1)n P (1)
.
= n
=k sin(βk − β )
k=0 sin βk
When P (w) is a Laurent polynomial consisting of terms wk with |k| ≤ n, we
may multiply by eiz across the equation displayed in Theorem 1. Let z = −M i and
M → +∞ in the resulting equation and then applying the limiting relation
lim
z=−M i
M →+∞
eiz
2ieM
= lim
= 2ieiβk ,
M
−iβ
k − e−M +iβk
sin(z − βk ) M →+∞ e
we derive the following trigonometric identity.
Proposition 3 (n-even with P (w) = P (−w) and n-odd with P (w) = −P (−w)).
Let P (w) be a Laurent polynomial consisting of terms wk with
|k| ≤ n and {βk }nk=0
n
distinct complex numbers with the sum being denoted by B = k=0 βk . Then there
holds the summation formula:
n
k=0
eiβk P (eiβk )
= (2i)n eBi [wn ]P (w).
=k sin(βk − β )
§2.2. P (w) and n have the opposite parity. Suppose throughout this subsection
that (−1)n P (−w) = −P (w), i.e., the natural number n and the Laurent polynomial
P (w) have the opposite parity.
Similarly, the second case of (4) may be expressed in terms of another trigonometric sum identity.
Theorem 4 (n-even with P (w) = −P (−w) and n-odd with P (w) = P (−w)). Let
P (w) be a Laurent polynomial consisting of terms wk with |k| ≤ n + 1 and {βk }nk=0
n
distinct complex numbers with the sum being denoted by B = k=0 βk . Then there
holds the summation formula:
n
k=0
P (eiz )
ei(βk −z) P (eiβk )
= n
− (2i)n+1 eBi [wn+1 ]P (w).
sin(z − βk ) =k sin(βk − β )
sin(z
−
β
)
k
k=0
In particular for z = 0, Theorem 4 reduces to the following identity.
Corollary 5 (n-even with P (w) = −P (−w) and n-odd with P (w) = P (−w)). Let
P (w) be a Laurent polynomial consisting of terms wk with |k| ≤ n + 1 and {βk }nk=0
distinct complex numbers with the sum being denoted by B = nk=0 βk . Then there
holds the summation formula:
n
k=0
(−1)n P (1)
eiβk P (eiβk )
= n
+ (2i)n+1 eBi [wn+1 ]P (w).
sin βk =k sin(βk − β )
k=0 sin βk
Multiplying by eiz across the equation displayed in Theorem 4, we find that
the resulting identity is consistent with that displayed in Theorem 1, but with the
degree of numerator polynomial being altered.
Analogously, we can derive from Theorem 4 by letting z = M i and M → +∞
the following limiting case.
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PARTIAL DECOMPOSITIONS AND TRIGONOMETRIC SUM IDENTITIES
233
Proposition 6 (n-even with P (w) = −P (−w) and n-odd with P (w) = P (−w)).
+ 1 and
Let P (w) be a Laurent polynomial consisting of terms wk with |k| ≤ n
n
{βk }nk=0 distinct complex numbers with the sum being denoted by B = k=0 βk .
Then there holds the summation formula:
n
k=0
P (eiβk )
= (2i)n eBi [wn+1 ]P (w) + (−2i)n e−Bi [w−n−1 ]P (w).
sin(β
−
β
)
k
=k
3. Examples of trigonometric identities
The general results displayed in the last section imply numerous identities on
trigonometric sums, which will be exhibited in this section.
§3.1. With the trigonometric polynomial P (w) being defined by
P (eiz ) = e−iz cos(z − β0 ) sin(α − 2z),
we have P (w) = P (−w) which follows from
w−4 −i(α+β0 ) 2iα
e
(e − w4 )(w2 + e2iβ0 ).
4i
When n = 2m with m ≥ 2, we get from Proposition 3 the following identity:
P (w) =
cot(βk − β0 ) sin(α − 2βk )
sin(α − 2β0 )
=
.
2m
=k sin(β − βk )
=1 sin(β − β0 )
k=1
2m
(5)
Putting β0 = 0 and m = 2, we reconfirm the identity anticipated in (1).
§3.2. With the trigonometric polynomial P (w) being defined by
iz
P (e ) =
n
sin(z − αk ) and
k=1
n
w−n −iαk 2
P (w) =
e
(w − e2iαk ),
(2i)n
k=1
the relation P (w) = (−1)n P (−w) implies that P (w) has the same parity as n. Then
Theorem 1 leads us to the following identity due to Gustafson [8, Lemma 2.14]:
n
n
n
j=1 sin(z − αj )
j=1 sin(αj − βk )
n
.
=
(6)
sin(z
−
β
sin(z
−
β
)
k)
j
j=k sin(βj − βk )
j=0
k=0
§3.3. Let the trigonometric polynomial P (w) be defined by
P (eiz ) =
n
k=0
sin(z − αk ) and
P (w) =
n
w−n−1 −iαk 2
e
(w − e2iαk ).
(2i)n+1
k=0
Then n and P (w) have the opposite parity for P (w) = (−1)n+1 P (−w). Consequently, we derive from Theorem 4 the following identity:
n
n
n
n sin(z − αj )
ei(βk −z) j=0 sin(αj − βk )
j=0
= exp i
(7)
.
(βk − αk ) − n
sin(z − βk ) j=k sin(βj − βk )
j=0 sin(z − βj )
k=0
k=0
Extracting the real part across the last equation, we get the identity
n
n
n
n sin(z − αj )
j=0 sin(αj − βk )
j=0
= cos
cot(z − βk ) (αk − βk ) − n
.
(8)
j=k sin(βj − βk )
j=0 sin(z − βj )
k=0
k=0
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234
WENCHANG CHU
The imaginary part recovers another trigonometric formula due to Gustafson [9,
Lemma 5.10]:
n n
n
sin(αj − βk )
j=0
(9)
= sin
(αk − βk ) .
j=k sin(βj − βk )
k=0
k=0
There exist different proofs for this last identity, for example, the induction proof
by Mohlenkamp-Monzón [11, Theorem 2] and the matrix spectrum method due
to Calogero [2, §2.4.5.3] where more identities can also be verified by means of
trigonometric interpolation (cf. Kress [10, §8.2]).
§3.4. Let P (w) and P(w) be two trigonometric polynomials defined respectively by
P (eiz ) = sin(α + mz)
and
P(eiz ) = cos(α + mz)
and
w−m iα 2m
e (w − e−2iα ),
2i
w−m iα 2m
e (w + e−2iα ).
P(w) =
2
P (w) =
When m and n are two natural numbers with the same parity and 0 ≤ m ≤ n, we
derive from Theorem 1 the following two trigonometric sum identities:
(10a)
(10b)
n
k=0
n
k=0
sin(α + mβk )
sin(z − βk ) =k sin(βk − β )
=
sin(α + mz)
n
,
k=0 sin(z − βk )
cos(α + mβk )
sin(z − βk ) =k sin(βk − β )
=
cos(α + mz)
n
.
k=0 sin(z − βk )
Further, Proposition 3 gives us two other summation formulae:
n
eiβk sin(α + mβk )
=k sin(βk − β )
(11a)
k=0
n
(11b)
k=0
eiβk cos(α + mβk )
=k sin(βk − β )
= (2i)n−1 ei(α+B) δm,n ,
= (2i)n−1 iei(α+B) δm,n ,
where δm,n stands for the usual Kronecker symbol.
§3.5. Let P (w) and P(w) be two trigonometric polynomials defined respectively by
P (eiz ) = sin(α + mz)
and
P(eiz ) = cos(α + mz)
and
w−m iα 2m
e (w − e−2iα ),
2i
w−m iα 2m
e (w + e−2iα ).
P(w) =
2
P (w) =
When m and n are two natural numbers with different parity and 0 ≤ m ≤ n, we
derive from Theorem 4 the following two trigonometric sum identities:
(12a)
(12b)
n
k=0
n
k=0
ei(βk −z) sin(α + mβk )
sin(z − βk ) =k sin(βk − β )
=
sin(α + mz)
n
,
k=0 sin(z − βk )
ei(βk −z) cos(α + mβk )
sin(z − βk ) =k sin(βk − β )
=
cos(α + mz)
n
.
k=0 sin(z − βk )
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PARTIAL DECOMPOSITIONS AND TRIGONOMETRIC SUM IDENTITIES
235
Their real and imaginary parts yield respectively the following identities:
(13a)
(13b)
(13c)
n
sin(α + mz)
cot(z − βk ) sin(α + mβk )
= n
,
sin(β
−
β
)
k
=k
k=0 sin(z − βk )
k=0
n
k=0
n
k=0
cot(z − βk ) cos(α + mβk )
cos(α + mz)
= n
;
sin(β
−
β
)
k
=k
k=0 sin(z − βk )
n
sin(α + mβk )
cos(α + mβk )
=
= 0.
=k sin(βk − β )
=k sin(βk − β )
k=0
§3.6. Let P (w) and P(w) be two trigonometric polynomials defined respectively by
P (eiz ) = sinm (α + z)
and
P(eiz ) = cosm (α + z)
and
w−m miα 2
e
(w − e−2iα )m ,
(2i)m
w−m
P(w) = m emiα (w2 + e−2iα )m .
2
P (w) =
When m and n are two natural numbers with the same parity and 0 ≤ m ≤ n, we
derive from Theorem 1 the following two trigonometric sum identities:
(14a)
(14b)
n
k=0
n
k=0
sinm (α + βk )
sin(z − βk ) =k sin(βk − β )
=
sinm (α + z)
n
,
k=0 sin(z − βk )
cosm (α + βk )
sin(z − βk ) =k sin(βk − β )
=
cosm (α + z)
n
.
k=0 sin(z − βk )
Two other summation formulae corresponding to Proposition 3 read as follows:
n
eiβk sinm (α + βk )
=k sin(βk − β )
(15a)
k=0
n
(15b)
k=0
eiβk cosm (α + βk )
=k sin(βk − β )
= ei(mα+B) δm,n ,
= in ei(mα+B) δm,n .
§3.7. Let P (w) and P(w) be two trigonometric polynomials defined respectively by
P (eiz ) = sinm (α + z)
and
P(eiz ) = cosm (α + z)
and
w−m miα 2
e
(w − e−2iα )m ,
(2i)m
w−m
P(w) = m emiα (w2 + e−2iα )m .
2
P (w) =
When m and n are two natural numbers with the different parity and 0 ≤ m ≤ n,
we derive from Theorem 4 the following two trigonometric sum identities:
(16a)
(16b)
n
k=0
n
k=0
ei(βk −z) sinm (α + βk )
sin(z − βk ) =k sin(βk − β )
=
sinm (α + z)
n
,
k=0 sin(z − βk )
ei(βk −z) cosm (α + βk )
sin(z − βk ) =k sin(βk − β )
=
cosm (α + z)
n
.
k=0 sin(z − βk )
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236
WENCHANG CHU
Their real and imaginary parts yield respectively the following identities:
n
sinm (α + z)
cot(z − βk ) sinm (α + βk )
= n
,
=k sin(βk − β )
k=0 sin(z − βk )
(17a)
k=0
n
(17b)
k=0
n
(17c)
k=0
cosm (α + z)
cot(z − βk ) cosm (α + βk )
= n
;
=k sin(βk − β )
k=0 sin(z − βk )
n
sinm (α + βk )
cosm (α + βk )
=
= 0.
=k sin(βk − β )
=k sin(βk − β )
k=0
§3.8. When P (eiz ) is a monomial of order m in sin z and cos z with 0 ≤ m ≤ n and
m ≡ n(mod 2), there holds the following general identity:
n
(18)
k=0
P (eiβk )
P (eiz )
= n
.
sin(z − βk ) j=k sin(βk − βj )
k=0 sin(z − βk )
For example, we have the following summation formulae:
n
(19a)
k=0
n
(19b)
k=0
n
(19c)
k=0
sin(α + mβk ) sin(γ + βk )
sin(z − βk ) j=k sin(βk − βj )
=
sin(α + mz) sin(γ + z)
n
,
k=0 sin(z − βk )
sin(α + mβk ) cos(γ + βk )
sin(z − βk ) j=k sin(βk − βj )
=
sin(α + mz) cos(γ + z)
n
,
k=0 sin(z − βk )
cos(α + mβk ) cos(γ + βk )
sin(z − βk ) j=k sin(βk − βj )
=
cos(α + mz) cos(γ + z)
n
,
k=0 sin(z − βk )
where , m, and n are three integers satisfying 0 ≤ |m| + || ≤ n and m + ≡ n
(mod 2).
§3.9. When P (eiz ) is a monomial of order m in sin z and cos z with 0 ≤ m ≤ n and
m ≡ n(mod 2), there holds the following general identity:
n
(20)
k=0
ei(βk −z) P (eiβk )
P (eiz )
= n
sin(z − βk ) j=k sin(βk − βj )
k=0 sin(z − βk )
where the imaginary and real parts yield two summation formulae:
n
k=0
P (eiβk )
= 0 and
j=k sin(βk − βj )
n
P (eiz )
P (eiβk ) cot(z − βk )
= n
.
j=k sin(βk − βj )
k=0 sin(z − βk )
k=0
For example, we have the following summation formulae:
(21)
n
n
n
sinm (α+βk ) sin (γ+βk )
sinm (α+βk ) cos (γ+βk )
cosm (α+βk ) cos (γ+βk )
=
=
=0
j=k sin(βk − βj )
j=k sin(βk − βj )
j=k sin(βk − βj )
k=0
k=0
k=0
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PARTIAL DECOMPOSITIONS AND TRIGONOMETRIC SUM IDENTITIES
237
and
(22a)
n
cot(z − βk )
sinm (α + z) sin (γ + z)
sinm (α + βk ) sin (γ + βk )
n
=
,
j=k sin(βk − βj )
k=0 sin(z − βk )
cot(z − βk )
sinm (α + z) cos (γ + z)
sinm (α + βk ) cos (γ + βk )
n
=
,
j=k sin(βk − βj )
k=0 sin(z − βk )
cot(z − βk )
cosm (α + z) cos (γ + z)
cosm (α + βk ) cos (γ + βk )
n
=
,
j=k sin(βk − βj )
k=0 sin(z − βk )
k=0
(22b)
n
k=0
(22c)
n
k=0
where , m, and n are three integers satisfying 0 ≤ |m| + || ≤ n and m + ≡
n
(mod 2). The examples exhibited here are far from exhausting.
There are other interesting trigonometric sum identities, for example, those appearing in Berndt [1], Chu [3], Chu-Marini [4], and Gessel [7]. The reader is encouraged to examine further.
References
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[3] W. Chu, Summations on trigonometric functions Applied Mathematics and Computation
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[4] W. Chu, A. Marini, Partial fractions and trigonometric identities Advances in Appl. Math. 23
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[9] R. A. Gustafson, Some q-beta and Mellin-Barnes integrals on compact Lie groups and Lie
algebras Trans. Amer. Math. Soc. 341 (1994), 69-119. MR1139492 (94c:33032)
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Department of Applied Mathematics, Dalian University of Technology, Dalian
116024, People’s Republic of China
Current address: Dipartimento di Matematica, Università degli Studi di Lecce, Lecce-Arnesano, P. O. Box 193, 73100 Lecce, Italia
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