A Note on Cosine Power Sums

Mircea Merca Department of Mathematics University of Craiova A. I. Cuza 13 Craiova, 200585 Romania

                 mircea.merca@profinfo.edu.ro

Abstract Using the multisection series method, we establish formulas for various power sums

of cosine functions. As corollaries we derive several combinatorial identities.

1 Introduction

In [2, 3], we presented two open problems concerning the asymptotic behaviour of cosine power sums,

\begin{matrix}⌊ & \frac{n−1}{2} & ⌋ \\ ∑ \\ k=1 \end{matrix} cos^{p} (\frac{kπ}{n} ) ,  (1)

where ⌊x⌋ denotes the largest integer not greater than x. Under certain conditions, these cosine power sums can be determined exactly, without approximations [5].

If

f(x) = a_{0} + a_{1}x + a_{2}x^{2} + ⋯ + a_{n}x^{n} + ⋯

is a finite or convergent infinite series, then for 0 ≤ r < n the sum

a_{r}x^{r} + a_{r+n}x^{r+n} + a_{r+2n}x^{r+2n} + ⋯

is given by the multisection formula (see [1, Ch. 16], [4, Ch. 4, S. 4.3] and [6])

∑_{k≥0} a_{r+kn}x^{r+kn} = \frac{1}{n} ∑_{k=0}^{n−1} z^{−kr}f(z^{k}x) ,  (2)

where z = e^{\frac{2πi}{n}} is the nth root of 1.

We shall use the multisection formula to prove:

Theorem 1. Let n and p be two positive integers. Then

\begin{matrix}⌊ & \frac{n−1}{2} & ⌋ \\ ∑ \\ k=1 \end{matrix} cos^{2p} (\frac{kπ}{n} ) = −\frac{1}{2} + \frac{n}{2^{2p+1}} \begin{matrix}⌊ & \frac{p}{n} & ⌋ \\ ∑ \\ k=−⌊\frac{p}{n}⌋ \end{matrix} \binom{2p}{p + kn} .

Theorem 2. Let n and p be two positive integers such that n ≡ p (mod 2). Then

\begin{matrix}⌊ & \frac{n−1}{2} & ⌋ \\ ∑ \\ k=1 \end{matrix}(−1)^{k} cos^{p} (\frac{kπ}{n} ) = −\frac{1}{2} + \frac{n}{2^{p+1}} \begin{matrix}[ & \frac{p}{2n} & ] \\ ∑ \\ k=−[\frac{p}{2n}] \end{matrix} \binom{p}{\frac{p+n}{2} + kn} ,

where [x] denotes the nearest integer to x, i.e., [x] = ⌊x + \frac{1}{2}⌋.

Theorem 3. Let n and p be two positive integers. Then

\begin{matrix}⌊ & \frac{n}{2} & ⌋ \\ ∑ \\ k=1 \end{matrix} cos^{2p} (\frac{2k − 1}{n} · \frac{π}{2} ) = \frac{n}{2^{2p+1}} \begin{matrix}⌊ & \frac{p}{n} & ⌋ \\ ∑ \\ k=−⌊\frac{p}{n}⌋ \end{matrix}(−1)^{k}\binom{2p}{p + kn} .

We note that the formula proved in [5] is the case p < n of Theorem 1. The conjecture published in [3] is solved by Theorem 2. In the final section of this paper we present some relations that can be obtained immediately by the above theorems.

2 Proof of Theorem 1

We apply the multisection formula (2) for

r = p − n ⌊\frac{p}{n}⌋ and f(x) = (1 + x)^{2p} .

Thus, when x = 1, we get

\begin{matrix}⌊ & \frac{p}{n} & ⌋ \\ ∑ \\ k=−⌊\frac{p}{n}⌋ \end{matrix} \binom{2p}{p + kn} = \frac{1}{n} ∑_{k=0}^{n−1} z^{−k(p−n⌊ \frac{p}{n} ⌋)}(1 + z^{k})^{2p} ,  (3)

where z = e^{\frac{2πi}{n}}. Having

e^{−irφ}(1 + e^{iφ})^{p} = e^{−irφ} (e^{\frac{iφ}{2} − \frac{iφ}{2}} + e^{\frac{iφ}{2} + \frac{iφ}{2}} )^{p}
= e^{−irφ}e^{\frac{ipφ}{2}} (e^{− \frac{iφ}{2}} + e^{\frac{iφ}{2}} )^{p}
= 2^{p} cos^{p} \frac{φ}{2} · e^{i(\frac{p}{2} −r)φ} ,  (4)

we can write

z^{−k(p−n⌊ \frac{p}{n} ⌋)}(1 + z^{k})^{2p} = 2^{2p} cos^{2p} (\frac{kπ}{n} ) · e^{i2k⌊\frac{p}{n}⌋π} .

So, we may rewrite (3) as

\begin{matrix}⌊ & \frac{p}{n} & ⌋ \\ ∑ \\ k=−⌊\frac{p}{n}⌋ \end{matrix} \binom{2p}{p + kn} = \frac{2^{2p}}{n} ∑_{k=0}^{n−1} cos^{2p} (\frac{kπ}{n} ) ,  (5)

Taking into account that

cos (\frac{(n − k)π}{n} ) = − cos (\frac{kπ}{n} ) , k = 0, 1, …, n ,

Theorem 1 is proved.

3 Proof of Theorem 2

To prove the theorem we proceed as in Theorem 1. By the multisection formula (2), with

x = 1 , f(x) = (1 + x)^{p} and r = \frac{p + n}{2} − n ⌊\frac{p + n}{2n}⌋ ,

we get

∑_{k=−⌊ \frac{p+n}{2n} ⌋}^{⌊ \frac{p+n}{2n} ⌋} \binom{p}{\frac{p+n}{2} + nk} = \frac{1}{n} ∑_{k=0}^{n−1} z^{−k(\frac{p+n}{2} −n⌊ \frac{p+n}{2n} ⌋)}(1 + z^{k})^{p} ,

where z = e^{\frac{2πi}{n}}. By (4), it immediately follows that

z^{−k(\frac{p+n}{2} −n⌊ \frac{p+n}{2n} ⌋)}(1 + z^{k})^{p} = 2^{p} cos^{p} (\frac{kπ}{n} ) · e^{ikπ(2⌊ \frac{p+n}{2n} ⌋−1)} .

Thus we can write

∑_{k=−⌊ \frac{p+n}{2n} ⌋}^{⌊ \frac{p+n}{2n} ⌋} \binom{p}{\frac{p+n}{2} + kn} = \frac{2^{p}}{n} ∑_{k=0}^{n−1}(−1)^{k} cos^{p} (\frac{kπ}{n} ) .  (6)

On the other hand, we have

∑_{k=0}^{n−1}(−1)^{k} cos^{p} (\frac{kπ}{n} )

= 1 + \begin{matrix}⌊ & \frac{n−1}{2} & ⌋ \\ ∑ \\ k=1 \end{matrix}(−1)^{k} cos^{p} (\frac{kπ}{n} ) + \begin{matrix}⌊ & \frac{n−1}{2} & ⌋ \\ ∑ \\ k=1 \end{matrix}(−1)^{n−k} cos^{p} (\frac{(n − k)π}{n} )

= 1 + \begin{matrix}⌊ & \frac{n−1}{2} & ⌋ \\ ∑ \\ k=1 \end{matrix}(−1)^{k} cos^{p} (\frac{kπ}{n} ) + \begin{matrix}⌊ & \frac{n−1}{2} & ⌋ \\ ∑ \\ k=1 \end{matrix}(−1)^{n+p−k} cos^{p} (\frac{kπ}{n} )

= 1 + 2 \begin{matrix}⌊ & \frac{n−1}{2} & ⌋ \\ ∑ \\ k=1 \end{matrix}(−1)^{k} cos^{p} (\frac{kπ}{n} ) .  (7)

According to (3) and (7), Theorem 2 is proved.

4 Proof of Theorem 3

To prove the theorem, we use the formula

\begin{matrix}⌊ & \frac{n}{2} & ⌋ \\ ∑ \\ k=1 \end{matrix} f(2k − 1) = \frac{1}{2} (∑_{k=1}^{n−1} f(k) − ∑_{k=1}^{n−1}(−1)^{k}f(k))  (8)

for f(k) = cos^{2p} (\frac{kπ}{2n}). By Theorem 1 (with n replaced by 2n) and Theorem 2 (with n and p replaced by 2n and 2p, respectively), we have

\begin{matrix}⌊ & \frac{n}{2} & ⌋ \\ ∑ \\ k=1 \end{matrix} f(2k − 1) = \frac{1}{2} (−\frac{1}{2} + \frac{2n}{2^{2p+1}} \begin{matrix}⌊ & \frac{p}{2n} & ⌋ \\ ∑ \\ k=−⌊\frac{p}{2n}⌋ \end{matrix} \binom{2p}{p + 2kn}

+\frac{1}{2} − \frac{2n}{2^{2p+1}} \begin{matrix}[ & \frac{p}{2n} & ] \\ ∑ \\ k=−[\frac{p}{2n}] \end{matrix} \binom{2p}{p + (2k + 1)n})

= \frac{n}{2^{2p+1}} \begin{matrix}[ & \frac{p}{2n} & ] \\ ∑ \\ k=−[\frac{p}{2n}] \end{matrix} (\binom{2p}{p + 2kn} − \binom{2p}{p + (2k + 1)n})
= \frac{n}{2^{2p+1}} \begin{matrix}⌊ & \frac{p}{n} & ⌋ \\ ∑ \\ k=−⌊\frac{p}{n}⌋ \end{matrix}(−1)^{k}\binom{2p}{p + kn} .

Theorem 3 is proved.

5 Special cases

We begin this section with two immediate consequences of Theorems 1, 2 and 3. Corollary 4. Let n be a positive integer. Then

1. \begin{matrix}⌊ & \frac{n−1}{2} & ⌋ \\ ∑ \\ k=1 \end{matrix} cos^{2n} (\frac{kπ}{n} ) = −\frac{1}{2} + \frac{n}{2^{2n}} + \frac{n}{2^{2n+1}} \binom{2n}{n};

2. \begin{matrix}⌊ & \frac{n−1}{2} & ⌋ \\ ∑ \\ k=1 \end{matrix}(−1)^{k} cos^{n} (\frac{kπ}{n} ) = −\frac{1}{2} + \frac{n}{2^{n}};

3. \begin{matrix}⌊ & \frac{n}{2} & ⌋ \\ ∑ \\ k=1 \end{matrix} cos^{2n} (\frac{2k − 1}{n} · \frac{π}{2} ) = −\frac{n}{2^{2n}} + \frac{n}{2^{2n+1}} \binom{2n}{n}.

Corollary 5. Let n and p be two positive integers such that p < n. Then

1. \begin{matrix}⌊ & \frac{n−1}{2} & ⌋ \\ ∑ \\ k=1 \end{matrix} cos^{2p} (\frac{kπ}{n} ) = −\frac{1}{2} + \frac{n}{2^{2p+1}} \binom{2p}{p};

2. \begin{matrix}⌊ & \frac{n−1}{2} & ⌋ \\ ∑ \\ k=1 \end{matrix}(−1)^{k} cos^{p} (\frac{kπ}{n} ) = −\frac{1}{2} , n ≡ p (mod 2);

3. \begin{matrix}⌊ & \frac{n}{2} & ⌋ \\ ∑ \\ k=1 \end{matrix} cos^{2p} (\frac{2k − 1}{n} · \frac{π}{2} ) = \frac{n}{2^{2p+1}} \binom{2p}{p}.

By Theorems 1, 2 and 3, for fixed values of n, we can obtain some combinatorial identities. Corollary 6. Let p be a positive integer. Then

1. ∑_{k=0}^{p} \binom{2p}{p − k} = \frac{1}{2}\binom{2p}{p} + 2^{2p−1} ;

2. ∑_{k=0}^{⌊p/2⌋} \binom{2p}{p − 2k} = \frac{1}{2}\binom{2p}{p} + 2^{2p−2} ;

3. ∑_{k=0}^{⌊p/3⌋} \binom{2p}{p − 3k} = \frac{1}{2}\binom{2p}{p} + \frac{1 + 2^{2p−1}}{3} ;

4. ∑_{k=0}^{⌊p/4⌋} \binom{2p}{p − 4k} = \frac{1}{2}\binom{2p}{p} + 2^{2p−3} + 2^{p−2} ;

5. ∑_{k=0}^{⌊p/5⌋} \binom{2p}{p − 5k} = \frac{1}{2}\binom{2p}{p} + \frac{(3 + \sqrt{5})^{p} + (3 − \sqrt{5})^{p} + 2^{3p−1}}{5 · 2^{p}} ;

6. ∑_{k=0}^{⌊p/6⌋} \binom{2p}{p − 6k} = \frac{1}{2}\binom{2p}{p} + \frac{3^{p} + 2^{2p−1} + 1}{6} .

Corollary 7. Let p be a positive integer. Then

1. ∑_{k=1}^{p} \binom{2p − 1}{p − k} = 4^{p−1} ;

2. ∑_{k=1}^{⌊p/3⌋} \binom{2p − 3}{p − 3k} = \frac{4^{p−2} − 1}{3} , p > 1 ;

3. ∑_{k=1}^{⌊p/5⌋} \binom{2p − 5}{p − 5k} = \frac{4^{p−3}}{5} − \frac{(\sqrt{5} + 1)^{2p−5} − (\sqrt{5} − 1)^{2p−5}}{5 · 2^{2p−5}} , p > 2 .

Corollary 8. Let p be a positive integer. Then

1. ∑_{k=0}^{p}(−1)^{k}\binom{2p}{p − k} = \frac{1}{2}\binom{2p}{p} ;

2. ∑_{k=0}^{⌊p/2⌋}(−1)^{k}\binom{2p}{p − 2k} = \frac{1}{2}\binom{2p}{p} + 2^{p−1} ;

3. ∑_{k=0}^{⌊p/3⌋}(−1)^{k}\binom{2p}{p − 3k} = \frac{1}{2}\binom{2p}{p} + 3^{p−1} ;

4. ∑_{k=0}^{⌊p/4⌋}(−1)^{k}\binom{2p}{p − 4k} = \frac{1}{2}\binom{2p}{p} + \frac{(2 + \sqrt{2})^{p} + (2 − \sqrt{2})^{p}}{4} ;

5. ∑_{k=0}^{⌊p/5⌋}(−1)^{k}\binom{2p}{p − 5k} = \frac{1}{2}\binom{2p}{p} + \frac{(5 + \sqrt{5})^{p} + (5 − \sqrt{5})^{p}}{5 · 2^{p}} .

Note that Corollary 6 is related in [7] with the sequences \underline{A032443}, \underline{A114121}, \underline{A007583}, \underline{A007582}, \underline{A078789}, \underline{A085282}, Corollary 7 with \underline{A000302}, \underline{A002450}, \underline{A095931}, Corollary 8 with \underline{A088218}, \underline{A005317}, \underline{A191993}, \underline{A007052}, \underline{A081567}, respectively.

6 Acknowledgements

The author would like to thank Professor Cecil C. Rousseau from the University of Memphis for his comments on the exact formulas of these cosine power sums. The author expresses his gratitude to Oana Merca for the careful reading of the manuscript and helpful remarks.

References

[1] R. Honsberger, Mathematical Gems III, Dolciani Math. Expositions No. 9, Mathematical Association of America, 1985.

[2] M. Merca, Asymptotic behavior of cosine power sums, SIAM, Problems and Solutions

    Online Archive, http://siam.org/journals/categories/11-002.php, 2011.

[3] M. Merca, Problem 89, Eur. Math. Soc. Newsl., 81 (2011), 59.

[4] J. Riordan, Combinatorial Identities, John Wiley & Sons, 1968.

[5] J. M. Quoniam, and M. G. Greening, A trigonometric summation, Amer. Math. Monthly, 75 (1968), 405–406.

[6] T. Simpson, The invention of a general method for determining the sum of every 2d, 3d, 4th, or 5th, &c. term of a series, taken in order; The sum of the whole series being known, Philosophical Transactions, 50 (1757–1758), 757–769. Available at

    http://www.jstor.org/stable/105328.

[7] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, published electronically

    at http://oeis.org, 2012.

2010 Mathematics Subject Classification: Primary 33B10; Secondary 05A19, 11B65. Keywords: trigonometric power sum, multisection formula, combinatorial identity.

(Concerned with sequences \underline{A000302}, \underline{A002450}, \underline{A005317}, \underline{A007052}, \underline{A007582}, \underline{A007583}, \underline{A032443}, \underline{A078789}, \underline{A081567}, \underline{A085282}, \underline{A088218}, \underline{A095931}, \underline{A114121}, and \underline{A191993}.)

Received February 15 2012; revised version received May 8 2012. Published in Journal of Integer Sequences, May 28 2012.

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