Math. Sci. VOL. · 2019. 8. 1. · l_zk with the same restrictions on 0 and x. Raxnanujan himself...

Internat. J. Math. & Math. Sci.

VOL. 14 NO. 4 (1991) 815-816

A TRIGONOMETRICAL IDENTITY

JOHN A. EWELLDepartment of Mathematical Sciences

Northern Illinois UniversityDeKalb, Illinois 60115

(Received September 4, 1990)

1. INTRODUCTION. The object of this note is to establish the following identity:

oo x2k+k= 1--x2k

21 zk sin

k= l--x2k

x2k cos k8 (3- 4 cos kS){(1/4) cot (0/2)}2 43- k 1(1- x2k)2 + kx2k

k= ll-x2k

/k: 1-x2k

3 xk(l+ x2k)gk (1 x2k)2

cos (k8/2), (1.1)

valid for 8ER, xEC, 8 not an even multiple of r and [x[ <1. The details of the proof are

supplied in section 2. In our concluding remarks we compare (1.1) with a celebrated identity of

Rarnanujan, and discuss a uniform method which reveals a total of four such trigonometricalidentities.

2. PROOF OF IDENTITY (1.1). Our argument is based on the following variant of the

quintuple-product identity:

(1 x2n) (1 a2 x2n 2) (1 a-2 x2n);::] - 4- a-1 x2n 1) -oo

xn(3n + 2) (a-3n a3n + 2), (1.2)

valid for a,x C, a # 0 and xl < 1. For a discussion of (1.2) and other forms of the quintuple-product identity see [1].

In (1.2) let a --. -a, x z3, and multiply the subsequent identity by -a-lx to get

(a a-1) (1 x6n) (1 a2 6n) (1 a-2 x6n) )2(1--az6n-3) (1--a-lx6"-3) -oo (--1)n x(3n+ (a3n+

816 J.A. EWELL

Let F(a, x) denote the left side of the foregoing identity, and for a complex variable z, regard zDzas an operator, where Dz denotes derivation with respect to z. Then,

(aDa)2 {F(a, x)} (-I)n (3n + 1)2 x(3n+ (a3n+ a-3n- l)

(D) {F(., )}.

We now use the technique of logarithmic differentiation to evaluate the leftmost and rightmostmembers of (2.2), cancel F(a, :r) in the resulting identity, and then let z --, xI/3 to get

=1+ 4 oo kx2k (a2k a_2k)_(a a-1)2 +

4 E z2/ + kzl: (al: + a-k)k= 11-- k= 11--z2k

oo kz2k oo :r.2k (a2k a_2k xk(1 x2k)--6 E m2k +6 E m2k)2 + +3 + (a +a-).k=ll-- k=1(1-- k=l (1--z2k)2

In the foregoing identity let a ei/2, subject to the stated restrictions. We simpfy the resultingidentity, d finly divide by -16 to rive at identity (1.1).

CONCLUDING REMAS. The forerunner of ] identities of type (1.1) is a celebrated one

due touj [2, p. 139], z.,2

(1/4) cot (0/2)+ xk sin kO1-xk J

{(1/4) cot (0/2)}2 + xk cos/cO

(I xk)2kxt (1 cos/cO), (2.4)q"1/2 l_zk

with the same restrictions on 0 and x. Raxnanujan himself made substantial applications of his

identity to the theory of elliptic modular functions. However, the most familiar application of the

identity is perhaps that of Hardy and Wright [3, pp. 311-314]. These authors use the identity to

establish Jacobi’s formula for the number r4(rt of representations of a natural number n by sums of

four squares. Ewell [4] shows that the method of this note permits an easy and straightforwardderivation of Ramanujan’s identity. Moreover, the method also reveals two additional

trigonometrical identities of this type.

REFERENCES1. SUBBARAO, M.V. and VIDYASAGAR, M., "On Watson’s quintuple-product identity", Proc.

Amer. Math. Soc. 26 (1970), 23-27.

2. RAMANUJAN, S., Collected Papers, Chelsea, New York, 1962.

3. HARDY, G.H. and WRIGHT, E.M., An Introduction to the Theory of Numbers, 4th ed.,Oxford, (1960).

4. EWELL, J.A., "Consequences of the triple- and quintriple-product identities", Houston J.Math. 14 (1988) 51-55.

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