Proof for a q -Trigonometric Identity of Gosper
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Proof for a q-Trigonometric Identity of Gosper Bing He , Fuli He and Hongcun Zhai Abstract. Gosper in 2001 introduced the q-trigonometric functions and conjectured many interesting q-trigonometric identities. In this paper, we apply Riemann’s addition formula to deduce two Jacobi theta function identities. From these theta function identities, we confirm a qtrigonometric identity conjectured by Gosper and establish two other similar results. As an application, two theta function analogues for Ptolemy’s theorem are given. Mathematics Subject Classification. 11F27, 33E05, 11E25. Keywords. q-Trigonometric identity, Riemann’s addition formula, Jacobi theta function, Ptolemy’s theorem.
1. Introduction
√ Throughout this paper, we use q to denote exp(πiτ ), where i = −1 and Im τ > 0. To carry out our study, we need the definition of the Jacobi theta functions. Definition 1.1. The Jacobi theta functions θj (z|τ ) for j = 1, 2, 3, 4 are defined as [12,17] 1
θ1 (z|τ ) = −iq 4
∞
(−1)k q k(k+1) e(2k+1)zi , θ3 (z|τ ) =
k=−∞ 1
θ2 (z|τ ) = q 4
∞
∞
2
q k e2kzi ,
k=−∞
q k(k+1) e(2k+1)zi , θ4 (z|τ ) =
k=−∞
∞
2
(−1)k q k e2kzi .
k=−∞
For convenience, we use the notation ϑj (τ ) to denote θj (0|τ ) and employ the familiar notation: ∞ (1 − aq n ). (a; q)∞ = n=0 0123456789().: V,-vol
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With this notation, the celebrated Jacobi triple product identity can be written as (see [6, (21)] and [8]) ∞
(−1)n q n(n−1)/2 z n = (q; q)∞ (z; q)∞ (q/z; q)∞ , z = 0.
n=−∞
With the Jacobi triple product identity, we can deduce the Jacobi infinite product expressions for theta functions: 1
θ1 (z|τ ) = 2q 4 (sin z)(q 2 ; q 2 )∞ (q 2 e2zi ; q 2 )∞ (q 2 e−2zi ; q 2 )∞ , 1
θ2 (z|τ ) = 2q 4 (cos z)(q 2 ; q 2 )∞ (−q 2 e2zi ; q 2 )∞ (−q 2 e−2zi ; q 2 )∞ , θ3 (z|τ ) = (q 2 ; q 2 )∞ (−qe2zi ; q 2 )∞ (−qe−2zi ; q 2 )∞ . We also have the following interesting relations: π θ1 z + τ = θ2 (z|τ ), 2 and
π + πτ τ = q −1/4 e−iz θ3 (z|τ ). θ1 z + 2
(1.1)
(1.2)
Gosper [7] introduced q-analogues of sin z and cos z, which are defined by (q 2−2z ; q 2 )∞ (q 2z ; q 2 )∞ (z−1/2)2 q , (q; q 2 )2∞ (q 1−2z ; q 2 )∞ (q 1+2z ; q 2 )∞ z2 q . cosq (πz) := (q; q 2 )2∞ sinq (πz) :=
It is known from the above definitions that cosq z = sinq ( π2 ±z), limq→1 sinq z = sin z and limq→1 cosq z = cos z. Two relations between sinq , cosq and the functions θ1 and θ2 are equivalent to the following identities: sinq z =
θ1 (z|τ ) , ϑ2 (τ )
(1.3)
cosq z =
θ2 (z|τ ) , ϑ2 (τ )
(1.4)
and
where τ = − τ1 . In [7], Gosper stated without proof many identities involving sinq z and cosq z. He conjectured these identities using a computer program called MACSYMA. Since then, finding theta function analogues for trigonometric identities and proving these q-trigonometric function conjectures of Gosper become interesting topics in number theory and the theory of special functions. Using a direct analysis of its two sides through logarithmic derivatives, Mez˝ o in [13] confirmed the conjecture (q-Double2 ):
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