On certain q-trigonometric identities

Bing He

arXiv:1907.00711·math.GM·July 2, 2019

On certain q-trigonometric identities

Bing He

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TL;DR

This paper introduces $q$-analogues of tangent and cotangent functions, establishing theta function identities and deriving new $q$-trigonometric identities related to classical identities.

Contribution

It defines $q$-analogues for tangent and cotangent and uses elliptic function theory to derive new theta function identities and $q$-trigonometric identities.

Findings

01

Derived two $q$-trigonometric identities involving $ an_q z$ and $ ext{cot}_q z$

02

Established theta function identities using elliptic functions

03

Presented additional $q$-trigonometric identities

Abstract

Finding theta function (or $q$ -)analogues for well-known trigonometric identities is an interesting topic. In this paper, we first introduce the definition of $q$ -analogues for $tan z$ and $cot z$ and then apply the theory of elliptic functions to establish a theta function identity. From this identity we deduce two $q$ -trigonometric identities involving $tan_{q} z$ and $cot_{q} z,$ which are theta function analogues for two well-known trigonometric identities concerning $tan z$ and $cot z .$ Some other $q$ -trigonometric identities are also given.

Equations69

θ_{1} (z ∣ τ)

θ_{1} (z ∣ τ)

θ_{2} (z ∣ τ)

θ_{1} (z ∣ τ) = 2 q^{\frac{1}{4}} sin z (q^{2}; q^{2})_{\infty} (q^{2} e^{2 z i}; q^{2})_{\infty} (q^{2} e^{- 2 z i}; q^{2})_{\infty},

θ_{1} (z ∣ τ) = 2 q^{\frac{1}{4}} sin z (q^{2}; q^{2})_{\infty} (q^{2} e^{2 z i}; q^{2})_{\infty} (q^{2} e^{- 2 z i}; q^{2})_{\infty},

θ_{2} (z ∣ τ) = 2 q^{\frac{1}{4}} cos z (q^{2}; q^{2})_{\infty} (- q^{2} e^{2 z i}; q^{2})_{\infty} (- q^{2} e^{- 2 z i}; q^{2})_{\infty},

θ_{3} (z ∣ τ) = (q^{2}; q^{2})_{\infty} (- q e^{2 z i}; q^{2})_{\infty} (- q e^{- 2 z i}; q^{2})_{\infty},

θ_{4} (z ∣ τ) = (q^{2}; q^{2})_{\infty} (q e^{2 z i}; q^{2})_{\infty} (q e^{- 2 z i}; q^{2})_{\infty},

(z; q)_{\infty} = n \geq 0 \prod (1 - z q^{n}) .

(z; q)_{\infty} = n \geq 0 \prod (1 - z q^{n}) .

θ_{1} (z + π ∣ τ) = \textminus θ_{1} (z ∣ τ), θ_{1} (z + π τ ∣ τ) = - q^{\textminus 1} e^{\textminus 2 z i} θ_{1} (z ∣ τ), θ_{2} (z + π ∣ τ) = \textminus θ_{2} (z ∣ τ), θ_{2} (z + π τ ∣ τ) = q^{\textminus 1} e^{\textminus 2 z i} θ_{2} (z ∣ τ), θ_{3} (z + π ∣ τ) = θ_{3} (z ∣ τ), θ_{3} (z + π τ ∣ τ) = q^{\textminus 1} e^{\textminus 2 z i} θ_{3} (z ∣ τ), θ_{4} (z + π ∣ τ) = θ_{4} (z ∣ τ), θ_{4} (z + π τ ∣ τ) = - q^{\textminus 1} e^{\textminus 2 z i} θ_{4} (z ∣ τ) .

θ_{1} (z + π ∣ τ) = \textminus θ_{1} (z ∣ τ), θ_{1} (z + π τ ∣ τ) = - q^{\textminus 1} e^{\textminus 2 z i} θ_{1} (z ∣ τ), θ_{2} (z + π ∣ τ) = \textminus θ_{2} (z ∣ τ), θ_{2} (z + π τ ∣ τ) = q^{\textminus 1} e^{\textminus 2 z i} θ_{2} (z ∣ τ), θ_{3} (z + π ∣ τ) = θ_{3} (z ∣ τ), θ_{3} (z + π τ ∣ τ) = q^{\textminus 1} e^{\textminus 2 z i} θ_{3} (z ∣ τ), θ_{4} (z + π ∣ τ) = θ_{4} (z ∣ τ), θ_{4} (z + π τ ∣ τ) = - q^{\textminus 1} e^{\textminus 2 z i} θ_{4} (z ∣ τ) .

\begin{gathered}\theta_{1}\bigg{(}z+\frac{\pi\tau}{2}\bigg{|}\tau\bigg{)}=iB\theta_{4}(z|\tau),\quad\theta_{2}\bigg{(}z+\frac{\pi\tau}{2}\bigg{|}\tau\bigg{)}=B\theta_{3}(z|\tau),\\ \theta_{3}\bigg{(}z+\frac{\pi\tau}{2}\bigg{|}\tau\bigg{)}=B\theta_{2}(z|\tau),\quad\theta_{4}\bigg{(}z+\frac{\pi\tau}{2}\bigg{|}\tau\bigg{)}=iB\theta_{1}(z|\tau),\end{gathered}

\begin{gathered}\theta_{1}\bigg{(}z+\frac{\pi\tau}{2}\bigg{|}\tau\bigg{)}=iB\theta_{4}(z|\tau),\quad\theta_{2}\bigg{(}z+\frac{\pi\tau}{2}\bigg{|}\tau\bigg{)}=B\theta_{3}(z|\tau),\\ \theta_{3}\bigg{(}z+\frac{\pi\tau}{2}\bigg{|}\tau\bigg{)}=B\theta_{2}(z|\tau),\quad\theta_{4}\bigg{(}z+\frac{\pi\tau}{2}\bigg{|}\tau\bigg{)}=iB\theta_{1}(z|\tau),\end{gathered}

sin_{q} (π z) := \frac{( q ^{2 - 2 z} ; q ^{2} ) _{\infty} ( q ^{2 z} ; q ^{2} ) _{\infty}}{( q ; q ^{2} ) _{\infty}^{2}} q^{(z - 1 /2)^{2}},

sin_{q} (π z) := \frac{( q ^{2 - 2 z} ; q ^{2} ) _{\infty} ( q ^{2 z} ; q ^{2} ) _{\infty}}{( q ; q ^{2} ) _{\infty}^{2}} q^{(z - 1 /2)^{2}},

cos_{q} (π z) := \frac{( q ^{1 - 2 z} ; q ^{2} ) _{\infty} ( q ^{1 + 2 z} ; q ^{2} ) _{\infty}}{( q ; q ^{2} ) _{\infty}^{2}} q^{z^{2}},

sin_{q} z = \frac{θ _{1} ( z ∣ τ ^{'} )}{ϑ _{2} ( τ ^{'} )}, cos_{q} z = \frac{θ _{2} ( z ∣ τ ^{'} )}{ϑ _{2} ( τ ^{'} )},

sin_{q} z = \frac{θ _{1} ( z ∣ τ ^{'} )}{ϑ _{2} ( τ ^{'} )}, cos_{q} z = \frac{θ _{2} ( z ∣ τ ^{'} )}{ϑ _{2} ( τ ^{'} )},

tan_{q} (π z) := \frac{sin _{q} ( π z )}{cos _{q} ( π z )} = \frac{( q ^{2 - 2 z} ; q ^{2} ) _{\infty} ( q ^{2 z} ; q ^{2} ) _{\infty}}{( q ^{1 - 2 z} ; q ^{2} ) _{\infty} ( q ^{1 + 2 z} ; q ^{2} ) _{\infty}} q^{1/4 - z}

tan_{q} (π z) := \frac{sin _{q} ( π z )}{cos _{q} ( π z )} = \frac{( q ^{2 - 2 z} ; q ^{2} ) _{\infty} ( q ^{2 z} ; q ^{2} ) _{\infty}}{( q ^{1 - 2 z} ; q ^{2} ) _{\infty} ( q ^{1 + 2 z} ; q ^{2} ) _{\infty}} q^{1/4 - z}

cot_{q} (π z) := \frac{cos _{q} ( π z )}{sin _{q} ( π z )} = \frac{( q ^{1 - 2 z} ; q ^{2} ) _{\infty} ( q ^{1 + 2 z} ; q ^{2} ) _{\infty}}{( q ^{2 - 2 z} ; q ^{2} ) _{\infty} ( q ^{2 z} ; q ^{2} ) _{\infty}} q^{z - 1/4} .

cot_{q} (π z) := \frac{cos _{q} ( π z )}{sin _{q} ( π z )} = \frac{( q ^{1 - 2 z} ; q ^{2} ) _{\infty} ( q ^{1 + 2 z} ; q ^{2} ) _{\infty}}{( q ^{2 - 2 z} ; q ^{2} ) _{\infty} ( q ^{2 z} ; q ^{2} ) _{\infty}} q^{z - 1/4} .

tan_{q} z = \frac{θ _{1} ( z ∣ τ ^{'} )}{θ _{2} ( z ∣ τ ^{'} )}

tan_{q} z = \frac{θ _{1} ( z ∣ τ ^{'} )}{θ _{2} ( z ∣ τ ^{'} )}

cot_{q} z = \frac{θ _{2} ( z ∣ τ ^{'} )}{θ _{1} ( z ∣ τ ^{'} )} .

cot_{q} z = \frac{θ _{2} ( z ∣ τ ^{'} )}{θ _{1} ( z ∣ τ ^{'} )} .

tan x + tan y + tan z = tan x tan y tan z,

tan x + tan y + tan z = tan x tan y tan z,

cot x cot y + cot y cot z + cot z cot x = 1.

ccs_{q} (x - y) tan_{q^{2}} x + ccs_{q} (x - y) tan_{q^{2}} y + ssn_{q} (x - y) tan_{q} z = ssn_{q} (x - y) tan_{q^{2}} x tan_{q^{2}} y tan_{q} z,

ccs_{q} (x - y) tan_{q^{2}} x + ccs_{q} (x - y) tan_{q^{2}} y + ssn_{q} (x - y) tan_{q} z = ssn_{q} (x - y) tan_{q^{2}} x tan_{q^{2}} y tan_{q} z,

ssn_{q} (x - y) cot_{q^{2}} x cot_{q^{2}} y + ccs_{q} (x - y) cot_{q^{2}} y cot_{q} z + ccs_{q} (x - y) cot_{q} z cot_{q^{2}} x = ssn_{q} (x - y),

ssn_{q} (x - y) cot_{q^{2}} x cot_{q^{2}} y + ccs_{q} (x - y) cot_{q^{2}} y cot_{q} z + ccs_{q} (x - y) cot_{q} z cot_{q^{2}} x = ssn_{q} (x - y),

ccs_{q} z := \frac{cos _{q^{2}} z}{cos _{q} z}, ssn_{q} z := \frac{sin _{q^{2}} z}{sin _{q} z} .

ccs_{q} z := \frac{cos _{q^{2}} z}{cos _{q} z}, ssn_{q} z := \frac{sin _{q^{2}} z}{sin _{q} z} .

\lim_{q\rightarrow 1}\big{\{}\mathrm{tan}_{q}z,\mathrm{ssn}_{q}z,\mathrm{ccs}_{q}z\big{\}}=\big{\{}\mathrm{tan}z,1,1\big{\}},

\lim_{q\rightarrow 1}\big{\{}\mathrm{tan}_{q}z,\mathrm{ssn}_{q}z,\mathrm{ccs}_{q}z\big{\}}=\big{\{}\mathrm{tan}z,1,1\big{\}},

ccs_{q} (x - y) cot_{q^{2}} x + ccs_{q} (x - y) cot_{q^{2}} y + ssn_{q} (x - y) cot_{q} z = ssn_{q} (x - y) cot_{q^{2}} x cot_{q^{2}} y cot_{q} z

ccs_{q} (x - y) cot_{q^{2}} x + ccs_{q} (x - y) cot_{q^{2}} y + ssn_{q} (x - y) cot_{q} z = ssn_{q} (x - y) cot_{q^{2}} x cot_{q^{2}} y cot_{q} z

ssn_{q} (x - y) tan_{q^{2}} x tan_{q^{2}} y + ccs_{q} (x - y) tan_{q^{2}} y tan_{q} z + ccs_{q} (x - y) tan_{q} z tan_{q^{2}} x = 1.

ssn_{q} (x - y) tan_{q^{2}} x tan_{q^{2}} y + ccs_{q} (x - y) tan_{q^{2}} y tan_{q} z + ccs_{q} (x - y) tan_{q} z tan_{q^{2}} x = 1.

cot x + cot y + cot z = cot x cot y cot z,

cot x + cot y + cot z = cot x cot y cot z,

tan x tan y + tan y tan z + tan z tan x = 1.

θ_{2} (x + y ∣2 τ) θ_{3} (x - y ∣2 τ) (θ_{1} (x ∣ τ) θ_{2} (y ∣ τ) + θ_{1} (y ∣ τ) θ_{2} (x ∣ τ)) = θ_{1} (x + y ∣2 τ) θ_{4} (x - y ∣2 τ) (θ_{2} (x ∣ τ) θ_{2} (y ∣ τ) - θ_{1} (x ∣ τ) θ_{1} (y ∣ τ)) .

θ_{2} (x + y ∣2 τ) θ_{3} (x - y ∣2 τ) (θ_{1} (x ∣ τ) θ_{2} (y ∣ τ) + θ_{1} (y ∣ τ) θ_{2} (x ∣ τ)) = θ_{1} (x + y ∣2 τ) θ_{4} (x - y ∣2 τ) (θ_{2} (x ∣ τ) θ_{2} (y ∣ τ) - θ_{1} (x ∣ τ) θ_{1} (y ∣ τ)) .

F (x) = f (x) / g (x),

F (x) = f (x) / g (x),

f (x)

f (x)

\leavevmode - θ_{2} (x + y ∣2 τ) θ_{3} (x - y ∣2 τ) (θ_{1} (x ∣ τ) θ_{2} (y ∣ τ) + θ_{1} (y ∣ τ) θ_{2} (x ∣ τ))

g (x) = θ_{1} (x + y ∣2 τ) θ_{4} (x - y ∣2 τ) θ_{1} (x ∣ τ) θ_{1} (y ∣ τ) .

g (x) = θ_{1} (x + y ∣2 τ) θ_{4} (x - y ∣2 τ) θ_{1} (x ∣ τ) θ_{1} (y ∣ τ) .

h (x) = h (x + π) = q^{8} e^{8 i x} h (x + 2 π τ) .

h (x) = h (x + π) = q^{8} e^{8 i x} h (x + 2 π τ) .

f (π - y) = f (y + π τ) = 0.

f (π - y) = f (y + π τ) = 0.

2 θ_{2} (z ∣2 τ) θ_{3} (z ∣2 τ)

2 θ_{2} (z ∣2 τ) θ_{3} (z ∣2 τ)

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TopicsAdvanced Mathematical Identities · Mathematical functions and polynomials · Analytic Number Theory Research

Full text

On certain $q$ -trigonometric identities

Bing He

School of Mathematics and Statistics, Central South University

Changsha 410083, Hunan, People’s Republic of China

[email protected]; [email protected]

Abstract.

Finding theta function (or $q$ -)analogues for well-known trigonometric identities is an interesting topic. In this paper, we first introduce the definition of $q$ -analogues for $\mathrm{tan}z$ and $\mathrm{cot}z$ and then apply the theory of elliptic functions to establish a theta function identity. From this identity we deduce two $q$ -trigonometric identities involving $\mathrm{tan}_{q}z$ and $\cot_{q}z,$ which are theta function analogues for two well-known trigonometric identities concerning $\mathrm{tan}z$ and $\cot z.$ Some other $q$ -trigonometric identities are also given.

Key words and phrases:

$q$ -trigonometric identity, elliptic function, theta function identity, trigonometric identity

2000 Mathematics Subject Classification:

33E05, 11F11, 11F12.

1. Introduction

To carry out our study, we need the definition of Jacobi’s theta functions [4, 11, 15]:

[TABLE]

where $q=\exp(\pi i\tau)$ and $\text{Im}\>\tau>0.$

Throughout this paper, we use the notation $\vartheta_{j}(\tau)$ to denote $\theta_{j}(0|\tau)$ for $j=2,3,4$ .

The Jacobi infinite product expressions for the theta functions are well-known:

[TABLE]

where $(z;q)_{\infty}$ is the $q$ -shifted factorial given by

[TABLE]

With respect to the (quasi) periods $\pi$ and $\pi\tau,$ we have

[TABLE]

For the half period $\pi\tau/2,$ we also have

[TABLE]

where $B=q^{-1/4}e^{-iz}.$

In [5] Gosper introduced $q$ -analogues of $\sin z$ and $\cos z:$

[TABLE]

and gave two relations between $\sin_{q},\;\cos_{q}$ and the functions $\theta_{1}$ and $\theta_{2},$ which are equivalent to

[TABLE]

where $\tau^{\prime}=-\frac{1}{\tau}.$ It is easily seen that $\cos_{q}z=\sin_{q}(\frac{\pi}{2}\pm z).$ In that paper, using empirical evidence based on a computer program called MACSYMA, Gosper conjectured without proofs various identities involving $\sin_{q}z$ and $\cos_{q}z.$ Some of these conjectures were confirmed by different authors [1, 2, 3, 12].

We now define the $q$ -analogues for $\mathrm{tan}z$ and $\mathrm{cot}z:$

[TABLE]

and

[TABLE]

Then

[TABLE]

and

[TABLE]

Trigonometric identities is a very importan topic. Two well-known trigonometric identity are as follows [16]: if $x+y+z=\pi,$ then

[TABLE]

Finding theta function analogues for the trigonometric identities (1.4) and (1.5) is also interesting. Our motivation for the present work emanates from [5]. In this paper we shall establish the following $q$ -trigonometric identities.

Theorem 1.1.

If $x+y+z=\pi,$ then

[TABLE]

and

[TABLE]

where $\mathrm{ccs}_{q}z$ and $\mathrm{ssn}_{q}z$ are also $q$ -trigonometric functions given by

[TABLE]

Setting $q\rightarrow 1$ in (1.6) and (1.7) and noting that

[TABLE]

we can easily obtain (1.4) and (1.5) respectively.

We replace $(x,y,z)$ by $(\pi/2-x,\pi/2-y,\pi/2-z)$ in (1.6) and (1.7) to get

Corollary 1.1.

If $x+y+z=\pi/2,$ then

[TABLE]

and

[TABLE]

These identities are $q$ -analogues of the trigonometric identities: if $x+y+z=\pi/2,$ then

[TABLE]

In order to prove Theorem 1.1 we need to establish the following theta function identity by employing the theory of elliptic functions. For more information dealing with formulas of theta functions by using elliptic functions, please, see Liu [7, 8, 9, 10] and Shen [13, 14].

Theorem 1.2.

For all complex numbers $x$ and $y,$ we have

[TABLE]

In the next section we first provide our proof of Theorem 1.2 and then prove Theorems 1.1 by using Theorem 1.2.

2. Proof of Theorems 1.1 and 1.2

Proof of Theorem 1.2. Let

[TABLE]

where

[TABLE]

and

[TABLE]

It is easily deduced from (1.1) that the entire functions $f$ and $g$ satisfy the functional equations:

[TABLE]

We have $F(x)=F(x+\pi)=F(x+2\pi\tau).$ This means that $F$ is an elliptic function with periods $\pi$ and $2\pi\tau.$

We temporarily assume that $0<y<\pi.$ In the fundamental period parallelogram $\varLambda=\{a\pi+2b\pi\tau|0\leq a,b<1\},$ the function $g(x)$ has only four zeros $0,\pi\tau,\pi-y,y+\pi\tau$ and all of them are simple. It is easily seen that

[TABLE]

It follows from the Jacobi infinite product expressions that

[TABLE]

Then, by (1.1) and (1.2),

[TABLE]

and

[TABLE]

Therefore, the points $0,\pi\tau,\pi-y,y+\pi\tau$ are also zeros of the function $f(x)$ and so the function $F(x)$ has no pole in $\varLambda.$ Namely, $F(x)$ is an entire function of $x$ and then $F(x)$ is a constant (independent of $x$ ). Set $F=C(y).$ Namely, $f(x)=C(y)g(x).$ By analytic continuation, this identity also holds for any complex number $y.$ That is, $F=C(y)$ holds for any complex number $y.$ Interchanging the role of $x$ and $y$ in this identity and noticing that $F$ is symmetric in $x$ and $y$ we can get $F=C(x).$ This means that $F$ is a constant independent of $x$ and $y,$ say, $C.$ Namely, $F=C.$ Putting $x=y=\pi/4$ in this identity and noticing that $\theta_{1}(\pi/4|\tau)=\theta_{2}(\pi/4|\tau)$ we can obtain $C=1$ and then (1.8) follows. This completes the proof of Theorem 1.2. ∎

Proof of Theorem 1.1. We first prove (1.6). According to [6, (8.7) and (8.8)] we have

[TABLE]

Replacing $\tau$ by $\tau^{\prime}/2$ in (1.8), dividing both sides of the resulting identity by $\vartheta_{3}(\tau^{\prime})\theta_{2}(x|\tau^{\prime}/2)\theta_{2}(y|\tau^{\prime}/2)\theta_{2}(x+y|\tau^{\prime})$ and then employing (1.3) and (2.1) we get

[TABLE]

Then (1.6) follows readily by substituting $x+y=\pi-z$ into this identity.

The identity (1.7) follows easily by dividing both sides of (1.6) by $\mathrm{tan}_{q^{2}}x\mathrm{tan}_{q^{2}}y$

$\cdot\mathrm{tan}_{q}z$ . ∎

Acknowledgement

This work was partially supported by the National Natural Science Foundation of China (Grant No. 11801451).

Bibliography16

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] S. Abo Touk, Z. Al Houchan and M. El Bachraoui, Proofs for two q 𝑞 q -trigonometric identities of Gosper. J. Math. Anal. Appl. 456 (2017), 662–670.
2[2] M. El Bachraoui, Confirming a q 𝑞 q -trigonometric conjecture of Gosper, Proc. Amer. Math. Soc. 146(4) (2018), 1619–1625.
3[3] M. El Bachraoui, Solving some q 𝑞 q -trigonometric conjectures of Gosper, J. Math. Anal. Appl. 460 (2018), 610–617.
4[4] G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990.
5[5] R.W. Gosper, Experiments and discoveries in q 𝑞 q -trigonometry, in: F.G. Garvan, M.E.H. Ismail (Eds.), Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics, Kluwer, Dordrecht, Netherlands, 2001, pp.79–105.
6[6] B. He and H.-C. Zhai, A three-term theta function identity with applications, ar Xiv:1805. 08648 v 1.
7[7] Z.-G. Liu, A theta function identity and its implications, Trans. Amer. Math. Soc. 357(2) (2005), 825–835.
8[8] Z.-G. Liu, A three-term theta function identity and its applications. Adv. Math. 195(1) (2005), 1–23.

TL;DR

Contribution

Findings

Abstract

Peer Reviews

Videos

Taxonomy

On certain qqq-trigonometric identities

Abstract.

Key words and phrases:

2000 Mathematics Subject Classification:

1. Introduction

Theorem 1.1**.**

Corollary 1.1**.**

Theorem 1.2**.**

2. Proof of Theorems 1.1 and 1.2

Acknowledgement

On certain $q$ -trigonometric identities

Theorem 1.1.

Corollary 1.1.

Theorem 1.2.