On Positivities of Certain q-Special Functions

TL;DR

This paper uses Bochner's theorem to establish positivity of specific q-special functions, leading to new inequalities for the Jacobi theta and q-Gamma functions.

Contribution

It introduces a novel application of Bochner's theorem to prove positivity of q-exponentials and derives new inequalities for important q-special functions.

Findings

Proved certain q-exponentials are positive definite functions.

Derived new inequalities for the Jacobi theta function.

Established properties of the q-Gamma function.

Abstract

In this work we shall apply the Bochner's theorem to prove certain combinations of Euler's q-exponentials are positive definite functions. Then we apply this positivity to prove curious inequalities for the Jacobi theta function $\vartheta_{4}$ and q-Gamma function $\Gamma_{q}$.

Equations

$$A={\left(a_{j,k}\right)}_{j,k=1}^n,B={\left(b_{j,k}\right)}_{j,k=1}^n,a_{j,k},b_{j,k}\in \mathbb{C},$$

$$\det \left(A\circ B\right)\ge \det (A)\cdot \det (B).$$

$$f(x)={\int }_{\mathbb{R}}{e}^{ixy}\mu \left(dy\right).$$

$$\prod _{\ell =1}^n\frac{\exp \left(\frac{x^2}{\log q_{\ell }^4}\right)}{{\left(y_{\ell }{e}^{ix};q_{\ell }\right)}_{\infty }}x\in \mathbb{R}$$

$${\left(\prod _{\ell =1}^n\frac{\exp \left(\frac{{\left(x_j-x_k\right)}^2}{\log q_{\ell }^4}\right)}{{\left(y_{\ell }{e}^{i(x_j-x_k)};q_{\ell }\right)}_{\infty }}\right)}_{j,k=1}^m$$

$$\prod _{\ell =1}^n\frac{{\left(y_{\ell };q_{\ell }\right)}_{\infty }}{\left|{\left(y_{\ell }{e}^{ix};q_{\ell }\right)}_{\infty }\right|}\exp \left(\frac{x^2}{\log q_{\ell }^4}\right)\le 1,x\in \mathbb{R}.$$

$$A_q(z)=\sum _{n=0}^{\infty }\frac{q^{n^2}}{(q;q{)}_n}(-z{)}^n,$$

$$q^{{\alpha }^2}{A}_q\left(-q^{2\alpha }z\right)>0,\alpha \in \mathbb{R}.$$

$$\frac{\exp \left(\frac{x^2}{\log q^4}\right)}{{\left(z{e}^{ix};q\right)}_{\infty }\sqrt{\log q^{-2}}}=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }{q}^{{\alpha }^2}{A}_q\left(-q^{2\alpha }z\right)\exp \left(-i\alpha x\right)d\alpha ,$$

$$\frac{1}{{\left(y;q\right)}_{\infty }}>0,y,q\in (0,1),$$

$$\frac{\exp \left(\frac{x^2}{\log q^4}\right)}{{\left(y{e}^{ix};q\right)}_{\infty }},y,q\in (0,1)$$

$${\left(\frac{\exp \left(\frac{(x_i-x_j{)}^2}{\log q^4}\right)}{{\left(y{e}^{i(x_i-x_j)};q\right)}_{\infty }}\right)}_{i,j=1}^m,q,y\in (0,1)$$

$$\det {\left(\prod _{\ell =1}^n\frac{\exp \left(\frac{{\left(x_j-x_k\right)}^2}{\log q_{\ell }^4}\right)}{{\left(y_{\ell }{e}^{i(x_i-x_j)};q_{\ell }\right)}_{\infty }}\right)}_{j,k=1}^m>0.$$

$${\Gamma }_q(x)=(1-q{)}^{1-x}\frac{(q;q{)}_{\infty }}{(q^x;q{)}_{\infty }},$$

$${\vartheta }_4\left(z;q\right)={\vartheta }_4\left(v|\tau \right)=\sum _{n=-\infty }^{\infty }{q}^{n^2}{\left(-z\right)}^n={\left(q^2,qz,q/z;q^2\right)}_{\infty },$$

$$z=e^{2\pi iv},q=e^{\pi i\tau },\Im (\tau )>0.$$

$$\left|{\left(q^{u+iv};q\right)}_{\infty }\right|\ge {\left(q^u;q\right)}_{\infty }{q}^{v^2/4},$$

$${\Gamma }_q(u)\ge \left|{\Gamma }_q(u+iv)\right|q^{v^2/4}$$

$$\frac{{\vartheta }_4\left(v|iu\right)}{{\vartheta }_4\left(0|iu\right)}\ge e^{-\pi v^2/u}.$$

$$\left|{\left(y{e}^{ix};q\right)}_{\infty }\right|\ge {\left(y;q\right)}_{\infty }\exp \left(\frac{x^2}{4\log q}\right),0<y,q<1,x\in \mathbb{R}.$$

$$x=v\log q,y=q^u,u\in (0,\infty ),v\in \mathbb{R}$$

$$\left|{\left(q^{u+iv};q\right)}_{\infty }\right|\ge {\left(q^u;q\right)}_{\infty }{q}^{v^2/4}$$

$${\Gamma }_q(u)\ge \left|{\Gamma }_q(u+iv)\right|q^{v^2/4},0<q<1,u>0,v\in \mathbb{R}.$$

$${\left(q,y{e}^{ix},y{e}^{-ix};q\right)}_{\infty }\ge {\left(q,y,y;q\right)}_{\infty }\exp \left(\frac{x^2}{2\log q}\right).$$

$${\left(q^2,q{e}^{ix},q{e}^{-ix};q^2\right)}_{\infty }\ge {\left(q^2,q,q;q^2\right)}_{\infty }\exp \left(\frac{x^2}{4\log q}\right),$$

$${\vartheta }_4(v|ui)\ge {\vartheta }_4(0|ui)e^{-\pi v^2/u}$$

$$q=e^{-\pi u},x=2\pi v,u>0,v\in \mathbb{R}.$$

$$\left|{\left(q^{f(z)};q\right)}_{\infty }\right|\ge {\left(q^{\Re (f(z))};q\right)}_{\infty }{q}^{(\Im (z){)}^2/4},z\in \Omega$$

$${\Gamma }_q\left(\Re (f(z))\right)\ge \left|{\Gamma }_q\left(f(z)\right)\right|q^{(\Im (z){)}^2/4},z\in \Omega .$$

$${\vartheta }_4\left(v+\tau |\tau \right)=-e^{-\pi i\tau -2\pi iv}{\vartheta }_4\left(v|\tau \right),\Im (\tau )>0,$$

Full text

The work is supported by the National Natural Science Foundation of China grants No. 11371294 and No. 11771355.

1. Introduction

For any $n\in\mathbb{N}$, the matrix $\left(a_{j,k}\right)_{j,k=1}^{n},\ a_{j,k}\in\mathbb{C}$ is called positive semidefinite if and only if ${\displaystyle\sum_{j,k=1}^{n}}a_{j,k}z_{j}\overline{z_{k}}\geq 0$ for all $z_{1},\,z_{2},\dots\,z_{n}\in\mathbb{C}$. A positive semidefinite matrix is positive definite if ${\displaystyle\sum_{j,k=1}^{n}}a_{j,k}z_{j}\overline{z_{k}}=0$ implies that $z_{1}=\dots=z_{n}=0$. Given two positive semidefinite matrices

[TABLE]

it is well-known that their Schur (Hadamard) product $A\circ B=\left(a_{j,k}b_{j,k}\right)_{j,k=1}^{n}$ is also positive semidefinite. Furthermore, it satisfies [4, 8]

[TABLE]

A continuous function $f(x)$ on $\mathbb{R}$ with $f(0)=1$ is called positive definite if and only if the matrices $\left(f(x_{j}-x_{k})\right)_{j,k=1}^{n}$ are positive semidefinite for all $n\in\mathbb{N}$ and $x_{1},\dots,x_{n}\in\mathbb{R}$. The Bochner’s theorem states that $f(x)$ is positive definite if and only if there exists a probability measure $\mu(dx)$ on $\mathbb{R}$ such that [1]

[TABLE]

The condition $f(0)=1$ can be replaced by $f(0)>0$, then the total mass of the measure $\mu(dx)$ is $f(0)$. If $f(x)$ is a positive definite function, then $\overline{f(x)},\,\Re(f(x)),\,\left|f(x)\right|^{2},\,f(ax)$ are all positive definite functions where $a$ is any nonzero real number. Furthermore, for any $m\in\mathbb{N}$ if $f_{1}(x),\dots,f_{m}(x)$ are positive definite functions, then $\prod_{n=1}^{m}f_{n}(x)$ and $\sum_{n=1}^{m}a_{n}f_{n}(x)$ are also positive definite functions where $a_{n}\geq 0,\ n=1,\dots,m$ and $\sum_{n=1}^{m}a_{n}=1$.

In [6] we derived numerous Fourier transformations for several q-special functions. In this work we shall apply the Bochner’s theorem to prove certain combinations of Euler’s q-exponentials are positive definite functions. As a corollary of the positivities we prove some curious inequalities for the Jacobi theta function $\vartheta_{4}$, q-Gamma function $\Gamma_{q}$ and Euler’s q-exponentials.

2. Main Results

Theorem 1**.**

Let $n\in\mathbb{N}$. If $0<y_{\ell},q_{\ell}<1$ for all $1\leq\ell\leq n$, then

[TABLE]

is a positive definite function. Furthermore, for all $m\in\mathbb{N}$ and all $x_{1},\dots,x_{m}\in\mathbb{R}$, the matrices

[TABLE]

are positive semidefinite. In particular, we have the inequality,

[TABLE]

Proof.

Since

[TABLE]

then for $z>0,\ q\in(0,1)$ we have

[TABLE]

Then by (5.37) of [6],

[TABLE]

where $|z|<1,\ x\in\mathbb{R}$, the inequality

[TABLE]

and by the Bochner’s theorem we see that the continuous function

[TABLE]

is positive semidefinite in variable $x\in\mathbb{R}$. Therefore, for all $m\in\mathbb{N}$ and all distinct $x_{1},x_{2},\dots,x_{m}\in\mathbb{R}$, the matrices

[TABLE]

are positive semidefinite. (2.1) and (2.2) are obtained by taking the Schur (Hadamard) product of (2.4) or (2.5) respectively. For all $m\in\mathbb{N}$, by (2.2) we get

[TABLE]

Then (2.3) is obtained by setting $m=2$ in the above inequality. ∎

The $q$-Gamma function is defined by [6]

[TABLE]

and the Jacobi theta function $\vartheta_{4}$ is [6]

[TABLE]

where

[TABLE]

Corollary 2**.**

If $0<q<1$, $u>0$ and $v\in\mathbb{R}$, then

[TABLE]

[TABLE]

and

[TABLE]

Proof.

From (2.3) we get

[TABLE]

Let

[TABLE]

in (2.9) we get

[TABLE]

and

[TABLE]

Multiply $(q;q)_{\infty}$ to the square of (2.9) to obtain,

[TABLE]

Then let $q\to q^{2},\ y\to q$ in the above inequality to obtain

[TABLE]

which gives

[TABLE]

by setting

[TABLE]

∎

Remark 3*.*

Given any simplify connected region $\Omega$ that is not the entire complex plane, then by Riemann’s mapping theorem, [2], there exists an analytic function $f(z)=u(z)+iv(z)$ in $\Omega$ defines a one-to-one mapping of $\Omega$ onto the right half-plane $H=\left\{u+iv|u>0,\ v\in\mathbb{R}\right\}$. The inequalities (2.6), (2.7) and (2.8) imply that

[TABLE]

[TABLE]

By [7]

[TABLE]

for $u>0,\ v\in\mathbb{R}$ we get

[TABLE]

hence,

[TABLE]

Then for $z\in\Omega$,

[TABLE]

In particular, since the function

[TABLE]

maps unit disk unto the right-half plane. Then for $z=x+iy$ with $x^{2}+y^{2}<1$ we have

[TABLE]

[TABLE]

and

[TABLE]

Theorem 4**.**

Let $n\in\mathbb{N}$. If $0<z_{\ell},b_{\ell},q_{\ell}<1,\ -1<a_{\ell}<1,$ for all $1\leq\ell\leq n$, then the function

[TABLE]

is positive definite in $x$. In particular, for all $m\in\mathbb{N}$ and distinct $x_{1},\dots,x_{m}\in\mathbb{R}$ the matrices

[TABLE]

are positive semidefinite. Furthermore, for $x\in\mathbb{R}$ we have

[TABLE]

Proof.

For $q\in(0,1)$, the confluent basic hypergeometric series

[TABLE]

satisfies

[TABLE]

By (5.83) in [6],

[TABLE]

we get

[TABLE]

Since for $q,b,z\in(0,1),\ a\in(-1,1),\ \alpha\in\mathbb{R}$ we have

[TABLE]

and

[TABLE]

Then by Bochner’s theorem we know that the continuous function,

[TABLE]

is a positive definite function in $x\in\mathbb{R}$. (2.19) and (2.20) are obtained by taking Schur (Hadamard) products, and (2.21) comes from the determinant of (2.20) at $m=2$. ∎

Corollary 5**.**

If $0<a,b,q<1$, $-1<c<1$ and $v\in\mathbb{R}$, then

[TABLE]

In particular,

[TABLE]

Proof.

For $0<z,b,q<1$, $-1<a<1$ and $x\in\mathbb{R}$, let $n=1$ in (2.21) we get

[TABLE]

In particular,

[TABLE]

Then set $x=v\log q,\ v\in\mathbb{R}$ to get

[TABLE]

and

[TABLE]

∎

Then the corollary is obtained by renaming variables.

Let $q=e^{-2k^{2}}$ and $|q|<1$, from a Ramanujan’s identity [3]

[TABLE]

we get

[TABLE]

Then we have the following:

Theorem 6**.**

For all $n\in\mathbb{N}$ and

[TABLE]

the function

[TABLE]

is positive definite. Furthermore, for all $m\in\mathbb{N}$ and $x_{1},\dots,x_{m}\in\mathbb{R}$ the matrices

[TABLE]

are positive semidefinite and for $0<c<q^{1/2},\ k>0,\ q=e^{-2k^{2}}$ we have

[TABLE]