Turán-type inequalities for generalized \(k\)-Bessel functions (Q6597871)

The paper studies some properties of the generalized \({k}\)-Bessel function defined by\N\[\N\mathrm{U}_{p, q, r}^{{k}}(z) = \sum_{n=0}^{\infty} \frac{(-r)^n}{\Gamma_{{k}}(n{k}+p+(q+1){k}/{2}) n!} \left(\frac{z}{2}\right)^{2n+p/{k}},\N\]\Nwhere \({k} > 0\), \(p, q, r \in \mathbb{C}\) and \(\Gamma_{{k}}\) is the \({k}\)-gamma function\N\[\N\Gamma_{{k}}(z) := \int_0^{\infty} \ell^{z-1} e^{-\ell^{{k}}/{k}} \,d \ell, \qquad \operatorname{Re}(z) > 0,\N\]\Nthat is related to the classical gamma function by \(\Gamma_{{k}}(z) ={k}^{z/{k}-1}\Gamma(z/{k})\).\N\NThe author discusses the uniform convergence of \(\mathrm{U}_{p, q, r}^{{k}}(z)\), proves that the function is entire and determines its growth order and type. He also finds its Weierstrass factorization, which turns out to be an infinite product uniformly convergent on a compact subset of the complex plane; under certain hypotheses, this factorization is\N\[\N\mathrm{U}_{p, q, r}^{{k}}(z) = \frac{\left(z/2\right)^{p/{k}}} {\Gamma_{{k}}(p+(q+1){k}/{2})} \prod_{n=1}^{\infty} \left(1-\frac{z^2}{(\mathrm{u}_{p,q,r,n}^{{k}})^2}\right),\N\]\Nwhere \(\mathrm{u}_{p,q,r,n}^{{k}}\) stands for the \(n\)-th positive zero of \(\mathrm{U}_{p, q, r}^{{k}}(z)\).\N\NThe author also finds an integral representation for \(\mathrm{U}_{p, q, r}^{{k}}(z)\), and proves that it is a solution of a second-order differential equation that generalizes certain well-known differential equations for the classical Bessel functions, showing some properties, such as recurrence and differential relations.\N\NSome of these properties are used to establish the Turán-type inequality\N\[\N(\mathrm{U}_{p,q,r}^{{k}}(z))^2 - \mathrm{U}_{p-{k},q,r}^{{k}}(z) \mathrm{U}_{p+{k},q,r}^{{k}}(z) \geq 0,\N\]\Nvalid under certain hypotheses.\N\NFinally, the author studies monotonicity and log-convexity of the normalized form of the modified \({k}\)-Bessel function \(\mathrm{T}_{p,q,1}^{{k}}(z) = i^{p/{k}} \mathrm{U}_{p,q,1}^{{k}}(iz)\), as well as the quotient of the modified \({k}\)-Bessel function, exponential, and \({k}\)-hypergeometric functions.