On properties of an entire function that is a generalization of the Wright function (Q2206262)

The author considers the function \[\Theta_{n,\alpha}(z;\mu)=\sum_{k=0}^\infty \frac{\sin n\omega_k}{\sin \omega_k}\, \frac{z^k}{k! \Gamma(\mu-\frac{\alpha k}{2n})}, \qquad \omega_k:=(k+1)\pi/(2n)\] for positive integer \(n\). This is a fundamental solution of the fractional differential equation \[\frac{\partial^\alpha u}{\partial t^\alpha}+(-1)^n \frac{\partial^{2n}u}{\partial x^{2n}}=f(x,t).\] Various properties of this function are established. It is shown that \(\Theta_{n,\alpha}(z;\mu)\) can be expressed as a finite sum of Wright functions \(\phi(-\rho,\mu,z)\), \(\rho\in (0,1)\) with suitably rotated arguments. Two integral representations are obtained in the form of a Fourier cosine integral involving a Mittag-Leffler function and a contour integral taken round a Hankel contour. An expression in terms of the Fox \(H\) function is also given. A differentiation formula and a partial differential formula are presented. The final section deals with a growth estimate for \(\Theta_{n,\alpha}(z;\mu)\) for large \(z>0\). This is obtained from the representation as a sum of Wright functions combined with the known asymptotics of this latter function.