Remarkable properties of the ${\text{sinc}}_{p,q}$ functions and related integrals

Different types of sinc integrals are investigated when the standard sine function is replaced by the generalised ${\sin }_{p,q}$ in two parameters. A striking generalisation of the improper Dirichlet integral is achieved. A second surprising but interesting generalisation of the identity between the Dirichlet integral and that of the integrand ${\text{sinc}}^2$ is discovered. Moreover, an asymptotically sharpened form of Ball’s integral inequality is obtained in terms of two parameters.